Linking Entanglement To Discord With State Extensions

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One of the fundamental problem to explore the potential and advantage of quantum technology is the characterization and quantification of the quantum correlations, especially entanglement. In [Phys. Rev. A 94, 032129 (2016)], the author proposed the minimal discord over state extensions as the measure of entanglement. In this work, we show that the minimal Bures distance of discord over state extensions is equivalent to the Bures distance of entanglement and its convex roof. This equivalence puts discord in a more primitive place than entanglement conceptually, that is, entanglement can be interpreted as the irreducible part of discord over all state extensions. Moreover, for bipartite state, we also show that the minimal quantum discord over a kind of “symmetric” state extensions is an entanglement monotone, i.e., non-increasing under local operation and classical communications. The results presented here show that a large class of discord measures can be used to construct entanglement measures. In particular, although Hilbert-Schmidt distance is not contractive, our result show that the corresponding quantification is an entanglement monotone in our framework.



Entanglement is perhaps the most well studied form of quantum correlation and plays a fundamental role in many useful quantum protocols, such as quantum cryptography Ekert (1991), quantum teleportation Bennett et al. (1993), superdense coding Bennett and Wiesner (1992), and Shor’s algorithm Shor (1994). It has also been shown to be an important resource in quantum information theory Horodecki et al. (2009) since the remarkable advantage it provided makes quantum information processing much more powerful than classical theory. Generalized notion of quantum correlation is also considered beyond entanglement, most prominent in the form quantum discord Ollivier and Zurek (2001); Henderson and Vedral (2001). It has been proved that quantum discord, rather than entanglement, is the genuine resource in the DQC1 algorithm Datta et al. (2005, 2008).



Entanglement and discord have both essential similarities and significant differences. In recent years, some remarkable relationships and interactions have been investigated between entanglement and discord Cubitt et al. (2003); Koashi and Winter (2004); Cen et al. (2011); Adesso and Datta (2010); Cavalcanti et al. (2011); Piani et al. (2011); Madhok and Datta (2011); Streltsov et al. (2011, 2012); Piani and Adesso (2012); Chuan et al. (2012). Especially, Li and Luo revealed the correspondence between classical states versus separable states Li and Luo (2008). Moreover, Luo proposed to treat the minimal discord of bipartite quantum state over state extensions as the quantification of entanglement Luo (2016). Besides, the minimal correlated coherence over symmetric state extensions was proved to be a good characterization of entanglement Tan et al. (2016); Tan and Jeong (2018). The framework that quantifying entanglement with quantum correlation, or coherence over state extension, is quite different from the existing entanglement measures, which is mostly based on operational meaning, information principles, and mathematics, like the entanglement of formation Bennett et al. (1996), the entanglement cost Bennett et al. (1996), the distillable entanglement Bennett et al. (1997), the relative entropy of entanglement Vedral et al. (1997); Vedral and Plenio (1998), Bures distance of entanglement Vedral et al. (1997); Vedral and Plenio (1998), the robustness of entanglement Vidal and Tarrach (1999), and the squashed entanglement Tucci (2002); Christandl and Winter (2004). So far, the interrelationship between these two different forms of entanglement measures is not clear. discord server



In this work, we will clarify the relationship for the Bures distance case. The Bures metric Bures (1969); Uhlmann (1976) provides a nice distance on the convex cone of density matrices. In particular, Bures distance is monotonous, Riemannian Petz (1996) and its metric coincides with the quantum Fisher information which plays an important role in high precision interferometry Braunstein and Caves (1994). As a consequence, the minimal Bures distance to separable states satisfies all criteria of an entanglement measure Vedral and Plenio (1998), which has been widely studied in Wei and Goldbart (2003); Streltsov et al. (2010). More importantly, Bures distance of entanglement is proved to be equal to its corresponding convex roof Streltsov et al. (2010), which plays a key role in our proof. Furthermore, Bures distance is also used to quantify quantum discord in Spehner and Orszag (2013); Aaronson et al. (2013b), and its physical operational meaning is revealed by establishing a connection to quantum state discriminations in Spehner and Orszag (2013). In fact, by proving that the Bures distance of entanglement is the lower bound of the minimal Bures distance of discord over state extensions and the corresponding convex roof is an upper bound, our first main result is obtained.



A necessary condition for a candidate to be considered as an entanglement measure is that it must be an entanglement monotone, that is, non-increasing under local operation and classical communications (LOCC). The class of LOCC plays an important role in quantum information theory, especially when studying entanglement. In fact, any separable state can be created with LOCC, but entangled states cannot. Moreover, LOCCs are regards as the class of operation that can not create entanglement and non-increasing under LOCC is a fundamental requirement for entanglement measures. Therefore, a natural question arises that whether the minimal discord over state extensions is an entanglement monotone.



Our results will give an affirmative answer to this question for several kind of quantum discord. In fact, we show that the minimal quantum discord over a kind of “symmetric” state extensions is an entanglement monotone. In particular, the Hilbert-Schmidt distance is proved to be not contractive and it is not clear whether this distance can be used to quantify entanglement Ozawa (2000). However, we show that the minimal Hilbert-Schmidt distance of discord over this “symmetric” state extensions is non-increasing under LOCC operations. Besides, we also prove that Hilbert-Schmidt distance of discord Dakić et al. (2010) reduces to an entanglement monotone on the pure states. These results indicate that Hilbert-Schmidt distance is a good candidate to quantify quantum correlations including entanglement and discord.



II Bures distance of entanglement



II.1 Preliminaries



Let ℋ=ℋa⊗ℋbℋtensor-productsubscriptℋ𝑎subscriptℋ𝑏\mathcalH=\mathcalH_a\otimes\mathcalH_bcaligraphic_H = caligraphic_H start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ caligraphic_H start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT and D(ℋ)𝐷ℋD(\mathcalH)italic_D ( caligraphic_H ) be the set of density matrices on ℋℋ\mathcalHcaligraphic_H. A state ρab∈D(ℋ)subscript𝜌𝑎𝑏𝐷ℋ\rho_ab\in D(\mathcalH)italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ∈ italic_D ( caligraphic_H ) shared by two parties a and b is called separable if it can be represented in a separable form



ρab=∑ipiρa⊗ρb,subscript𝜌𝑎𝑏subscript𝑖tensor-productsubscript𝑝𝑖subscript𝜌𝑎subscript𝜌𝑏\displaystyle\rho_ab=\sum_ip_i\rho_a\otimes\rho_b,italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_ρ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT , where pi≥0,∑ipi=1formulae-sequencesubscript𝑝𝑖0subscript𝑖subscript𝑝𝑖1p_i\geq 0,\sum_ip_i=1italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0 , ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 and ρasubscript𝜌𝑎\rho_aitalic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT, ρbsubscript𝜌𝑏\rho_bitalic_ρ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT are local state for parties a and b, respectively. Otherwise, it is called entangled. In fact, any separable state can be created by the following procedure: Alice prepares the state ρasubscript𝜌𝑎\rho_aitalic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT with the probability pisubscript𝑝𝑖p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, and lets Bob knows which state she prepared. Based on this information, Bob prepares the corresponding state ρbsubscript𝜌𝑏\rho_bitalic_ρ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT. However, it is not possible to create entangled states in this way. The process for creating separable states presented there belongs to the class of LOCC.



Moreover, a bipartite state ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT is classically correlated if there exists a quantum measurement Π=Πai⊗ΠbjΠtensor-productsubscriptsuperscriptΠ𝑖𝑎subscriptsuperscriptΠ𝑗𝑏\Pi=\\Pi^i_a\otimes\Pi^j_b\roman_Π = roman_Π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT which does not disturb the state in the sense that



ρab=∑i,j(Πai⊗Πbj)ρab(Πai⊗Πbj),subscript𝜌𝑎𝑏subscript𝑖𝑗tensor-productsubscriptsuperscriptΠ𝑖𝑎subscriptsuperscriptΠ𝑗𝑏subscript𝜌𝑎𝑏tensor-productsubscriptsuperscriptΠ𝑖𝑎subscriptsuperscriptΠ𝑗𝑏\displaystyle\rho_ab=\sum_i,j(\Pi^i_a\otimes\Pi^j_b)\rho_ab(\Pi^% i_a\otimes\Pi^j_b),italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT ( roman_Π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ( roman_Π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) , where ΠaisubscriptsuperscriptΠ𝑖𝑎\\Pi^i_a\ roman_Π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT and ΠbjsubscriptsuperscriptΠ𝑗𝑏\\Pi^j_b\ roman_Π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT are von Neumann measurements on parties a and b, respectively. Equivalently, each classical correlated state can be written as



ρab=∑i,jpij|i⟩a⟨i|⊗|j⟩b⟨j|,pij≥0.formulae-sequencesubscript𝜌𝑎𝑏subscript𝑖𝑗tensor-productsubscript𝑝𝑖𝑗subscriptket𝑖𝑎bra𝑖subscriptket𝑗𝑏bra𝑗subscript𝑝𝑖𝑗0\displaystyle\rho_ab=\sum_i,jp_ij\keti_a\brai\otimes\ketj_b% \braj,p_ij\geq 0.italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ | start_ARG italic_j end_ARG ⟩ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⟨ start_ARG italic_j end_ARG | , italic_p start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ≥ 0 . where ∑ijpij=1subscript𝑖𝑗subscript𝑝𝑖𝑗1\sum_ijp_ij=1∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = 1 and subscriptket𝑖𝑎\\keti_a\ and subscriptket𝑗𝑏\\ketj_b\ are orthonormal basis of parties a and b, respectively.



Next, we consider the quantification of entanglement and discord with Bures distance, i.e.,



dB(ρ,σ):=2-2F(ρ,σ),assignsubscript𝑑𝐵𝜌𝜎22𝐹𝜌𝜎\displaystyle d_B(\rho,\sigma):=\sqrt2-2F(\rho,\sigma),italic_d start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ , italic_σ ) := square-root start_ARG 2 - 2 italic_F ( italic_ρ , italic_σ ) end_ARG , where fidelity F(ρ,σ):=trσρσassign𝐹𝜌𝜎𝑡𝑟𝜎𝜌𝜎F(\rho,\sigma):=tr\sqrt\sqrt\sigma\rho\sqrt\sigmaitalic_F ( italic_ρ , italic_σ ) := italic_t italic_r square-root start_ARG square-root start_ARG italic_σ end_ARG italic_ρ square-root start_ARG italic_σ end_ARG end_ARG. As F(ρ,σ)∈[0,1]𝐹𝜌𝜎01F(\rho,\sigma)\in[0,1]italic_F ( italic_ρ , italic_σ ) ∈ [ 0 , 1 ] and gets to 1111 iff ρ=σ𝜌𝜎\rho=\sigmaitalic_ρ = italic_σ, then dB(ρ,σ)subscript𝑑𝐵𝜌𝜎d_B(\rho,\sigma)italic_d start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ , italic_σ ) is nonnegative and vanishes iff ρ=σ𝜌𝜎\rho=\sigmaitalic_ρ = italic_σ. Moreover, the monotonicity and jointly concavity of fidelity Nielsen and Chuang (2010), implies that dB2subscriptsuperscript𝑑2𝐵d^2_Bitalic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is contractive and jointly convex.



Denote the set of separable states and classical correlated states by 𝒮𝒮\mathcalScaligraphic_S and 𝒞𝒞𝒞𝒞\mathcalC\mathcalCcaligraphic_C caligraphic_C respectively, then the Bures distance of entanglement is defined as the minimal square of Bures distance to separable states Vedral et al. (1997), i.e.,



EB(ρab):=minσ∈𝒮dB2(ρab,σ).assignsubscript𝐸𝐵subscript𝜌𝑎𝑏subscript𝜎𝒮subscriptsuperscript𝑑2𝐵subscript𝜌𝑎𝑏𝜎\displaystyle E_B(\rho_ab):=\min_\sigma\in\mathcalSd^2_B(\rho_ab% ,\sigma).italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) := roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_S end_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , italic_σ ) . Therefore, EB(ρab)subscript𝐸𝐵subscript𝜌𝑎𝑏E_B(\rho_ab)italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is nonnegative and vanishes iff ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT is separable. Furthermore, EBsubscript𝐸𝐵E_Bitalic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is convex and non-increasing under LOCC operations Vedral et al. (1997); Vedral and Plenio (1998).



Moreover, the Bures distance of discord is defined as



DB(ρab):=minσ∈𝒞𝒞dB2(ρab,σ),assignsubscript𝐷𝐵subscript𝜌𝑎𝑏subscript𝜎𝒞𝒞subscriptsuperscript𝑑2𝐵subscript𝜌𝑎𝑏𝜎\displaystyle D_B(\rho_ab):=\min_\sigma\in\mathcalC\mathcalCd^2_B% (\rho_ab,\sigma),italic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) := roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_C caligraphic_C end_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , italic_σ ) , where the minimal is taken over all classical correlated states. By definition, DB(ρab)subscript𝐷𝐵subscript𝜌𝑎𝑏D_B(\rho_ab)italic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is nonnegative, vanishes iff ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT is classical correlated, and invariant under local unitary operations.



II.2 Bures distance of entanglement and it’s convex roof



In Luo (2016), Luo proposed that the minimal discord of bipartite quantum state over state extensions can be used to characterize entanglement. In this part, we will consider the Bures distance case.



For bipartite state ρab∈ℋsubscript𝜌𝑎𝑏ℋ\rho_ab\in\mathcalHitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ∈ caligraphic_H, the minimal Bures distance of discord over state extensions is defined as



ℰB(ρab):=mintra′b′ρaa′bb′=ρabDB(ρaa′bb′),assignsubscriptℰ𝐵subscript𝜌𝑎𝑏subscript𝑡subscript𝑟superscript𝑎′superscript𝑏′subscript𝜌𝑎superscript𝑎′𝑏superscript𝑏′subscript𝜌𝑎𝑏subscript𝐷𝐵subscript𝜌𝑎superscript𝑎′𝑏superscript𝑏′\displaystyle\mathcalE_B(\rho_ab):=\min_tr_a^\primeb^\prime\rho_% aa^\primebb^\prime=\rho_abD_B(\rho_aa^\primebb^\prime),caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) := roman_min start_POSTSUBSCRIPT italic_t italic_r start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) , where the minimal discord is taken respect to bipartite aa′:bb′:𝑎superscript𝑎′𝑏superscript𝑏′aa^\prime:bb^\primeitalic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT.



By the result of Li and Luo (2008), any separable state ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT can be embedded into a larger classical state ρaa′:bb′subscript𝜌:𝑎superscript𝑎′𝑏superscript𝑏′\rho_aa^\prime:bb^\primeitalic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT such that ρab=tra′b′ρaa′:bb′subscript𝜌𝑎𝑏𝑡subscript𝑟superscript𝑎′superscript𝑏′subscript𝜌:𝑎superscript𝑎′𝑏superscript𝑏′\rho_ab=tr_a^\primeb^\prime\rho_aa^\prime:bb^\primeitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = italic_t italic_r start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, where a′superscript𝑎′a^\primeitalic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and b′superscript𝑏′b^\primeitalic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT are two ancillary systems pertinent to parties a and b, respectively. However, any entangled state does not admit such an extension. Consequently, a bipartite state is separable if and only if it is a reduced state of a classical correlated state. Therefore, ℰBsubscriptℰ𝐵\mathcalE_Bcaligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is faithful in separable states.



To illustrate our main results, let us first list a few fundamental properties. First, we observe that ℰB(ρab)subscriptℰ𝐵subscript𝜌𝑎𝑏\mathcalE_B(\rho_ab)caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is a convex function of quantum states.



ℰBsubscriptℰ𝐵\mathcalE_Bcaligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is convex in the sense that



See Appendix A. ∎



Now we consider the pure state case. Let bipartite pure state |ψ⟩=∑iλi|xi⟩a|yi⟩bket𝜓subscript𝑖subscript𝜆𝑖subscriptketsubscript𝑥𝑖𝑎subscriptketsubscript𝑦𝑖𝑏\ket\psi=\sum_i\sqrt\lambda_i\ketx_i_a\kety_i_b| start_ARG italic_ψ end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | start_ARG italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | start_ARG italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT with λ1≥…≥λnsubscript𝜆1…subscript𝜆𝑛\lambda_1\geq...\geq\lambda_nitalic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ … ≥ italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, then EB(|Ψ⟩)=2(1-λ1)subscript𝐸𝐵ketΨ21subscript𝜆1E_B(\ket\Psi)=2(1-\sqrt\lambda_1)italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( | start_ARG roman_Ψ end_ARG ⟩ ) = 2 ( 1 - square-root start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) Streltsov et al. (2010). In fact, assume σab∈𝒮subscript𝜎𝑎𝑏𝒮\sigma_ab\in\mathcalSitalic_σ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ∈ caligraphic_S has a separable pure state decomposition σab=∑jqj|ϕj⟩ab⟨ϕj|subscript𝜎𝑎𝑏subscript𝑗subscript𝑞𝑗subscriptketsubscriptitalic-ϕ𝑗𝑎𝑏brasubscriptitalic-ϕ𝑗\sigma_ab=\sum_jq_j\ket\phi_j_ab\bra\phi_jitalic_σ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⟨ start_ARG italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG |, then



F(|ψ⟩,σab)=∑jqj|⟨ψ|ϕj⟩|2𝐹ket𝜓subscript𝜎𝑎𝑏fragmentssubscript𝑗subscript𝑞𝑗|bra𝜓subscriptitalic-ϕ𝑗⟩|2\displaystyle F(\ket\psi,\sigma_ab)=\sqrt^2italic_F ( | start_ARG italic_ψ end_ARG ⟩ , italic_σ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = square-root start_ARG ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | ⟨ start_ARG italic_ψ end_ARG | italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG



≤\displaystyle\leq≤ ∑jqj|⟨ψ|ϕmax⟩|2=|⟨ψ|ϕmax⟩|,fragmentsfragmentssubscript𝑗subscript𝑞𝑗|bra𝜓subscriptitalic-ϕ𝑚𝑎𝑥⟩|2|bra𝜓subscriptitalic-ϕ𝑚𝑎𝑥⟩|,\displaystyle\sqrt\bra\psi\phi_max\rangle=|\bra\psi% \phi_max\rangle|,square-root start_ARG ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | ⟨ start_ARG italic_ψ end_ARG | italic_ϕ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = | ⟨ start_ARG italic_ψ end_ARG | italic_ϕ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT ⟩ | , where |⟨ψ|ϕmax⟩|fragments|bra𝜓subscriptitalic-ϕ𝑚𝑎𝑥⟩||\bra\psi\phi_max\rangle|| ⟨ start_ARG italic_ψ end_ARG | italic_ϕ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT ⟩ | is the maximal over all j𝑗jitalic_j. Therefore, one has that EB(|ψ⟩)=2(1-λ1)subscript𝐸𝐵ket𝜓21subscript𝜆1E_B(\ket\psi)=2(1-\sqrt\lambda_1)italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( | start_ARG italic_ψ end_ARG ⟩ ) = 2 ( 1 - square-root start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) and the corresponding closest separable state is |x1,y1⟩absubscriptketsubscript𝑥1subscript𝑦1𝑎𝑏\ketx_1,y_1_ab| start_ARG italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT.



On the other hand, with the definition for pure states, the convex roof of Bures distance of entanglement is defined as



EBcr(ρab):=minpi,|ψi⟩∑ipiEB(|ψi⟩),assignsubscriptsuperscript𝐸𝑐𝑟𝐵subscript𝜌𝑎𝑏subscriptsubscript𝑝𝑖ketsubscript𝜓𝑖subscript𝑖subscript𝑝𝑖subscript𝐸𝐵ketsubscript𝜓𝑖\displaystyle E^cr_B(\rho_ab):=\min_p_i,\ket\psi_i\sum_ip_iE% _B(\ket\psi_i),italic_E start_POSTSUPERSCRIPT italic_c italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) := roman_min start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , | start_ARG italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( | start_ARG italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ ) , where the minimal is taken over all pure state decomposition ρab=∑ipi|ψi⟩⟨ψi|subscript𝜌𝑎𝑏subscript𝑖subscript𝑝𝑖ketsubscript𝜓𝑖brasubscript𝜓𝑖\rho_ab=\sum_ip_i\ket\psi_i\bra\psi_iitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG |.



With above Theorem, one has the following result.



For ρab∈D(ℋ)subscript𝜌𝑎𝑏𝐷ℋ\rho_ab\in D(\mathcalH)italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ∈ italic_D ( caligraphic_H ), one has



EB(ρab)≤ℰB(ρab)≤EBcr(ρab).subscript𝐸𝐵subscript𝜌𝑎𝑏subscriptℰ𝐵subscript𝜌𝑎𝑏subscriptsuperscript𝐸𝑐𝑟𝐵subscript𝜌𝑎𝑏\displaystyle E_B(\rho_ab)\leq\mathcalE_B(\rho_ab)\leq E^cr_B(% \rho_ab).italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≤ caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≤ italic_E start_POSTSUPERSCRIPT italic_c italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) .



On one hand, supposing ρaa′bb′⋆subscriptsuperscript𝜌⋆𝑎superscript𝑎′𝑏superscript𝑏′\rho^\star_aa^\primebb^\primeitalic_ρ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is the optimal state extensions of ρabsuperscript𝜌𝑎𝑏\rho^abitalic_ρ start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT and σ⋆superscript𝜎⋆\sigma^\staritalic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT is the corresponding closest classical correlated state, then



ℰB(ρab)=dB2(ρaa′bb′⋆,σ⋆)≥dB2(ρab,tra′b′σ⋆)≥EB(ρab).subscriptℰ𝐵subscript𝜌𝑎𝑏subscriptsuperscript𝑑2𝐵subscriptsuperscript𝜌⋆𝑎superscript𝑎′𝑏superscript𝑏′superscript𝜎⋆subscriptsuperscript𝑑2𝐵subscript𝜌𝑎𝑏𝑡subscript𝑟superscript𝑎′superscript𝑏′superscript𝜎⋆subscript𝐸𝐵subscript𝜌𝑎𝑏\displaystyle\mathcalE_B(\rho_ab)=d^2_B(\rho^\star_aa^\primebb^% \prime,\sigma^\star)\geq d^2_B(\rho_ab,tr_a^\primeb^\prime% \sigma^\star)\geq E_B(\rho_ab).caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ) ≥ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , italic_t italic_r start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ) ≥ italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) . The first "≥""""\geq"" ≥ " is the result of the contractibility of Bures distance and the second "≥""""\geq"" ≥ " is because tra′b′σ⋆𝑡subscript𝑟superscript𝑎′superscript𝑏′superscript𝜎⋆tr_a^\primeb^\prime\sigma^\staritalic_t italic_r start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT is a separable state. In fact, suppose σ⋆=∑i,jpij|i⟩aa′⟨i|⊗|j⟩bb′⟨j|superscript𝜎⋆subscript𝑖𝑗tensor-productsubscript𝑝𝑖𝑗subscriptket𝑖𝑎superscript𝑎′bra𝑖subscriptket𝑗𝑏superscript𝑏′bra𝑗\sigma^\star=\sum_i,jp_ij\keti_aa^\prime\brai\otimes\ketj_bb^% \prime\brajitalic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ | start_ARG italic_j end_ARG ⟩ start_POSTSUBSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_j end_ARG |, then directly tracing out the subsystems a′superscript𝑎′a^\primeitalic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and b′superscript𝑏′b^\primeitalic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT will lead to a decomposition of the form tra′b′σ⋆=∑i,j,k,lpijλkiμlj|aki⟩a⟨aki|⊗|blj⟩b⟨blj|𝑡subscript𝑟superscript𝑎′superscript𝑏′superscript𝜎⋆subscript𝑖𝑗𝑘𝑙tensor-productsubscript𝑝𝑖𝑗subscriptsuperscript𝜆𝑖𝑘subscriptsuperscript𝜇𝑗𝑙subscriptketsubscriptsuperscript𝑎𝑖𝑘𝑎brasubscriptsuperscript𝑎𝑖𝑘subscriptketsubscriptsuperscript𝑏𝑗𝑙𝑏brasubscriptsuperscript𝑏𝑗𝑙tr_a^\primeb^\prime\sigma^\star=\sum_i,j,k,lp_ij\lambda^i_k\mu% ^j_l\keta^i_k_a\braa^i_k\otimes\ketb^j_l_b\brab^j% _litalic_t italic_r start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i , italic_j , italic_k , italic_l end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_μ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT | start_ARG italic_a start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟨ start_ARG italic_a start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG | ⊗ | start_ARG italic_b start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⟨ start_ARG italic_b start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG |, that must be a separable state.



In particular, let us consider the pure state case. Suppose |Ψ⟩=∑iλi|xi⟩a|yi⟩bketΨsubscript𝑖subscript𝜆𝑖subscriptketsubscript𝑥𝑖𝑎subscriptketsubscript𝑦𝑖𝑏\ket\Psi=\sum_i\sqrt\lambda_i\ketx_i_a\kety_i_b| start_ARG roman_Ψ end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | start_ARG italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | start_ARG italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT with λ1≥…≥λnsubscript𝜆1…subscript𝜆𝑛\lambda_1\geq...\geq\lambda_nitalic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ … ≥ italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, then



ℰB(|Ψ⟩)≤minσ∈𝒞𝒞subscriptℰ𝐵ketΨsubscript𝜎𝒞𝒞\displaystyle\mathcalE_B(\ket\Psi)\leq\min_\sigma\in\mathcalC\mathcal% Ccaligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( | start_ARG roman_Ψ end_ARG ⟩ ) ≤ roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_C caligraphic_C end_POSTSUBSCRIPT dB2(|Ψ⟩ab⟨Ψ|⊗|u,v⟩a′b′⟨u,v|,σ)subscriptsuperscript𝑑2𝐵tensor-productsubscriptketΨ𝑎𝑏braΨsubscriptket𝑢𝑣superscript𝑎′superscript𝑏′bra𝑢𝑣𝜎\displaystyle d^2_B(\ket\Psi_ab\bra\Psi\otimes\ketu,v_a^\primeb% ^\prime\brau,v,\sigma)italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( | start_ARG roman_Ψ end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⟨ start_ARG roman_Ψ end_ARG | ⊗ | start_ARG italic_u , italic_v end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_u , italic_v end_ARG | , italic_σ )



≤dB2(fragmentssubscriptsuperscript𝑑2𝐵(\displaystyle\leq d^2_B(≤ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( |Ψ⟩ab⟨Ψ|⊗|u,v⟩a′b′⟨u,v|,tensor-productsubscriptketΨ𝑎𝑏braΨsubscriptket𝑢𝑣superscript𝑎′superscript𝑏′bra𝑢𝑣\displaystyle\ket\Psi_ab\bra\Psi\otimes\ketu,v_a^\primeb^\prime% \brau,v,| start_ARG roman_Ψ end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⟨ start_ARG roman_Ψ end_ARG | ⊗ | start_ARG italic_u , italic_v end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_u , italic_v end_ARG | ,



|x1,u⟩aa′⟨x1,u|⊗|y1,v⟩bb′⟨y1,v|)fragmentssubscriptketsubscript𝑥1𝑢𝑎superscript𝑎′brasubscript𝑥1𝑢tensor-productsubscriptketsubscript𝑦1𝑣𝑏superscript𝑏′brasubscript𝑦1𝑣)\displaystyle\ketx_1,u_aa^\prime\brax_1,u\otimes\kety_1,v_bb^% \prime\bray_1,v)| start_ARG italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_u end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_u end_ARG | ⊗ | start_ARG italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v end_ARG ⟩ start_POSTSUBSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v end_ARG | )



=EB(fragmentssubscript𝐸𝐵(\displaystyle=E_B(= italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( |Ψ⟩).fragmentsketΨ).\displaystyle\ket\Psi).| start_ARG roman_Ψ end_ARG ⟩ ) . Combining with above inequality, one has that ℰB(|Ψ⟩)=EB(|Ψ⟩)subscriptℰ𝐵ketΨsubscript𝐸𝐵ketΨ\mathcalE_B(\ket\Psi)=E_B(\ket\Psi)caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( | start_ARG roman_Ψ end_ARG ⟩ ) = italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( | start_ARG roman_Ψ end_ARG ⟩ ) for any pure state |Ψ⟩ketΨ\ket\Psi| start_ARG roman_Ψ end_ARG ⟩.



On the other hand, for any mixed state with pure state decomposition ρab=∑ipi|Ψi⟩ab⟨Ψi|subscript𝜌𝑎𝑏subscript𝑖subscript𝑝𝑖subscriptketsubscriptΨ𝑖𝑎𝑏brasubscriptΨ𝑖\rho_ab=\sum_ip_i\ket\Psi_i_ab\bra\Psi_iitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG roman_Ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⟨ start_ARG roman_Ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG |,



ℰB(ρab)≤∑ipiℰB(|Ψi⟩)=∑ipiEB(|Ψi⟩).subscriptℰ𝐵subscript𝜌𝑎𝑏subscript𝑖subscript𝑝𝑖subscriptℰ𝐵ketsubscriptΨ𝑖subscript𝑖subscript𝑝𝑖subscript𝐸𝐵ketsubscriptΨ𝑖\displaystyle\mathcalE_B(\rho_ab)\leq\sum_ip_i\mathcalE_B(\ket% \Psi_i)=\sum_ip_iE_B(\ket\Psi_i).caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≤ ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( | start_ARG roman_Ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ ) = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( | start_ARG roman_Ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ ) . Take the minimal over all pure state decomposition, one has



ℰB(ρab)≤EBcr(ρab).subscriptℰ𝐵subscript𝜌𝑎𝑏subscriptsuperscript𝐸𝑐𝑟𝐵subscript𝜌𝑎𝑏\displaystyle\mathcalE_B(\rho_ab)\leq E^cr_B(\rho_ab).caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≤ italic_E start_POSTSUPERSCRIPT italic_c italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) .



Thanks to the result that EB(ρab)=EBcr(ρab)subscript𝐸𝐵subscript𝜌𝑎𝑏subscriptsuperscript𝐸𝑐𝑟𝐵subscript𝜌𝑎𝑏E_B(\rho_ab)=E^cr_B(\rho_ab)italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = italic_E start_POSTSUPERSCRIPT italic_c italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) Streltsov et al. (2010), we actually get that



EB(ρab)=ℰB(ρab)=EBcr(ρab).subscript𝐸𝐵subscript𝜌𝑎𝑏subscriptℰ𝐵subscript𝜌𝑎𝑏subscriptsuperscript𝐸𝑐𝑟𝐵subscript𝜌𝑎𝑏\displaystyle E_B(\rho_ab)=\mathcalE_B(\rho_ab)=E^cr_B(\rho_ab).italic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = italic_E start_POSTSUPERSCRIPT italic_c italic_r end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) . (1)



With Eq.(1), we show the miniaml Bures distance of discord over state extensions is equivalent to the Bures distance of entanglement. This equivalence links the former to the existing entanglement measure, which also offer another interesting perspective on entanglement, that is, entanglement can be regards as the irreducible part of quantum discord from the perspective of a small system.



Luo first proposes the framework to quantify entanglement as the minimal quantum discord over state extensions Luo (2016). Above theorem offers the first affirmative evidence that this kind of entanglement quantification is consistent with the previous entanglement measures. So far, it is not clear whether this equivalence still holds for other entanglement measures, like relative entropy of entanglement Vedral et al. (1997), and entanglement measures based on distance Horodecki et al. (2009).



III entanglement as the minimal discord over symmetric state extensions



Even though the relationship between these two kind of entanglement measures is not clear in general, we will show that a large class of discord measures can be used to construct entanglement measures. Let us first introduce the following definition.



III.1 cross-symmetric state extension



Definition 2.



A cross-symmetric extension (CSE) of a bipartite state ρa1b1subscript𝜌subscript𝑎1subscript𝑏1\rho_a_1b_1italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is an extension ρa1a1′a2′b1b1′b2′subscript𝜌subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝑏1subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2\rho_a_1a^\prime_1a^\prime_2b_1b^\prime_1b^\prime_2italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT satisfying tra1′a2′b1′b2′ρa1a1′a2′b1b1′b2′=ρa1b1𝑡subscript𝑟subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2subscript𝜌subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝑏1subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2subscript𝜌subscript𝑎1subscript𝑏1tr_a^\prime_1a^\prime_2b^\prime_1b^\prime_2\rho_a_1a^% \prime_1a^\prime_2b_1b^\prime_1b^\prime_2=\rho_a_1b_1italic_t italic_r start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and up to an local unitary, invariant under the swap operation ΨSa1↔b1′(ΨSa1′↔b1)subscriptsuperscriptΨ↔subscript𝑎1subscriptsuperscript𝑏′1𝑆subscriptsuperscriptΨ↔subscriptsuperscript𝑎′1subscript𝑏1𝑆\Psi^a_1\leftrightarrow b^\prime_1_S(\Psi^a^\prime_1% \leftrightarrow b_1_S)roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ) between subsystems a1(a1′)subscript𝑎1subscriptsuperscript𝑎′1a_1(a^\prime_1)italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) of Alice and b1′(b1)subscriptsuperscript𝑏′1subscript𝑏1b^\prime_1(b_1)italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) of Bob, i.e., there exists some unitary Ua1a1′a2′subscript𝑈subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2U_a_1a^\prime_1a^\prime_2italic_U start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT such that



ΨSa1↔b1′(Ua1a1′a2′ρa1a1′a2′b1b1′b2′Ua1a1′a2′†)subscriptsuperscriptΨ↔subscript𝑎1subscriptsuperscript𝑏′1𝑆subscript𝑈subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝜌subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝑏1subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2subscriptsuperscript𝑈†subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2\displaystyle\Psi^a_1\leftrightarrow b^\prime_1_S(U_a_1a^\prime% _1a^\prime_2\rho_a_1a^\prime_1a^\prime_2b_1b^\prime_1b% ^\prime_2U^\dagger_a_1a^\prime_1a^\prime_2)roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )



=\displaystyle== Ua1a1′a2′ρa1a1′a2′b1b1′b2′Ua1a1′a2′†,subscript𝑈subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝜌subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝑏1subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2subscriptsuperscript𝑈†subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2\displaystyle U_a_1a^\prime_1a^\prime_2\rho_a_1a^\prime_1a^% \prime_2b_1b^\prime_1b^\prime_2U^\dagger_a_1a^\prime_1% a^\prime_2,italic_U start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , and similarly for ΨSa1′↔b1subscriptsuperscriptΨ↔subscriptsuperscript𝑎′1subscript𝑏1𝑆\Psi^a^\prime_1\leftrightarrow b_1_Sroman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT.



By definition, the extended subsystems are label with ′′\prime′, like a1′subscriptsuperscript𝑎′1a^\prime_1italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, b2′subscriptsuperscript𝑏′2b^\prime_2italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, and there exist a subsystem b1′subscriptsuperscript𝑏′1b^\prime_1italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT in party b is symmetric with the original subsystem a1subscript𝑎1a_1italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT in party a in the sense of invariance under swap operation up to a local operation, and the same for b1′subscriptsuperscript𝑏′1b^\prime_1italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Since the symmetry is “cross” two subsystems, we call it cross-symmetric extension (Fig.1).



ρa1a1′a2′b1b1′b2′subscript𝜌subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝑏1subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2\rho_a_1a^\prime_1a^\prime_2b_1b^\prime_1b^\prime_2italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPTρa1subscript𝜌subscript𝑎1\rho_a_1italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPTρa1′subscript𝜌subscriptsuperscript𝑎′1\rho_a^\prime_1italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPTρb1subscript𝜌subscript𝑏1\rho_b_1italic_ρ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPTρb1′subscript𝜌subscriptsuperscript𝑏′1\rho_b^\prime_1italic_ρ start_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPTρa2′subscript𝜌subscriptsuperscript𝑎′2\rho_a^\prime_2italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPTρb2′subscript𝜌subscriptsuperscript𝑏′2\rho_b^\prime_2italic_ρ start_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPTρa1b1subscript𝜌subscript𝑎1subscript𝑏1\rho_a_1b_1italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPTSWAPSWAP Fig.1: The original state is ρa1b1subscript𝜌subscript𝑎1subscript𝑏1\rho_a_1b_1italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT (red color) shared by party a and b. The cross-symmetric extension ρa1a1′a2′b1b1′b2′subscript𝜌subscript𝑎1subscriptsuperscript𝑎normal-′1subscriptsuperscript𝑎normal-′2subscript𝑏1subscriptsuperscript𝑏normal-′1subscriptsuperscript𝑏normal-′2\rho_a_1a^\prime_1a^\prime_2b_1b^\prime_1b^\prime_2italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is invariant under swap operation between local subsystem a1(b1)subscript𝑎1subscript𝑏1a_1(b_1)italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and extended local subsystem in b1′(a1′)subscriptsuperscript𝑏normal-′1subscriptsuperscript𝑎normal-′1b^\prime_1(a^\prime_1)italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ), up to local unitary operation.



Furthermore, we claim that CSE will always exist for bipartite quantum state. Let us consider a quantum state with spectral decomposition ρab=∑iλi|ψi⟩ab⟨ψi|subscript𝜌𝑎𝑏subscript𝑖subscript𝜆𝑖subscriptketsubscript𝜓𝑖𝑎𝑏brasubscript𝜓𝑖\rho_ab=\sum_i\lambda_i\ket\psi_i_ab\bra\psi_iitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⟨ start_ARG italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG |. Assume |ψi⟩ab=∑jμji|xji⟩a|yji⟩bsubscriptketsubscript𝜓𝑖𝑎𝑏subscript𝑗subscriptsuperscript𝜇𝑖𝑗subscriptketsubscriptsuperscript𝑥𝑖𝑗𝑎subscriptketsubscriptsuperscript𝑦𝑖𝑗𝑏\ket\psi_i_ab=\sum_j\sqrt\mu^i_j\ketx^i_j_a\kety^i_j% _b| start_ARG italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT square-root start_ARG italic_μ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG | start_ARG italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | start_ARG italic_y start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT, then |Ψ⟩=∑iλi|ψi⟩ab|i,i⟩a′b′ketΨsubscript𝑖subscript𝜆𝑖subscriptketsubscript𝜓𝑖𝑎𝑏subscriptket𝑖𝑖superscript𝑎′superscript𝑏′\ket\Psi=\sum_i\sqrt\lambda_i\ket\psi_i_ab\keti,i_a^\primeb% ^\prime| start_ARG roman_Ψ end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | start_ARG italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT | start_ARG italic_i , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is a CSE of ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT. In fact, rewrite |Ψ⟩=∑ijλiμji|xji,i⟩aa′|yji,i⟩bb′ketΨsubscript𝑖𝑗subscript𝜆𝑖subscriptsuperscript𝜇𝑖𝑗subscriptketsubscriptsuperscript𝑥𝑖𝑗𝑖𝑎superscript𝑎′subscriptketsubscriptsuperscript𝑦𝑖𝑗𝑖𝑏superscript𝑏′\ket\Psi=\sum_ij\sqrt\lambda_i\mu^i_j\ketx^i_j,i_aa^\prime% \kety^i_j,i_bb^\prime| start_ARG roman_Ψ end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_μ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG | start_ARG italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG italic_y start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT and define Uaa′=∑i,j|i,yji⟩⟨xji,i|subscript𝑈𝑎superscript𝑎′subscript𝑖𝑗ket𝑖subscriptsuperscript𝑦𝑖𝑗brasubscriptsuperscript𝑥𝑖𝑗𝑖U_aa^\prime=\sum_i,j\keti,y^i_j\brax^i_j,iitalic_U start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT | start_ARG italic_i , italic_y start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_i end_ARG |, one has



Uaa′|Ψ⟩=∑i,jλiμji|i,yji⟩aa′|yji,i⟩bb′subscript𝑈𝑎superscript𝑎′ketΨsubscript𝑖𝑗subscript𝜆𝑖subscriptsuperscript𝜇𝑖𝑗subscriptket𝑖subscriptsuperscript𝑦𝑖𝑗𝑎superscript𝑎′subscriptketsubscriptsuperscript𝑦𝑖𝑗𝑖𝑏superscript𝑏′\displaystyle U_aa^\prime\ket\Psi=\sum_i,j\sqrt\lambda_i\mu^i_j% \keti,y^i_j_aa^\prime\kety^i_j,i_bb^\primeitalic_U start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG roman_Ψ end_ARG ⟩ = ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_μ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG | start_ARG italic_i , italic_y start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | start_ARG italic_y start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is invariant under swap operation between subsystem a(a′)𝑎superscript𝑎′a(a^\prime)italic_a ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) and b′(b)superscript𝑏′𝑏b^\prime(b)italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_b ).



III.2 Quantifying entanglement



In this part, we will employ the minimal discord over cross-symmetric state extensions to quantify entanglement from the geometric perspective. A pseudo-distance d𝑑ditalic_d in state space D(H)𝐷𝐻D(H)italic_D ( italic_H ) is a non-negative bivariate function with with d(ρ,σ)=0𝑑𝜌𝜎0d(\rho,\sigma)=0italic_d ( italic_ρ , italic_σ ) = 0 iff ρ=σ𝜌𝜎\rho=\sigmaitalic_ρ = italic_σ. We call d𝑑ditalic_d contractive, if it satisfy d(ρ,σ)≥d(Φ(ρ),Φ(σ))𝑑𝜌𝜎𝑑Φ𝜌Φ𝜎d(\rho,\sigma)\geq d(\Phi(\rho),\Phi(\sigma))italic_d ( italic_ρ , italic_σ ) ≥ italic_d ( roman_Φ ( italic_ρ ) , roman_Φ ( italic_σ ) ) for any quantum operation ΦΦ\Phiroman_Φ and ρ,σ∈D(H)𝜌𝜎𝐷𝐻\rho,\sigma\in D(H)italic_ρ , italic_σ ∈ italic_D ( italic_H ). For simplicity, we say “distance” in the following instead of “pseudo-distance”. Since d(ρ,σ)≥d(UρU†,UσU†)≥d(ρ,σ)𝑑𝜌𝜎𝑑𝑈𝜌superscript𝑈†𝑈𝜎superscript𝑈†𝑑𝜌𝜎d(\rho,\sigma)\geq d(U\rho U^\dagger,U\sigma U^\dagger)\geq d(\rho,\sigma)italic_d ( italic_ρ , italic_σ ) ≥ italic_d ( italic_U italic_ρ italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT , italic_U italic_σ italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) ≥ italic_d ( italic_ρ , italic_σ ), d𝑑ditalic_d is also unitary invariant in the sense that d(ρ,σ)=d(UρU†,UσU†)𝑑𝜌𝜎𝑑𝑈𝜌superscript𝑈†𝑈𝜎superscript𝑈†d(\rho,\sigma)=d(U\rho U^\dagger,U\sigma U^\dagger)italic_d ( italic_ρ , italic_σ ) = italic_d ( italic_U italic_ρ italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT , italic_U italic_σ italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ).



A key deviation of the quantum from classical is the fundamental fact that the information of a classical system can be extracted with measurement without destroying the system, while the quantum system cannot. Therefore, the classical correlated states can be regards as classical, and quantum correlation of bipartite system can be characterized in the following ways.



Definition 3.



For contractive distance d𝑑ditalic_d, the corresponding geometric discord is defined as Vedral et al. (1997)



DdG(ρab):=minσ∈𝒞𝒞d(ρab,σ),assignsubscriptsuperscript𝐷𝐺𝑑subscript𝜌𝑎𝑏subscript𝜎𝒞𝒞𝑑subscript𝜌𝑎𝑏𝜎\displaystyle D^G_d(\rho_ab):=\min_\sigma\in\mathcalC\mathcalCd(% \rho_ab,\sigma),italic_D start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) := roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_C caligraphic_C end_POSTSUBSCRIPT italic_d ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , italic_σ ) , where the minimal is taken over all classical correlated states.



By definition, it is easy to check that Ddsubscript𝐷𝑑D_ditalic_D start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT is faithful in classical correlated states and local unitary invariant.



Definition 4.



For contractive distance d𝑑ditalic_d, the measurement-induced discord (MID) is defined as Luo (2008)



DdM(ρab):=minΠd(ρab,Π(ρab)),assignsubscriptsuperscript𝐷𝑀𝑑subscript𝜌𝑎𝑏subscriptΠ𝑑subscript𝜌𝑎𝑏Πsubscript𝜌𝑎𝑏\displaystyle D^M_d(\rho_ab):=\min_\Pid(\rho_ab,\Pi(\rho_ab)),italic_D start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) := roman_min start_POSTSUBSCRIPT roman_Π end_POSTSUBSCRIPT italic_d ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , roman_Π ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ) , where the minimal is taken over all local von Neumann measurement πai⊗πbjtensor-productsubscriptsuperscript𝜋𝑖𝑎subscriptsuperscript𝜋𝑗𝑏\\pi^i_a\otimes\pi^j_b\ italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT , i.e., Π(ρab)=∑ijπai⊗πbjρabπai⊗πbjΠsubscript𝜌𝑎𝑏subscript𝑖𝑗tensor-producttensor-productsubscriptsuperscript𝜋𝑖𝑎subscriptsuperscript𝜋𝑗𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝜋𝑖𝑎subscriptsuperscript𝜋𝑗𝑏\Pi(\rho_ab)=\sum_ij\pi^i_a\otimes\pi^j_b\rho_ab\pi^i_a% \otimes\pi^j_broman_Π ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT.



Moreover, the measurement-induced discord is called asymmetric, if the minimal measurement is taken over all one-side von Neumann measurement tensor-productsubscriptketsubscript𝛼𝑖𝑎brasubscript𝛼𝑖subscript𝐼𝑏\\ket\alpha_i_a\bra\alpha_i\otimes I_b\ . The measurement-induced asymmetric discord with the Bures, Hellinger, trace, and Hilbert-Schmidt distances, have been considered in Luo and Fu (2010); Dakić et al. (2010); Nakano et al. (2013); Ciccarello et al. (2014); Piani et al. (2014); Roga et al. (2016). Actually, though the minimal involved in the above definition is intuitively appealing, it is usually intractable to find the measurement with minimal disturbance.



Definition 5.



For contractive distance d𝑑ditalic_d, define Luo (2008)



DdL(ρab):=d(ρab,Π(ρab)),assignsubscriptsuperscript𝐷𝐿𝑑subscript𝜌𝑎𝑏𝑑subscript𝜌𝑎𝑏Πsubscript𝜌𝑎𝑏\displaystyle D^L_d(\rho_ab):=d(\rho_ab,\Pi(\rho_ab)),italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) := italic_d ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , roman_Π ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ) , as the measure of quantum correlations, where the measurement ΠΠ\Piroman_Π will preserve the local reduced states ρasubscript𝜌𝑎\rho_aitalic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT and ρbsubscript𝜌𝑏\rho_bitalic_ρ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT invariant, i.e., Π(ρab)=∑ijπai⊗πbjρabπai⊗πbjΠsubscript𝜌𝑎𝑏subscript𝑖𝑗tensor-producttensor-productsubscriptsuperscript𝜋𝑖𝑎subscriptsuperscript𝜋𝑗𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝜋𝑖𝑎subscriptsuperscript𝜋𝑗𝑏\Pi(\rho_ab)=\sum_ij\pi^i_a\otimes\pi^j_b\rho_ab\pi^i_a% \otimes\pi^j_broman_Π ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT with ρa=∑ipiπaisubscript𝜌𝑎subscript𝑖subscript𝑝𝑖subscriptsuperscript𝜋𝑖𝑎\rho_a=\sum_ip_i\pi^i_aitalic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT, ρb=∑ipiπbisubscript𝜌𝑏subscript𝑖subscript𝑝𝑖subscriptsuperscript𝜋𝑖𝑏\rho_b=\sum_ip_i\pi^i_bitalic_ρ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT.



In fact, the measurement induced by the spectral resolution is the least disturbing in the sense that it leaves the marginal information invariant. Moreover, this characterization of quantum correlation is in fact equivalent to the subsequent correlated coherence of bipartite system Tan et al. (2016); Tan and Jeong (2018). It can be verified that the DdMsubscriptsuperscript𝐷𝑀𝑑D^M_ditalic_D start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT and DdLsubscriptsuperscript𝐷𝐿𝑑D^L_ditalic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT are both faithful in 𝒞𝒞𝒞𝒞\mathcalC\mathcalCcaligraphic_C caligraphic_C and invariant under local unitary operations.



Let D𝐷Ditalic_D be DdGsubscriptsuperscript𝐷𝐺𝑑D^G_ditalic_D start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT, DdMsubscriptsuperscript𝐷𝑀𝑑D^M_ditalic_D start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT or DdLsubscriptsuperscript𝐷𝐿𝑑D^L_ditalic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT, then D𝐷Ditalic_D is faithful in 𝒞𝒞𝒞𝒞\mathcalC\mathcalCcaligraphic_C caligraphic_C and invariant under local unitary operations and we also have the following fundamental property.



If d𝑑ditalic_d satisfying d(ρ,σ)=d(ρ⊗|0⟩⟨0|,σ⊗|0⟩⟨0|)𝑑𝜌𝜎𝑑tensor-product𝜌ket0bra0tensor-product𝜎ket0bra0d(\rho,\sigma)=d(\rho\otimes\ket0\bra0,\sigma\otimes\ket0\bra0)italic_d ( italic_ρ , italic_σ ) = italic_d ( italic_ρ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | , italic_σ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | ), then D𝐷Ditalic_D will not increase when adding a pure state ancilla, that is,



D(ρab⊗|0⟩b1⟨0|)≤D(ρab),𝐷tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0𝐷subscript𝜌𝑎𝑏\displaystyle D(\rho_ab\otimes\ket0_b_1\bra0)\leq D(\rho_ab),italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ) ≤ italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) , (2) where ρab⊗|0⟩b1⟨0|tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0\rho_ab\otimes\ket0_b_1\bra0italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | is a bipartite state between a𝑎aitalic_a and bb1𝑏subscript𝑏1bb_1italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.



Firstly, assume σ⋆superscript𝜎⋆\sigma^\staritalic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT is the closest classical correlated state, then



DdG(ρab)subscriptsuperscript𝐷𝐺𝑑subscript𝜌𝑎𝑏\displaystyle D^G_d(\rho_ab)italic_D start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) =d(ρab,σ⋆)=d(ρab⊗|0⟩b1⟨0|,σ⋆⊗|0⟩b1⟨0|)absent𝑑subscript𝜌𝑎𝑏superscript𝜎⋆𝑑tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0tensor-productsuperscript𝜎⋆subscriptket0subscript𝑏1bra0\displaystyle=d(\rho_ab,\sigma^\star)=d(\rho_ab\otimes\ket0_b_1% \bra0,\sigma^\star\otimes\ket0_b_1\bra0)= italic_d ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ) = italic_d ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | , italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | )



≥minσ∈𝒞𝒞d(ρab⊗|0⟩b1⟨0|,σ)=DdG(ρab⊗|0⟩b1⟨0|).absentsubscript𝜎𝒞𝒞𝑑tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0𝜎subscriptsuperscript𝐷𝐺𝑑tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0\displaystyle\geq\min_\sigma\in\mathcalC\mathcalCd(\rho_ab\otimes\ket% 0_b_1\bra0,\sigma)=D^G_d(\rho_ab\otimes\ket0_b_1\bra0).≥ roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_C caligraphic_C end_POSTSUBSCRIPT italic_d ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | , italic_σ ) = italic_D start_POSTSUPERSCRIPT italic_G end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ) .



Secondly, assume Π⋆superscriptΠ⋆\Pi^\starroman_Π start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT is the optimal measurement, then



DdM(ρab)subscriptsuperscript𝐷𝑀𝑑subscript𝜌𝑎𝑏\displaystyle D^M_d(\rho_ab)italic_D start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) =d(ρab,Π⋆(ρab))absent𝑑subscript𝜌𝑎𝑏superscriptΠ⋆subscript𝜌𝑎𝑏\displaystyle=d(\rho_ab,\Pi^\star(\rho_ab))= italic_d ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , roman_Π start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) )



=d(ρab⊗|0⟩b1⟨0|,Π⋆(ρab)⊗|0⟩b1⟨0|)absent𝑑tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0tensor-productsuperscriptΠ⋆subscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0\displaystyle=d(\rho_ab\otimes\ket0_b_1\bra0,\Pi^\star(\rho_ab)% \otimes\ket0_b_1\bra0)= italic_d ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | , roman_Π start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | )



≥minΠd(ρab⊗|0⟩b1⟨0|,Π(ρab⊗|0⟩b1⟨0|))absentsubscriptΠ𝑑tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0Πtensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0\displaystyle\geq\min_\Pid(\rho_ab\otimes\ket0_b_1\bra0,\Pi(\rho_% ab\otimes\ket0_b_1\bra0))≥ roman_min start_POSTSUBSCRIPT roman_Π end_POSTSUBSCRIPT italic_d ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | , roman_Π ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ) )



=DdM(ρab⊗|0⟩b1⟨0|).absentsubscriptsuperscript𝐷𝑀𝑑tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0\displaystyle=D^M_d(\rho_ab\otimes\ket0_b_1\bra0).= italic_D start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ) .



At last, if the reduced states have spectral decomposition ρa=∑ipiπaisubscript𝜌𝑎subscript𝑖subscript𝑝𝑖subscriptsuperscript𝜋𝑖𝑎\rho_a=\sum_ip_i\pi^i_aitalic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT, ρb=∑jpjπbjsubscript𝜌𝑏subscript𝑗subscript𝑝𝑗subscriptsuperscript𝜋𝑗𝑏\rho_b=\sum_jp_j\pi^j_bitalic_ρ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT, then ρbb1=∑ipiπbi⊗|0⟩b1⟨0|subscript𝜌𝑏subscript𝑏1subscript𝑖tensor-productsubscript𝑝𝑖subscriptsuperscript𝜋𝑖𝑏subscriptket0subscript𝑏1bra0\rho_bb_1=\sum_ip_i\pi^i_b\otimes\ket0_b_1\bra0italic_ρ start_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | and it is easy to check that



DdL(ρab⊗|0⟩b1⟨0|)subscriptsuperscript𝐷𝐿𝑑tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0\displaystyle D^L_d(\rho_ab\otimes\ket0_b_1\bra0)italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | )



=\displaystyle== d(ρab⊗|0⟩b1⟨0|,Π(ρab)⊗|0⟩b1⟨0|)=DdL(ρab).𝑑tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0tensor-productΠsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0subscriptsuperscript𝐷𝐿𝑑subscript𝜌𝑎𝑏\displaystyle d(\rho_ab\otimes\ket0_b_1\bra0,\Pi(\rho_ab)\otimes% \ket0_b_1\bra0)=D^L_d(\rho_ab).italic_d ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | , roman_Π ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ) = italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) . ∎



Remark 2.



It is easy to check that d(ρ,σ)=d(ρ⊗|0⟩⟨0|,σ⊗|0⟩⟨0|)𝑑𝜌𝜎𝑑tensor-product𝜌ket0bra0tensor-product𝜎ket0bra0d(\rho,\sigma)=d(\rho\otimes\ket0\bra0,\sigma\otimes\ket0\bra0)italic_d ( italic_ρ , italic_σ ) = italic_d ( italic_ρ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | , italic_σ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | ) is satisfied by Bures distance, Hellinger distance Luo and Zhang (2004), trace distance, Hilbert-Schmidt distance and relative entropy (See Appendix B).



Definition 6.



For bipartite state ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT, the minimal discord over cross-symmetric extensions is defined as



ℰ~D(ρab):=mina′b′D(ρaa′:bb′),assignsubscript~ℰ𝐷subscript𝜌𝑎𝑏subscriptsuperscript𝑎′superscript𝑏′𝐷subscript𝜌:𝑎superscript𝑎′𝑏superscript𝑏′\displaystyle\tilde\mathcalE_D(\rho_ab):=\min_a^\primeb^\primeD(% \rho_aa^\prime:bb^\prime),over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) := roman_min start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) , where the minimal is performed over all possible CSE of ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT.



Based on the above discussion, we have the following results.



If D(ρab)𝐷subscript𝜌𝑎𝑏D(\rho_ab)italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is faithful in 𝒞𝒞𝒞𝒞\mathcalC\mathcalCcaligraphic_C caligraphic_C, local unitary invariant and contractive when adding a pure state ancilla, then it can be verified that ℰ~Dsubscriptnormal-~ℰ𝐷\tilde\mathcalE_Dover~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT satisfies the following elementary properties:



1. ℰ~D(ρab)≥0subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏0\tilde\mathcalE_D(\rho_ab)\geq 0over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≥ 0, the equality holds if and only if ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT is separable.



2. ℰ~D(ρab)subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is locally unitary invariant on parties a and b.



3. ℰ~D(ρab)subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is nonincreasing under local partial trace in the sense that



4. ℰ~D(ρab)subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is contractive adding a pure state ancilla, i.e.,



ℰ~D(ρa1b1⊗|0⟩b2⟨0|)≤ℰ~D(ρa1b1).subscript~ℰ𝐷tensor-productsubscript𝜌subscript𝑎1subscript𝑏1subscriptket0subscript𝑏2bra0subscript~ℰ𝐷subscript𝜌subscript𝑎1subscript𝑏1\displaystyle\tilde\mathcalE_D(\rho_a_1b_1\otimes\ket0_b_2% \bra0)\leq\tilde\mathcalE_D(\rho_a_1b_1).over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ) ≤ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) .



5. ℰ~D(ρab)subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is nonincreasing under local operations, i.e.,



ℰ~D(ρab)≤ℰ~D(∑iKaiρabKai†).subscript~ℰ𝐷subscript𝜌𝑎𝑏subscript~ℰ𝐷subscript𝑖subscriptsuperscript𝐾𝑖𝑎subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑎\displaystyle\tilde\mathcalE_D(\rho_ab)\leq\tilde\mathcalE_D(% \sum_iK^i_a\rho_abK^i\dagger_a).over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≤ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) .



6. ℰ~D(ρab)subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is non-increasing under classical communications between parties a and b.



7. ℰ~D(ρab)subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is nonincreasing under LOCC type operations.



See Appendix C. ∎



As a result, we conclude our second main result.



Corollary.



If D(ρab)𝐷subscript𝜌𝑎𝑏D(\rho_ab)italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is faithful in 𝒞𝒞𝒞𝒞\mathcalC\mathcalCcaligraphic_C caligraphic_C, local unitary invariant and contractive when adding a pure state ancilla, then ℰ~D(ρab)subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is an entanglement monotone in the sense that (1) ℰ~D(ρab)subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is faithful in classical correlated states. (2) ℰ~D(ρab)subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is invariant under local unitary operations. (3) ℰ~D(ρab)≥ℰ~D(ΠLOCCρab)subscriptnormal-~ℰ𝐷subscript𝜌𝑎𝑏subscriptnormal-~ℰ𝐷subscriptnormal-Π𝐿𝑂𝐶𝐶subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)\geq\tilde\mathcalE_D(\Pi_LOCC\rho_% ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≥ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_L italic_O italic_C italic_C end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) for any LOCC type operations.



It follows directly from properties 1, 2, and 7 above. ∎



III.3 Hilbert Schmidt distance case



It is well known that Hilbert-Schmidt distance is not contractive and it is not clear whether this distance can be used to quantify entanglement Ozawa (2000). In this part, we will show that the minimal Hilbert-Schmidt distance of discord over cross-symmetric state extensions is an entanglement monotone, and give the analytic formula for pure states.



Consider Hilbert-Schmidt distance



dHS(ρ,σ):=||ρ-σ||2=tr(ρ2+σ2-2ρσ),assignsubscript𝑑𝐻𝑆𝜌𝜎subscriptnorm𝜌𝜎2𝑡𝑟superscript𝜌2superscript𝜎22𝜌𝜎\displaystyle d_HS(\rho,\sigma):=||\rho-\sigma||_2=\sqrttr(\rho^2+% \sigma^2-2\rho\sigma),italic_d start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT ( italic_ρ , italic_σ ) := | | italic_ρ - italic_σ | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = square-root start_ARG italic_t italic_r ( italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_ρ italic_σ ) end_ARG , let us define the corresponding discord measure by DHSL:=||ρ-σ||22=tr(ρ2+σ2-2ρσ)assignsubscriptsuperscript𝐷𝐿𝐻𝑆subscriptsuperscriptnorm𝜌𝜎22𝑡𝑟superscript𝜌2superscript𝜎22𝜌𝜎D^L_HS:=||\rho-\sigma||^2_2=\sqrttr(\rho^2+\sigma^2-2\rho\sigma)italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT := | | italic_ρ - italic_σ | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = square-root start_ARG italic_t italic_r ( italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_ρ italic_σ ) end_ARG and denotes the corresponding entanglement measure by ℰ~DHSLsubscript~ℰsubscriptsuperscript𝐷𝐿𝐻𝑆\tilde\mathcalE_D^L_HSover~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT end_POSTSUBSCRIPT.



By definition, DHSLsubscriptsuperscript𝐷𝐿𝐻𝑆D^L_HSitalic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT is faithful in 𝒞𝒞𝒞𝒞\mathcalC\mathcalCcaligraphic_C caligraphic_C, local unitary invariant. Since dHS(ρ,σ)=dp(ρ⊗|0⟩⟨0|,σ⊗|0⟩⟨0|)subscript𝑑𝐻𝑆𝜌𝜎subscript𝑑𝑝tensor-product𝜌ket0bra0tensor-product𝜎ket0bra0d_HS(\rho,\sigma)=d_p(\rho\otimes\ket0\bra0,\sigma\otimes\ket0\bra0)italic_d start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT ( italic_ρ , italic_σ ) = italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_ρ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | , italic_σ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | ) (See Appendix B), Theorem 3 implies that DHSLsubscriptsuperscript𝐷𝐿𝐻𝑆D^L_HSitalic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT is contractive when adding a pure state ancilla and then Theorem 4 tells us that ℰ~DHSLsubscript~ℰsubscriptsuperscript𝐷𝐿𝐻𝑆\tilde\mathcalE_D^L_HSover~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT end_POSTSUBSCRIPT is an entanglement monotone.



Next, we consider the minimal Hilbert-Schmidt distance of discord over CSE on pure states. Firstly, observing that |ψ⟩ab|0,0⟩a′b′subscriptket𝜓𝑎𝑏subscriptket00superscript𝑎′superscript𝑏′\ket\psi_ab\ket0,0_a^\primeb^\prime| start_ARG italic_ψ end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is a CSE for |ψ⟩absubscriptket𝜓𝑎𝑏\ket\psi_ab| start_ARG italic_ψ end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT as pure state |ϕ⟩ketitalic-ϕ\ket\phi| start_ARG italic_ϕ end_ARG ⟩ has trivial extensions, i.e., |ϕ⟩⟨ϕ|⊗ρtensor-productketitalic-ϕbraitalic-ϕ𝜌\ket\phi\bra\phi\otimes\rho| start_ARG italic_ϕ end_ARG ⟩ ⟨ start_ARG italic_ϕ end_ARG | ⊗ italic_ρ. Then, for |ψ⟩ab=∑iλi|i⟩a|i⟩bsubscriptket𝜓𝑎𝑏subscript𝑖subscript𝜆𝑖subscriptket𝑖𝑎subscriptket𝑖𝑏\ket\psi_ab=\sum_i\sqrt\lambda_i\keti_a\keti_b| start_ARG italic_ψ end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT with λi≥…≥λnsubscript𝜆𝑖…subscript𝜆𝑛\lambda_i\geq...\geq\lambda_nitalic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ … ≥ italic_λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, one has



ℰ~DHSL(|ψ⟩ab)≤DHSL(|ψ⟩ab|0,0⟩a′b′)=1-∑iλi2.subscript~ℰsubscriptsuperscript𝐷𝐿𝐻𝑆subscriptket𝜓𝑎𝑏subscriptsuperscript𝐷𝐿𝐻𝑆subscriptket𝜓𝑎𝑏subscriptket00superscript𝑎′superscript𝑏′1subscript𝑖subscriptsuperscript𝜆2𝑖\displaystyle\tilde\mathcalE_D^L_HS(\ket\psi_ab)\leq D^L_HS(% \ket\psi_ab\ket0,0_a^\primeb^\prime)=1-\sum_i\lambda^2_i.over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | start_ARG italic_ψ end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≤ italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT ( | start_ARG italic_ψ end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = 1 - ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT .



Secondly, note that



DHSL(|ψ⟩ab⟨ψ|⊗ρa′b′)subscriptsuperscript𝐷𝐿𝐻𝑆tensor-productsubscriptket𝜓𝑎𝑏bra𝜓subscript𝜌superscript𝑎′superscript𝑏′\displaystyle D^L_HS(\ket\psi_ab\bra\psi\otimes\rho_a^\primeb^% \prime)italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT ( | start_ARG italic_ψ end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⟨ start_ARG italic_ψ end_ARG | ⊗ italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT )



=\displaystyle== 1-∑iλi2(trρa′b′Πa′b′ρa′b′Πa′b′)1subscript𝑖subscriptsuperscript𝜆2𝑖𝑡𝑟subscript𝜌superscript𝑎′superscript𝑏′subscriptΠsuperscript𝑎′superscript𝑏′subscript𝜌superscript𝑎′superscript𝑏′subscriptΠsuperscript𝑎′superscript𝑏′\displaystyle 1-\sum_i\lambda^2_i(tr\rho_a^\primeb^\prime\Pi_a^% \primeb^\prime\rho_a^\primeb^\prime\Pi_a^\primeb^\prime)1 - ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t italic_r italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT )



≥\displaystyle\geq≥ 1-∑iλi2,1subscript𝑖subscriptsuperscript𝜆2𝑖\displaystyle 1-\sum_i\lambda^2_i,1 - ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , where Πa′b′(⋅)=∑ijπai⊗πjb(⋅)πai⊗πjbsubscriptΠsuperscript𝑎′superscript𝑏′⋅subscript𝑖𝑗tensor-producttensor-productsubscriptsuperscript𝜋𝑖𝑎subscriptsuperscript𝜋𝑏𝑗⋅subscriptsuperscript𝜋𝑖𝑎subscriptsuperscript𝜋𝑏𝑗\Pi_a^\primeb^\prime(\cdot)=\sum_ij\pi^i_a\otimes\pi^b_j(\cdot% )\pi^i_a\otimes\pi^b_jroman_Π start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( ⋅ ) = ∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_π start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( ⋅ ) italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_π start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT with πaisubscriptsuperscript𝜋𝑖𝑎\\pi^i_a\ italic_π start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT and πbjsubscriptsuperscript𝜋𝑗𝑏\\pi^j_b\ italic_π start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT are spectral of reduced states ρasubscript𝜌𝑎\rho_aitalic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT and ρbsubscript𝜌𝑏\rho_bitalic_ρ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT, respectively. The equality holds if and only if ρa′b′subscript𝜌superscript𝑎′superscript𝑏′\rho_a^\primeb^\primeitalic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is a separate pure state. Then, ℰ~DHSL(|ψ⟩ab)subscript~ℰsubscriptsuperscript𝐷𝐿𝐻𝑆subscriptket𝜓𝑎𝑏\tilde\mathcalE_D^L_HS(\ket\psi_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | start_ARG italic_ψ end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is also lower bounded by 1-∑iλi21subscript𝑖subscriptsuperscript𝜆2𝑖1-\sum_i\lambda^2_i1 - ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT.



On the other hand, recall the Hilbert-Schmidt distance of asymmetric discord with Dakić et al. (2010b), DHSA(ρab):=minσ∈𝒞𝒬||ρab-σ||22assignsubscriptsuperscript𝐷𝐴𝐻𝑆subscript𝜌𝑎𝑏subscript𝜎𝒞𝒬subscriptsuperscriptnormsubscript𝜌𝑎𝑏𝜎22D^A_HS(\rho_ab):=\min_\sigma\in\mathcalCQ||\rho_ab-\sigma||^2_2italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) := roman_min start_POSTSUBSCRIPT italic_σ ∈ caligraphic_C caligraphic_Q end_POSTSUBSCRIPT | | italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT - italic_σ | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, where the minimal is taken over all classical-quantum states, which can be written as the form ∑ipi|αi⟩⟨αi|⊗ρisubscript𝑖tensor-productsubscript𝑝𝑖ketsubscript𝛼𝑖brasubscript𝛼𝑖subscript𝜌𝑖\sum_ip_i\ket\alpha_i\bra\alpha_i\otimes\rho_i∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | ⊗ italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT with αi⟩ketsubscript𝛼𝑖\\ket\alpha_i\ are othogonal. Then, DHSAsubscriptsuperscript𝐷𝐴𝐻𝑆D^A_HSitalic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT is faithful and local unitary invariant. Furthermore, for pure states |ψ⟩ab=∑iλi|xi⟩a|yi⟩bsubscriptket𝜓𝑎𝑏subscript𝑖subscript𝜆𝑖subscriptketsubscript𝑥𝑖𝑎subscriptketsubscript𝑦𝑖𝑏\ket\psi_ab=\sum_i\sqrt\lambda_i\ketx_i_a\kety_i_b| start_ARG italic_ψ end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | start_ARG italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | start_ARG italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT, Chang and Luo (2013)



DHSA=1-∑iλi2.subscriptsuperscript𝐷𝐴𝐻𝑆1subscript𝑖subscriptsuperscript𝜆2𝑖\displaystyle D^A_HS=1-\sum_i\lambda^2_i.italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT = 1 - ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT .



Hence, we show that DHSA(ρab)subscriptsuperscript𝐷𝐴𝐻𝑆subscript𝜌𝑎𝑏D^A_HS(\rho_ab)italic_D start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is a fide bona measure of quantum correlation, that is, it is faithful in classical-quantum states, invariant under local operations and reduces to ℰ~DHSLsubscript~ℰsubscriptsuperscript𝐷𝐿𝐻𝑆\tilde\mathcalE_D^L_HSover~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_H italic_S end_POSTSUBSCRIPT end_POSTSUBSCRIPT on pure states.



In this paper, we introduce the minimal Bures distance of quantum discord over state extensions to characterize entanglement. By proving that this quantification is equivalent to the Bures distance of entanglement, we establish the link between entanglement and discord in the sense of state extensions. This result provides the first evidence that the minimal discords over state extensions are equivalent to the previous entanglement measures.



Moreover, we construct a large class of entanglement measures with the quantum discord measures. Actually, we show that the minimal discord over cross-symmetric state extensions is non-increasing under local operations and classical communications, if the discord measure is contractive when adding pure state ancilla. This result establishes an interesting connection between entanglement and discord.



In particular, even though Hilbert-Schmidt distance is not contractive and it is not clear whether it can be used to quantify entanglement directly, our results indicate that the corresponding quantification is an entanglement monotone in our framework.



Acknowledgements.This project is supported in part by the National Natural Science Foundation of China (Grants No. 61876195), the Natural Science Foundation of Guangdong Province of China (Grant No. 2017B030311011), the Fundamental Research Funds for the Central Universities of China (Grant No. 17lgjc24), and Jiangxi Provincial Natural Science Foundation (20202BAB201001).



Appendix A Proof of Theorem 1



Theorem 1. ℰBsubscriptℰ𝐵\mathcalE_Bcaligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is convex in the sense that



ℰB(∑ipiρabi)≤∑ipiℰB(ρabi),subscriptℰ𝐵subscript𝑖subscript𝑝𝑖subscriptsuperscript𝜌𝑖𝑎𝑏subscript𝑖subscript𝑝𝑖subscriptℰ𝐵subscriptsuperscript𝜌𝑖𝑎𝑏\displaystyle\mathcalE_B(\sum_ip_i\rho^i_ab)\leq\sum_ip_i% \mathcalE_B(\rho^i_ab),caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≤ ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) , where pisubscript𝑝𝑖p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are probabilities and ρabisubscriptsuperscript𝜌𝑖𝑎𝑏\rho^i_abitalic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT are bipartite states shared by parties a𝑎aitalic_a and b𝑏bitalic_b.



Noting that



ρaa′a′′:bb′b′′:=∑ipiρaa′:bb′i⊗|i,i⟩a′′b′′⟨i,i|assignsubscript𝜌:𝑎superscript𝑎′superscript𝑎′′𝑏superscript𝑏′superscript𝑏′′subscript𝑖tensor-productsubscript𝑝𝑖subscriptsuperscript𝜌𝑖:𝑎superscript𝑎′𝑏superscript𝑏′subscriptket𝑖𝑖superscript𝑎′′superscript𝑏′′bra𝑖𝑖\displaystyle\rho_aa^\primea^\prime\prime:bb^\primeb^\prime\prime:=% \sum_ip_i\rho^i_aa^\prime:bb^\prime\otimes\keti,i_a^\prime% \primeb^\prime\prime\brai,iitalic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT := ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG italic_i , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i , italic_i end_ARG | is a state extension of ρab=∑ipiρabisubscript𝜌𝑎𝑏subscript𝑖subscript𝑝𝑖subscriptsuperscript𝜌𝑖𝑎𝑏\rho_ab=\sum_ip_i\rho^i_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT whenever ρaa′:bb′isubscriptsuperscript𝜌𝑖:𝑎superscript𝑎′𝑏superscript𝑏′\rho^i_aa^\prime:bb^\primeitalic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is a state extension of ρabisubscriptsuperscript𝜌𝑖𝑎𝑏\rho^i_abitalic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT for all i. Without loss of generality, suppose ρaa′:bb′isubscriptsuperscript𝜌𝑖:𝑎superscript𝑎′𝑏superscript𝑏′\rho^i_aa^\prime:bb^\primeitalic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is the optimal state extension of each ρabisubscriptsuperscript𝜌𝑖𝑎𝑏\rho^i_abitalic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT and σi⋆subscriptsuperscript𝜎⋆𝑖\sigma^\star_iitalic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the corresponding closest classical correlated state, then we have



∑ipiℰB(ρabi)=∑ipidB2(ρaa′:bb′i,σi⋆)subscript𝑖subscript𝑝𝑖subscriptℰ𝐵subscriptsuperscript𝜌𝑖𝑎𝑏subscript𝑖subscript𝑝𝑖subscriptsuperscript𝑑2𝐵subscriptsuperscript𝜌𝑖:𝑎superscript𝑎′𝑏superscript𝑏′subscriptsuperscript𝜎⋆𝑖\displaystyle\sum_ip_i\mathcalE_B(\rho^i_ab)=\sum_ip_id^2_B% (\rho^i_aa^\prime:bb^\prime,\sigma^\star_i)∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT , italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )



=\displaystyle== ∑ipidB2(ρaa′:bb′i⊗|i,i⟩a′′b′′⟨i,i|,σi⋆⊗|i,i⟩a′′b′′⟨i,i|)subscript𝑖subscript𝑝𝑖subscriptsuperscript𝑑2𝐵tensor-productsubscriptsuperscript𝜌𝑖:𝑎superscript𝑎′𝑏superscript𝑏′subscriptket𝑖𝑖superscript𝑎′′superscript𝑏′′bra𝑖𝑖tensor-productsubscriptsuperscript𝜎⋆𝑖subscriptket𝑖𝑖superscript𝑎′′superscript𝑏′′bra𝑖𝑖\displaystyle\sum_ip_id^2_B(\rho^i_aa^\prime:bb^\prime\otimes% \keti,i_a^\prime\primeb^\prime\prime\brai,i,\sigma^\star_i% \otimes\keti,i_a^\prime\primeb^\prime\prime\brai,i)∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG italic_i , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i , italic_i end_ARG | , italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ | start_ARG italic_i , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i , italic_i end_ARG | )



≥\displaystyle\geq≥ dB2(∑ipiρaa′:bb′i⊗|i,i⟩a′′b′′⟨i,i|,∑ipiσi⋆⊗|i,i⟩a′′b′′⟨i,i|)subscriptsuperscript𝑑2𝐵subscript𝑖tensor-productsubscript𝑝𝑖subscriptsuperscript𝜌𝑖:𝑎superscript𝑎′𝑏superscript𝑏′subscriptket𝑖𝑖superscript𝑎′′superscript𝑏′′bra𝑖𝑖subscript𝑖tensor-productsubscript𝑝𝑖subscriptsuperscript𝜎⋆𝑖subscriptket𝑖𝑖superscript𝑎′′superscript𝑏′′bra𝑖𝑖\displaystyle d^2_B(\sum_ip_i\rho^i_aa^\prime:bb^\prime\otimes% \keti,i_a^\prime\primeb^\prime\prime\brai,i,\sum_ip_i\sigma^% \star_i\otimes\keti,i_a^\prime\primeb^\prime\prime\brai,i)italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG italic_i , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i , italic_i end_ARG | , ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ | start_ARG italic_i , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i , italic_i end_ARG | )



≥\displaystyle\geq≥ ℰB(∑ipiρaa′:bb′i⊗|i,i⟩a′′b′′⟨i,i|)subscriptℰ𝐵subscript𝑖tensor-productsubscript𝑝𝑖subscriptsuperscript𝜌𝑖:𝑎superscript𝑎′𝑏superscript𝑏′subscriptket𝑖𝑖superscript𝑎′′superscript𝑏′′bra𝑖𝑖\displaystyle\mathcalE_B(\sum_ip_i\rho^i_aa^\prime:bb^\prime% \otimes\keti,i_a^\prime\primeb^\prime\prime\brai,i)caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG italic_i , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i , italic_i end_ARG | )



≥\displaystyle\geq≥ ℰB(∑ipiρabi),subscriptℰ𝐵subscript𝑖subscript𝑝𝑖subscriptsuperscript𝜌𝑖𝑎𝑏\displaystyle\mathcalE_B(\sum_ip_i\rho^i_ab),caligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) , where the first inequality follows from the jointly convexity of dB2subscriptsuperscript𝑑2𝐵d^2_Bitalic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT and the second inequality is based on the definition of ℰBsubscriptℰ𝐵\mathcalE_Bcaligraphic_E start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. The last inequality follows from the fact that ∑ipiρaa′:bb′i⊗|i,i⟩a′′b′′⟨i,i|subscript𝑖tensor-productsubscript𝑝𝑖subscriptsuperscript𝜌𝑖:𝑎superscript𝑎′𝑏superscript𝑏′subscriptket𝑖𝑖superscript𝑎′′superscript𝑏′′bra𝑖𝑖\sum_ip_i\rho^i_aa^\prime:bb^\prime\otimes\keti,i_a^\prime% \primeb^\prime\prime\brai,i∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG italic_i , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i , italic_i end_ARG | is a state extension of ∑ipiρabisubscript𝑖subscript𝑝𝑖subscriptsuperscript𝜌𝑖𝑎𝑏\sum_ip_i\rho^i_ab∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT. ∎



Appendix B Invariance of quantum correlation when adding a pure state ancilla



We prove that



d(ρ⊗|0⟩b2⟨0|,σ⊗|0⟩b2⟨0|)=d(ρ,σ),𝑑tensor-product𝜌subscriptket0subscript𝑏2bra0tensor-product𝜎subscriptket0subscript𝑏2bra0𝑑𝜌𝜎\displaystyle d(\rho\otimes\ket0_b_2\bra0,\sigma\otimes\ket0_b_2% \bra0)=d(\rho,\sigma),italic_d ( italic_ρ ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | , italic_σ ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ) = italic_d ( italic_ρ , italic_σ ) , is true for the Bures distance, Hellinger distance, the relative entropy, trace distance and Hilbert distance.



Note that F(ρ1⊗σ1,ρ2⊗σ2)=F(ρ1,ρ2)F(σ1,σ2)𝐹tensor-productsubscript𝜌1subscript𝜎1tensor-productsubscript𝜌2subscript𝜎2𝐹subscript𝜌1subscript𝜌2𝐹subscript𝜎1subscript𝜎2F(\rho_1\otimes\sigma_1,\rho_2\otimes\sigma_2)=F(\rho_1,\rho_2)F(% \sigma_1,\sigma_2)italic_F ( italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_F ( italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_F ( italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ), then



dB(ρ⊗|0⟩⟨0|,σ⊗|0⟩⟨0|)subscript𝑑𝐵tensor-product𝜌ket0bra0tensor-product𝜎ket0bra0\displaystyle d_B(\rho\otimes\ket0\bra0,\sigma\otimes\ket0\bra0)italic_d start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | , italic_σ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | )



=\displaystyle== 2-2F(ρ⊗|0⟩⟨0|,σ⊗|0⟩⟨0|)22𝐹tensor-product𝜌ket0bra0tensor-product𝜎ket0bra0\displaystyle\sqrt2-2F(\rho\otimes\ket0\bra0,\sigma\otimes\ket0\bra0)square-root start_ARG 2 - 2 italic_F ( italic_ρ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | , italic_σ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | ) end_ARG



=\displaystyle== 2-2F(ρ,σ)=dB(ρ,σ).22𝐹𝜌𝜎subscript𝑑𝐵𝜌𝜎\displaystyle\sqrt2-2F(\rho,\sigma)=d_B(\rho,\sigma).square-root start_ARG 2 - 2 italic_F ( italic_ρ , italic_σ ) end_ARG = italic_d start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ , italic_σ ) . Similarly, we can prove the same result for the Hellinger distances and relative entropy.



Note that trace distance and Hilbert-Schmidt distance are the special case of



dp(ρ,σ):=||ρ-σ||p=(tr|ρ-σ|p)1/p,assignsubscript𝑑𝑝𝜌𝜎subscriptnorm𝜌𝜎𝑝superscript𝑡𝑟superscript𝜌𝜎𝑝1𝑝\displaystyle d_p(\rho,\sigma):=||\rho-\sigma||_p=(tr|\rho-\sigma|^p)^1% /p,italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_ρ , italic_σ ) := | | italic_ρ - italic_σ | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = ( italic_t italic_r | italic_ρ - italic_σ | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 1 / italic_p end_POSTSUPERSCRIPT , where |A|=A†A𝐴superscript𝐴†𝐴|A|=\sqrtA^\daggerA| italic_A | = square-root start_ARG italic_A start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_A end_ARG. By definition, one has that



dp(ρ⊗|0⟩⟨0|,σ⊗|0⟩⟨0|)subscript𝑑𝑝tensor-product𝜌ket0bra0tensor-product𝜎ket0bra0\displaystyle d_p(\rho\otimes\ket0\bra0,\sigma\otimes\ket0\bra0)italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_ρ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | , italic_σ ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | )



=[tr(|ρ-σ|p⊗|0⟩⟨0|)]1/p=dp(ρ,σ).absentsuperscriptdelimited-[]𝑡𝑟tensor-productsuperscript𝜌𝜎𝑝ket0bra01𝑝subscript𝑑𝑝𝜌𝜎\displaystyle=[tr(|\rho-\sigma|^p\otimes\ket0\bra0)]^1/p=d_p(\rho,% \sigma).= [ italic_t italic_r ( | italic_ρ - italic_σ | start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG | ) ] start_POSTSUPERSCRIPT 1 / italic_p end_POSTSUPERSCRIPT = italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_ρ , italic_σ ) . ∎



Appendix C The proof of Theorem 4



The proofs are presented in the same order as they are mentioned in the main text.



Property 1 (Faithfulness). ℰ~D(ρab)≥0subscript~ℰ𝐷subscript𝜌𝑎𝑏0\tilde\mathcalE_D(\rho_ab)\geq 0over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≥ 0, the equality holds if and only if ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT is separable.



First, note that discord is nonnegative over valid quantum states, therefore, ℰ~D(ρab)subscript~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ), being the discord of extended state, must also be nonnegative.



Suppose bipartite state ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT is separable, then it can be written as ρab=∑ipiρai⊗σbisubscript𝜌𝑎𝑏subscript𝑖tensor-productsubscript𝑝𝑖subscriptsuperscript𝜌𝑖𝑎subscriptsuperscript𝜎𝑖𝑏\rho_ab=\sum_ip_i\rho^i_a\otimes\sigma^i_bitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT. Assuming ρai=∑jqi,j|ai,j⟩⟨ai,j|subscriptsuperscript𝜌𝑖𝑎subscript𝑗subscript𝑞𝑖𝑗ketsubscript𝑎𝑖𝑗brasubscript𝑎𝑖𝑗\rho^i_a=\sum_jq_i,j\keta_i,j\braa_i,jitalic_ρ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT | start_ARG italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_a start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT end_ARG | and σbi=∑jri,k|bi,k⟩⟨bi,k|subscriptsuperscript𝜎𝑖𝑏subscript𝑗subscript𝑟𝑖𝑘ketsubscript𝑏𝑖𝑘brasubscript𝑏𝑖𝑘\sigma^i_b=\sum_jr_i,k\ketb_i,k\brab_i,kitalic_σ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i , italic_k end_POSTSUBSCRIPT | start_ARG italic_b start_POSTSUBSCRIPT italic_i , italic_k end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_b start_POSTSUBSCRIPT italic_i , italic_k end_POSTSUBSCRIPT end_ARG |, then, up to a rebelling of variables, the separable state can be written by a convex sum of separable pure states of the form ρab=∑lsl|al⟩⟨al|⊗|bl⟩⟨bl|subscript𝜌𝑎𝑏subscript𝑙tensor-productsubscript𝑠𝑙ketsubscript𝑎𝑙brasubscript𝑎𝑙ketsubscript𝑏𝑙brasubscript𝑏𝑙\rho_ab=\sum_ls_l\keta_l\braa_l\otimes\ketb_l\brab_litalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT | start_ARG italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG | ⊗ | start_ARG italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT end_ARG |. Therefore, ρaa′a1′bb′b1′=∑lsl|al,bl,l⟩aa′a1′⟨al,bl,l|⊗|bl,al,l⟩bb′b1′⟨bl,al,l|subscript𝜌𝑎superscript𝑎′subscriptsuperscript𝑎′1𝑏superscript𝑏′subscriptsuperscript𝑏′1subscript𝑙tensor-productsubscript𝑠𝑙subscriptketsubscript𝑎𝑙subscript𝑏𝑙𝑙𝑎superscript𝑎′subscriptsuperscript𝑎′1brasubscript𝑎𝑙subscript𝑏𝑙𝑙subscriptketsubscript𝑏𝑙subscript𝑎𝑙𝑙𝑏superscript𝑏′subscriptsuperscript𝑏′1brasubscript𝑏𝑙subscript𝑎𝑙𝑙\rho_aa^\primea^\prime_1bb^\primeb^\prime_1=\sum_ls_l\keta_% l,b_l,l_aa^\primea^\prime_1\braa_l,b_l,l\otimes\ketb_l,a% _l,l_bb^\primeb^\prime_1\brab_l,a_l,litalic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT | start_ARG italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_l end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_l end_ARG | ⊗ | start_ARG italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_l end_ARG ⟩ start_POSTSUBSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_b start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT , italic_l end_ARG | is a CSE and ℰ~D(ρab)≤D(ρaa′a1′bb′b1′)=0subscript~ℰ𝐷subscript𝜌𝑎𝑏𝐷subscript𝜌𝑎superscript𝑎′subscriptsuperscript𝑎′1𝑏superscript𝑏′subscriptsuperscript𝑏′10\tilde\mathcalE_D(\rho_ab)\leq D(\rho_aa^\primea^\prime_1bb^% \primeb^\prime_1)=0over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≤ italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = 0.



On the converse, ℰ~D(ρab)=0subscript~ℰ𝐷subscript𝜌𝑎𝑏0\tilde\mathcalE_D(\rho_ab)=0over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = 0 implies that there exist an extension for which D(ρaa′bb′)=0𝐷subscript𝜌𝑎superscript𝑎′𝑏superscript𝑏′0D(\rho_aa^\primebb^\prime)=0italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = 0, i.e., ρaa′bb′=∑iti|αi⟩aa′⟨αi|⊗|βi⟩bb′⟨βi|subscript𝜌𝑎superscript𝑎′𝑏superscript𝑏′subscript𝑖tensor-productsubscript𝑡𝑖subscriptketsubscript𝛼𝑖𝑎superscript𝑎′brasubscript𝛼𝑖subscriptketsubscript𝛽𝑖𝑏superscript𝑏′brasubscript𝛽𝑖\rho_aa^\primebb^\prime=\sum_it_i\ket\alpha_i_aa^\prime\bra% \alpha_i\otimes\ket\beta_i_bb^\prime\bra\beta_iitalic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | ⊗ | start_ARG italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_b italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG |. Tracing out the subsystem a′b′superscript𝑎′superscript𝑏′a^\primeb^\primeitalic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT will lead to the decomposition of the form ρab=∑iti|ai⟩a⟨ai|⊗|bi⟩⟨bi|subscript𝜌𝑎𝑏subscript𝑖tensor-productsubscript𝑡𝑖subscriptketsubscript𝑎𝑖𝑎brasubscript𝑎𝑖ketsubscript𝑏𝑖brasubscript𝑏𝑖\rho_ab=\sum_it_i\keta_i_a\braa_i\otimes\ketb_i\brab_iitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟨ start_ARG italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG | ⊗ | start_ARG italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG |, so ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT is separable.



In conclusion, ℰ~D(ρab)subscript~ℰ𝐷subscript𝜌𝑎𝑏\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is nonnegative and vanishes iff ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT is separable, then it must be strictly positive for entangled states. ∎



Property 2 (Invariance under local unitaries). ℰ~D(⋅)subscript~ℰ𝐷⋅\tilde\mathcalE_D(\cdot)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ⋅ ) is locally unitary invariant on parties a and b in the sense that,



ℰ~D[(Ua⊗Ub)ρab(Ua⊗Ub)]=ℰ~D(ρab).subscript~ℰ𝐷delimited-[]tensor-productsubscript𝑈𝑎subscript𝑈𝑏subscript𝜌𝑎𝑏tensor-productsubscript𝑈𝑎subscript𝑈𝑏subscript~ℰ𝐷subscript𝜌𝑎𝑏\displaystyle\tilde\mathcalE_D[(U_a\otimes U_b)\rho_ab(U_a% \otimes U_b)]=\tilde\mathcalE_D(\rho_ab).over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT [ ( italic_U start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) ] = over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) .



This property follows from the fact that D(ρab)𝐷subscript𝜌𝑎𝑏D(\rho_ab)italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) is invariant under local unitary operations. ∎



Property 3 (Contraction under local partial trace). ℰ~D(⋅)subscript~ℰ𝐷⋅\tilde\mathcalE_D(\cdot)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ⋅ ) is nonincreasing under local partial trace in the sense that



ℰ~D(ρab)≤ℰ~D(ρaa1bb1)subscript~ℰ𝐷subscript𝜌𝑎𝑏subscript~ℰ𝐷subscript𝜌𝑎subscript𝑎1𝑏subscript𝑏1\displaystyle\tilde\mathcalE_D(\rho_ab)\leq\tilde\mathcalE_D(% \rho_aa_1bb_1)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≤ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) for any state extension ρaa1bb1subscript𝜌𝑎subscript𝑎1𝑏subscript𝑏1\rho_aa_1bb_1italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT of ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT.



Since any CSE ρaa1a′a1′a2′bb1b′b1′b2′subscript𝜌𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2\rho_aa_1a^\primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_% 1b^\prime_2italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT of ρaa1bb1subscript𝜌𝑎subscript𝑎1𝑏subscript𝑏1\rho_aa_1bb_1italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is necessarily a state extension of the reduced state ρab=tra1b1ρaa1bb1subscript𝜌𝑎𝑏𝑡subscript𝑟subscript𝑎1subscript𝑏1subscript𝜌𝑎subscript𝑎1𝑏subscript𝑏1\rho_ab=tr_a_1b_1\rho_aa_1bb_1italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = italic_t italic_r start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT which satisfies



ΨSaa1↔b′b1′(Uaa1a′a1′a2′ρaa1a′a1′a2′bb1b′b1′b2′Uaa1a′a1′a2′†)subscriptsuperscriptΨ↔𝑎subscript𝑎1superscript𝑏′subscriptsuperscript𝑏′1𝑆subscript𝑈𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝜌𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2subscriptsuperscript𝑈†𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2\displaystyle\Psi^aa_1\leftrightarrow b^\primeb^\prime_1_S(U_aa_% 1a^\primea^\prime_1a^\prime_2\rho_aa_1a^\primea^\prime_1a% ^\prime_2bb_1b^\primeb^\prime_1b^\prime_2U^\dagger_aa_1a% ^\primea^\prime_1a^\prime_2)roman_Ψ start_POSTSUPERSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )



=\displaystyle== Uaa1a′a1′a2′ρaa1a′a1′a2′bb1b′b1′b2′Uaa1a′a1′a2′†,subscript𝑈𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝜌𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2subscriptsuperscript𝑈†𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2\displaystyle U_aa_1a^\primea^\prime_1a^\prime_2\rho_aa_1a^% \primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_1b^\prime_2% U^\dagger_aa_1a^\primea^\prime_1a^\prime_2,italic_U start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , for some local unitary Uaa1a′a1′a2′subscript𝑈𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2U_aa_1a^\primea^\prime_1a^\prime_2italic_U start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and a similar equality for ΨSa′a1′↔bb1subscriptsuperscriptΨ↔superscript𝑎′subscriptsuperscript𝑎′1𝑏subscript𝑏1𝑆\Psi^a^\primea^\prime_1\leftrightarrow bb_1_Sroman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT, then we have



ℰ~D(ρaa1bb1)=subscript~ℰ𝐷subscript𝜌𝑎subscript𝑎1𝑏subscript𝑏1absent\displaystyle\tilde\mathcalE_D(\rho_aa_1bb_1)=over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = minρaa1bb1,ΨSa′a1′↔bb1D(ρaa1a′a1′a2′bb1b′b1′b2′)subscriptsubscript𝜌𝑎subscript𝑎1𝑏subscript𝑏1subscriptsuperscriptΨ↔superscript𝑎′subscriptsuperscript𝑎′1𝑏subscript𝑏1𝑆𝐷subscript𝜌𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2\displaystyle\min_\\rho_aa_1bb_1,\Psi^a^\primea^\prime_1% \leftrightarrow bb_1_S\D(\rho_aa_1a^\primea^\prime_1a^\prime% _2bb_1b^\primeb^\prime_1b^\prime_2)roman_min start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )



≥\displaystyle\geq≥ minρab,ΨSa′↔bD(ρaa1a′a1′a2′bb1b′b1′b2′)≥ℰ~D(ρab),subscriptsubscript𝜌𝑎𝑏subscriptsuperscriptΨ↔superscript𝑎′𝑏𝑆𝐷subscript𝜌𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2subscript~ℰ𝐷subscript𝜌𝑎𝑏\displaystyle\min_\\rho_ab,\Psi^a^\prime\leftrightarrow b_S\D(\rho% _aa_1a^\primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_1b^% \prime_2)\geq\tilde\mathcalE_D(\rho_ab),roman_min start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT , roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ↔ italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ≥ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) , where the first minimal is taken over all ρaa1a′a1′a2′bb1b′b1′b2′subscript𝜌𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2\rho_aa_1a^\primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_% 1b^\prime_2italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT which is the extension of ρaa1bb1subscript𝜌𝑎subscript𝑎1𝑏subscript𝑏1\rho_aa_1bb_1italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and invariant under swap operation between subsystem aa1(a′a1′)𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1aa_1(a^\primea^\prime_1)italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and b′b1′(bb1)superscript𝑏′subscriptsuperscript𝑏′1𝑏subscript𝑏1b^\primeb^\prime_1(bb_1)italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ), while the second minimum is taken over all ρaa1a′a1′a2′bb1b′b1′b2′subscript𝜌𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2\rho_aa_1a^\primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_% 1b^\prime_2italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT that is the extension of ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT and invariant under swap operation between subsystem a(a′)𝑎superscript𝑎′a(a^\prime)italic_a ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) and b′(b)superscript𝑏′𝑏b^\prime(b)italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_b ). Therefore, the chain of inequalities holds by definition. ∎



Property 4. ℰ~D(⋅)subscript~ℰ𝐷⋅\tilde\mathcalE_D(\cdot)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ⋅ ) is contractive adding a pure state ancilla, i.e.,



ℰ~D(ρa1b1⊗|0⟩b2⟨0|)≤ℰ~D(ρa1b1).subscript~ℰ𝐷tensor-productsubscript𝜌subscript𝑎1subscript𝑏1subscriptket0subscript𝑏2bra0subscript~ℰ𝐷subscript𝜌subscript𝑎1subscript𝑏1\displaystyle\tilde\mathcalE_D(\rho_a_1b_1\otimes\ket0_b_2% \bra0)\leq\tilde\mathcalE_D(\rho_a_1b_1).over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ) ≤ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) . (3)



Note that the CSE of ρa1b1⊗|0⟩b2⟨0|tensor-productsubscript𝜌subscript𝑎1subscript𝑏1subscriptket0subscript𝑏2bra0\rho_a_1b_1\otimes\ket0_b_2\bra0italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | has the form ρa1a1′a2′a3′b1′b1b3′⊗|0⟩b2⟨0|tensor-productsubscript𝜌subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscriptsuperscript𝑎′3subscriptsuperscript𝑏′1subscript𝑏1subscriptsuperscript𝑏′3subscriptket0subscript𝑏2bra0\rho_a_1a^\prime_1a^\prime_2a^\prime_3b^\prime_1b_1b^% \prime_3\otimes\ket0_b_2\bra0italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG |. Therefore,



ℰ~D(ρa1b1)subscript~ℰ𝐷subscript𝜌subscript𝑎1subscript𝑏1\displaystyle\tilde\mathcalE_D(\rho_a_1b_1)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) =minρa1b1,ΨSa1↔b1′D(ρa1a1′a3′b1b1′b3′)absentsubscriptsubscript𝜌subscript𝑎1subscript𝑏1subscriptsuperscriptΨ↔subscript𝑎1subscriptsuperscript𝑏′1𝑆𝐷subscript𝜌subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′3subscript𝑏1subscriptsuperscript𝑏′1subscriptsuperscript𝑏′3\displaystyle=\min_\\rho_a_1b_1,\Psi^a_1\leftrightarrow b^\prime_% 1_S\D(\rho_a_1a^\prime_1a^\prime_3b_1b^\prime_1b^% \prime_3)= roman_min start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )



≥minρa1b1,ΨSa1↔b1′D(ρa1a1′a3′b1b1′b3′⊗|0,0⟩a2′b2⟨0,0|)absentsubscriptsubscript𝜌subscript𝑎1subscript𝑏1subscriptsuperscriptΨ↔subscript𝑎1subscriptsuperscript𝑏′1𝑆𝐷tensor-productsubscript𝜌subscript𝑎1subscriptsuperscript𝑎′1subscriptsuperscript𝑎′3subscript𝑏1subscriptsuperscript𝑏′1subscriptsuperscript𝑏′3subscriptket00subscriptsuperscript𝑎′2subscript𝑏2bra00\displaystyle\geq\min_\\rho_a_1b_1,\Psi^a_1\leftrightarrow b^% \prime_1_S\D(\rho_a_1a^\prime_1a^\prime_3b_1b^\prime_1% b^\prime_3\otimes\ket0,0_a^\prime_2b_2\bra0,0)≥ roman_min start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_D ( italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | )



≥ℰ~D(ρa1b1⊗|0⟩b2⟨0|),absentsubscript~ℰ𝐷tensor-productsubscript𝜌subscript𝑎1subscript𝑏1subscriptket0subscript𝑏2bra0\displaystyle\geq\tilde\mathcalE_D(\rho_a_1b_1\otimes\ket0_b_2% \bra0),≥ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ) , where the minimal is taken over all ρaa1a′a1′a2′bb1b′b1′b2′subscript𝜌𝑎subscript𝑎1superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′1subscriptsuperscript𝑏′2\rho_aa_1a^\primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_% 1b^\prime_2italic_ρ start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT which is the extension of ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT and invariant under swap operation between subsystem a(a′)𝑎superscript𝑎′a(a^\prime)italic_a ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) and b′(b)superscript𝑏′𝑏b^\prime(b)italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_b ). ∎



Property 5 (Contraction under local operations). ℰ~D(⋅)subscript~ℰ𝐷⋅\tilde\mathcalE_D(\cdot)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ⋅ ) is nonincreasing under local operations, i.e.,



ℰ~D(ρab)≥ℰ~D(∑iKaiρabKai†).subscript~ℰ𝐷subscript𝜌𝑎𝑏subscript~ℰ𝐷subscript𝑖subscriptsuperscript𝐾𝑖𝑎subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑎\displaystyle\tilde\mathcalE_D(\rho_ab)\geq\tilde\mathcalE_D(% \sum_iK^i_a\rho_abK^i\dagger_a).over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≥ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) .



Thanks to Stinespring representations Stingspring (1955), the local operation can be realized by adding a pure state ancilla, follows a global unitary operation and tracing out the ancilla system, i.e., ∑iKaiρabKai†=tra1Uaa1(ρab⊗|0⟩a1⟨0|)Uaa1†subscript𝑖subscriptsuperscript𝐾𝑖𝑎subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑎𝑡subscript𝑟subscript𝑎1subscript𝑈𝑎subscript𝑎1tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑎1bra0subscriptsuperscript𝑈†𝑎subscript𝑎1\sum_iK^i_a\rho_abK^i\dagger_a=tr_a_1U_aa_1(\rho_ab% \otimes\ket0_a_1\bra0)U^\dagger_aa_1∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = italic_t italic_r start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ) italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT. Therefore, one has



ℰ~D(ρab)subscript~ℰ𝐷subscript𝜌𝑎𝑏\displaystyle\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≥ℰ~D(ρab⊗|0⟩a1⟨0|)absentsubscript~ℰ𝐷tensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑎1bra0\displaystyle\geq\tilde\mathcalE_D(\rho_ab\otimes\ket0_a_1\bra0)≥ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | )



=ℰ~D(Uaa1ρab⊗|0⟩a1⟨0|Uaa1†)absentsubscript~ℰ𝐷tensor-productsubscript𝑈𝑎subscript𝑎1subscript𝜌𝑎𝑏subscriptket0subscript𝑎1bra0subscriptsuperscript𝑈†𝑎subscript𝑎1\displaystyle=\tilde\mathcalE_D(U_aa_1\rho_ab\otimes\ket0_a_1% \bra0U^\dagger_aa_1)= over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )



≥ℰ~D(∑iKaiρabKai†),absentsubscript~ℰ𝐷subscript𝑖subscriptsuperscript𝐾𝑖𝑎subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑎\displaystyle\geq\tilde\mathcalE_D(\sum_iK^i_a\rho_abK^i\dagger% _a),≥ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) , where the first inequality follows from above property 4. Note that the state in second line is an extension of ∑iKaiρabKai†subscript𝑖subscriptsuperscript𝐾𝑖𝑎subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑎\sum_iK^i_a\rho_abK^i\dagger_a∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT, then property 3 implies the last inequality.



Property 6 (Contraction under classical communications). For a bipartite state ∑ipiρab⊗|i⟩b1⟨i|subscript𝑖tensor-productsubscript𝑝𝑖subscript𝜌𝑎𝑏subscriptket𝑖subscript𝑏1bra𝑖\sum_ip_i\rho_ab\otimes\keti_b_1\brai∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | is the initial state in Bob’s side, and b1subscript𝑏1b_1italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is a classical register storing classical information. Then ℰ~Dsubscript~ℰ𝐷\tilde\mathcalE_Dover~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT is non-increasing in the sense that



ℰ~D(∑ipiρab⊗|i⟩b1⟨i|)≥ℰ~D(∑ipiρab⊗|i⟩Ma⟨i|⊗|i⟩Mb⟨i|),subscript~ℰ𝐷subscript𝑖tensor-productsubscript𝑝𝑖subscript𝜌𝑎𝑏subscriptket𝑖subscript𝑏1bra𝑖subscript~ℰ𝐷subscript𝑖tensor-producttensor-productsubscript𝑝𝑖subscript𝜌𝑎𝑏subscriptket𝑖subscript𝑀𝑎bra𝑖subscriptket𝑖subscript𝑀𝑏bra𝑖\displaystyle\tilde\mathcalE_D(\sum_ip_i\rho_ab\otimes\keti_b_% 1\brai)\geq\tilde\mathcalE_D(\sum_ip_i\rho_ab\otimes\keti_M% _a\brai\otimes\keti_M_b\brai),over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ) ≥ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ) , where ∑ipiρab⊗|i⟩Ma⟨i|⊗|i⟩Mb⟨i|subscript𝑖tensor-producttensor-productsubscript𝑝𝑖subscript𝜌𝑎𝑏subscriptket𝑖subscript𝑀𝑎bra𝑖subscriptket𝑖subscript𝑀𝑏bra𝑖\sum_ip_i\rho_ab\otimes\keti_M_a\brai\otimes\keti_M_b\brai∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | is equivalent to the state after Bob send the classical information stored in the computational basis of the register b1subscript𝑏1b_1italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to Alice.



Tan and Jeong (2018) Assuming that σaa′a1′a2′bb1b′b2′⋆subscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2\sigma^\star_aa^\primea^\prime_1a^\prime_2bb_1b^\primeb^% \prime_2italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT is the optimal CSE of ∑ipiρab⊗|i⟩b1⟨i|subscript𝑖tensor-productsubscript𝑝𝑖subscript𝜌𝑎𝑏subscriptket𝑖subscript𝑏1bra𝑖\sum_ip_i\rho_ab\otimes\keti_b_1\brai∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG |, by definition, there exists a local unitary that Alice can perform such that



Uaa′a1′a2′σaa′a1′a2′bb1b′b2′⋆Uaa′a1′a2′†=ΨSa1′↔b1(Uaa′a1′a2′σaa′a1′a2′bb1b′b2′⋆Uaa′a1′a2′†).subscript𝑈𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2subscriptsuperscript𝑈†𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscriptsuperscriptΨ↔subscriptsuperscript𝑎′1subscript𝑏1𝑆subscript𝑈𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2subscriptsuperscript𝑈†𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2\displaystyle U_aa^\primea^\prime_1a^\prime_2\sigma^\star_aa^% \primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_2U^\dagger_% aa^\primea^\prime_1a^\prime_2=\Psi^a^\prime_1\leftrightarrow b% _1_S(U_aa^\primea^\prime_1a^\prime_2\sigma^\star_aa^% \primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_2U^\dagger_% aa^\primea^\prime_1a^\prime_2).italic_U start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) .



Suppose we add registers Ma,Ma′subscript𝑀𝑎subscriptsuperscript𝑀′𝑎M_a,M^\prime_aitalic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT and Mb,Mb′subscript𝑀𝑏subscriptsuperscript𝑀′𝑏M_b,M^\prime_bitalic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT , italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT, initialized in the state |0,0⟩MaMa′subscriptket00subscript𝑀𝑎subscriptsuperscript𝑀′𝑎\ket0,0_M_aM^\prime_a| start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT and |0,0⟩MbMb′subscriptket00subscript𝑀𝑏subscriptsuperscript𝑀′𝑏\ket0,0_M_bM^\prime_b| start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT, and locally copy the classical information on registers a1′subscriptsuperscript𝑎′1a^\prime_1italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and b1subscript𝑏1b_1italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT via CNOT operations 𝒰Ca1′Masubscriptsuperscript𝒰subscriptsuperscript𝑎′1subscript𝑀𝑎𝐶\mathcalU^a^\prime_1M_a_Ccaligraphic_U start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, 𝒰Cb1Mbsubscriptsuperscript𝒰subscript𝑏1subscript𝑀𝑏𝐶\mathcalU^b_1M_b_Ccaligraphic_U start_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, with 𝒰Cab(⋅)=UCab(⋅)UCab†subscriptsuperscript𝒰𝑎𝑏𝐶⋅subscriptsuperscript𝑈𝑎𝑏𝐶⋅subscriptsuperscript𝑈𝑎𝑏†𝐶\mathcalU^ab_C(\cdot)=U^ab_C(\cdot)U^ab\dagger_Ccaligraphic_U start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( ⋅ ) = italic_U start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( ⋅ ) italic_U start_POSTSUPERSCRIPT italic_a italic_b † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT and UCab=∑ij|i,i+j(moddℋb)⟩⟨i,j|subscriptsuperscript𝑈𝑎𝑏𝐶subscript𝑖𝑗ket𝑖𝑖𝑗modsubscript𝑑subscriptℋ𝑏bra𝑖𝑗U^ab_C=\sum_ij\keti,i+j(\mathrmmod~d_\mathcalH_b)\brai,jitalic_U start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT | start_ARG italic_i , italic_i + italic_j ( roman_mod italic_d start_POSTSUBSCRIPT caligraphic_H start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_ARG ⟩ ⟨ start_ARG italic_i , italic_j end_ARG |. It can be verified that 𝒰Cabsubscriptsuperscript𝒰𝑎𝑏𝐶\mathcalU^ab_Ccaligraphic_U start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT is unitary and satisfies UCab|i,0⟩ab=|i,i⟩absubscriptsuperscript𝑈𝑎𝑏𝐶subscriptket𝑖0𝑎𝑏subscriptket𝑖𝑖𝑎𝑏U^ab_C\keti,0_ab=\keti,i_abitalic_U start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT | start_ARG italic_i , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = | start_ARG italic_i , italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT.



Assume that this unitary is already performed and included in the definition of σaa′a1′a2′bb1b′b2′⋆subscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2\sigma^\star_aa^\primea^\prime_1a^\prime_2bb_1b^\primeb^% \prime_2italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, then the invariance under SWAP operations implies that



𝒰Ca1′Ma∘𝒰Cb1Mb(|0,0⟩MaMa′⟨0,0|⊗σaa′a1′a2′bb1b′b2′⋆⊗|0,0⟩MbMb′⟨0,0|)subscriptsuperscript𝒰subscriptsuperscript𝑎′1subscript𝑀𝑎𝐶subscriptsuperscript𝒰subscript𝑏1subscript𝑀𝑏𝐶tensor-productsubscriptket00subscript𝑀𝑎subscriptsuperscript𝑀′𝑎bra00subscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2subscriptket00subscript𝑀𝑏subscriptsuperscript𝑀′𝑏bra00\displaystyle\mathcalU^a^\prime_1M_a_C\circ\mathcalU^b_1M_b% _C(\ket0,0_M_aM^\prime_a\bra0,0\otimes\sigma^\star_aa^% \primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_2\otimes\ket% 0,0_M_bM^\prime_b\bra0,0)caligraphic_U start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ∘ caligraphic_U start_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ⊗ italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ) (4)



=\displaystyle== ΨSa1′↔b1∘ΨSa1′↔b1∘𝒰Ca1′Ma∘𝒰Cb1Mb(|0,0⟩MaMa′⟨0,0|⊗ΨSa1′↔b1(σaa′a1′a2′bb1b′b2′⋆)⊗|0,0⟩MbMb′⟨0,0|)subscriptsuperscriptΨ↔subscriptsuperscript𝑎′1subscript𝑏1𝑆subscriptsuperscriptΨ↔subscriptsuperscript𝑎′1subscript𝑏1𝑆subscriptsuperscript𝒰subscriptsuperscript𝑎′1subscript𝑀𝑎𝐶subscriptsuperscript𝒰subscript𝑏1subscript𝑀𝑏𝐶tensor-producttensor-productsubscriptket00subscript𝑀𝑎subscriptsuperscript𝑀′𝑎bra00subscriptsuperscriptΨ↔subscriptsuperscript𝑎′1subscript𝑏1𝑆subscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2subscriptket00subscript𝑀𝑏subscriptsuperscript𝑀′𝑏bra00\displaystyle\Psi^a^\prime_1\leftrightarrow b_1_S\circ\Psi^a^% \prime_1\leftrightarrow b_1_S\circ\mathcalU^a^\prime_1M_a_C% \circ\mathcalU^b_1M_b_C(\ket0,0_M_aM^\prime_a\bra0,0% \otimes\Psi^a^\prime_1\leftrightarrow b_1_S(\sigma^\star_aa^% \primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_2)\otimes\ket% 0,0_M_bM^\prime_b\bra0,0)roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ∘ roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ∘ caligraphic_U start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ∘ caligraphic_U start_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ⊗ roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ⊗ | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ) (5)



=\displaystyle== ΨSa1′↔b1∘𝒰Cb1Ma∘𝒰Ca1′Mb(|0,0⟩MaMa′⟨0,0|⊗σaa′a1′a2′bb1b′b2′⋆⊗|0,0⟩MbMb′⟨0,0|),subscriptsuperscriptΨ↔subscriptsuperscript𝑎′1subscript𝑏1𝑆subscriptsuperscript𝒰subscript𝑏1subscript𝑀𝑎𝐶subscriptsuperscript𝒰subscriptsuperscript𝑎′1subscript𝑀𝑏𝐶tensor-productsubscriptket00subscript𝑀𝑎subscriptsuperscript𝑀′𝑎bra00subscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2subscriptket00subscript𝑀𝑏subscriptsuperscript𝑀′𝑏bra00\displaystyle\Psi^a^\prime_1\leftrightarrow b_1_S\circ\mathcalU^b% _1M_a_C\circ\mathcalU^a^\prime_1M_b_C(\ket0,0_M_aM^% \prime_a\bra0,0\otimes\sigma^\star_aa^\primea^\prime_1a^\prime% _2bb_1b^\primeb^\prime_2\otimes\ket0,0_M_bM^\prime_b\bra% 0,0),roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ∘ caligraphic_U start_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ∘ caligraphic_U start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ⊗ italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ) , (6) where 𝒰Cab(ρab)=UCabρabUCab†subscriptsuperscript𝒰𝑎𝑏𝐶subscript𝜌𝑎𝑏subscriptsuperscript𝑈𝑎𝑏𝐶subscript𝜌𝑎𝑏subscriptsuperscript𝑈𝑎𝑏†𝐶\mathcalU^ab_C(\rho_ab)=U^ab_C\rho_abU^ab\dagger_Ccaligraphic_U start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = italic_U start_POSTSUPERSCRIPT italic_a italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT italic_a italic_b † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT. The last equality follows from the fact that ΨSa↔b(ρab)=USa↔bρabUSa↔b†subscriptsuperscriptΨ↔𝑎𝑏𝑆subscript𝜌𝑎𝑏subscriptsuperscript𝑈↔𝑎𝑏𝑆subscript𝜌𝑎𝑏subscriptsuperscript𝑈↔𝑎𝑏†𝑆\Psi^a\leftrightarrow b_S(\rho_ab)=U^a\leftrightarrow b_S\rho_abU^% a\leftrightarrow b\dagger_Sroman_Ψ start_POSTSUPERSCRIPT italic_a ↔ italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = italic_U start_POSTSUPERSCRIPT italic_a ↔ italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT italic_a ↔ italic_b † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT, USa↔b=USa↔b†subscriptsuperscript𝑈↔𝑎𝑏𝑆subscriptsuperscript𝑈↔𝑎𝑏†𝑆U^a\leftrightarrow b_S=U^a\leftrightarrow b\dagger_Sitalic_U start_POSTSUPERSCRIPT italic_a ↔ italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = italic_U start_POSTSUPERSCRIPT italic_a ↔ italic_b † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT and USa↔bUCbcUSa↔b†=UCacsubscriptsuperscript𝑈↔𝑎𝑏𝑆subscriptsuperscript𝑈𝑏𝑐𝐶subscriptsuperscript𝑈↔𝑎𝑏†𝑆subscriptsuperscript𝑈𝑎𝑐𝐶U^a\leftrightarrow b_SU^bc_CU^a\leftrightarrow b\dagger_S=U^ac_Citalic_U start_POSTSUPERSCRIPT italic_a ↔ italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT italic_b italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT italic_U start_POSTSUPERSCRIPT italic_a ↔ italic_b † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = italic_U start_POSTSUPERSCRIPT italic_a italic_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT.



In fact, it can be verified that Eq.(C2) is a CSE of ∑i|i⟩Ma⟨i|⊗KbiρabKbi†⊗|i⟩Mb⟨i|subscript𝑖tensor-producttensor-productsubscriptket𝑖subscript𝑀𝑎bra𝑖subscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏subscriptket𝑖subscript𝑀𝑏bra𝑖\sum_i\keti_M_a\brai\otimes K^i_b\rho_abK^i\dagger_b% \otimes\keti_M_b\brai∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG |, which is equivalent to the state after Bob send the classical information to Alice. To see this, let us perform the following partial trace on Eq.(C4):



tra′a1′a2′b1b′b2′(ΨSa1′↔b1∘𝒰Cb1Ma∘𝒰Ca1′Mb(|0,0⟩MaMa′⟨0,0|⊗σaa′a1′a2′bb1b′b2′⋆⊗|0,0⟩MbMb′⟨0,0|)fragmentstsubscript𝑟superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2fragments(subscriptsuperscriptΨ↔subscriptsuperscript𝑎′1subscript𝑏1𝑆subscriptsuperscript𝒰subscript𝑏1subscript𝑀𝑎𝐶subscriptsuperscript𝒰subscriptsuperscript𝑎′1subscript𝑀𝑏𝐶fragments(subscriptket00subscript𝑀𝑎subscriptsuperscript𝑀′𝑎bra00tensor-productsubscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2tensor-productsubscriptket00subscript𝑀𝑏subscriptsuperscript𝑀′𝑏bra00)\displaystyle tr_a^\primea^\prime_1a^\prime_2b_1b^\primeb^% \prime_2(\Psi^a^\prime_1\leftrightarrow b_1_S\circ\mathcalU^b% _1M_a_C\circ\mathcalU^a^\prime_1M_b_C(\ket0,0_M_aM^% \prime_a\bra0,0\otimes\sigma^\star_aa^\primea^\prime_1a^\prime% _2bb_1b^\primeb^\prime_2\otimes\ket0,0_M_bM^\prime_b\bra% 0,0)italic_t italic_r start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_Ψ start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ∘ caligraphic_U start_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ∘ caligraphic_U start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ⊗ italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ) (7)



=\displaystyle== tra′a1′a2′b1b′b2′(𝒰Cb1Ma∘𝒰Ca1′Mb(|0,0⟩MaMa′⟨0,0|⊗σaa′a1′a2′bb1b′b2′⋆⊗|0,0⟩MbMb′⟨0,0|),fragmentstsubscript𝑟superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2fragments(subscriptsuperscript𝒰subscript𝑏1subscript𝑀𝑎𝐶subscriptsuperscript𝒰subscriptsuperscript𝑎′1subscript𝑀𝑏𝐶fragments(subscriptket00subscript𝑀𝑎subscriptsuperscript𝑀′𝑎bra00tensor-productsubscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2tensor-productsubscriptket00subscript𝑀𝑏subscriptsuperscript𝑀′𝑏bra00),\displaystyle tr_a^\primea^\prime_1a^\prime_2b_1b^\primeb^% \prime_2(\mathcalU^b_1M_a_C\circ\mathcalU^a^\prime_1M_b% _C(\ket0,0_M_aM^\prime_a\bra0,0\otimes\sigma^\star_aa^% \primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_2\otimes\ket% 0,0_M_bM^\prime_b\bra0,0),italic_t italic_r start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( caligraphic_U start_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ∘ caligraphic_U start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ⊗ italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ) , (8) where the equality follows from the fact that the SWAP does not affect the partial trace, i.e., trab∘ΨSa↔b=trab𝑡subscript𝑟𝑎𝑏subscriptsuperscriptΨ↔𝑎𝑏𝑆𝑡subscript𝑟𝑎𝑏tr_ab\circ\Psi^a\leftrightarrow b_S=tr_abitalic_t italic_r start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ∘ roman_Ψ start_POSTSUPERSCRIPT italic_a ↔ italic_b end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT = italic_t italic_r start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT. In Eq.(C6), note that the register Masubscript𝑀𝑎M_aitalic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT contains a copy of the information in register b1subscript𝑏1b_1italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT in the computational basis, even though Alice just performed a local operation in Eq.(C2). In Eq.(C2), we also see that register Mbsubscript𝑀𝑏M_bitalic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT just contains a copy of the classical information in register b1subscript𝑏1b_1italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Therefore, this implies that Eq.(C2) is in fact a CSE of the state ∑i|i⟩Ma⟨i|⊗KbiρabKbi†⊗|i⟩Mb⟨i|subscript𝑖tensor-producttensor-productsubscriptket𝑖subscript𝑀𝑎bra𝑖subscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏subscriptket𝑖subscript𝑀𝑏bra𝑖\sum_i\keti_M_a\brai\otimes K^i_b\rho_abK^i\dagger_b% \otimes\keti_M_b\brai∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG |.



In conclusion,



ℰ~D(∑iKbiρabKbi†⊗|i⟩b1⟨i|)=D(σaa′a1′a2′bb1b′b2′⋆)≥D(|0,0⟩MaMa′⟨0,0|⊗σaa′a1′a2′bb1b′b2′⋆⊗|0,0⟩MbMb′⟨0,0|)subscript~ℰ𝐷subscript𝑖tensor-productsubscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏subscriptket𝑖subscript𝑏1bra𝑖𝐷subscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2𝐷tensor-productsubscriptket00subscript𝑀𝑎subscriptsuperscript𝑀′𝑎bra00subscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2subscriptket00subscript𝑀𝑏subscriptsuperscript𝑀′𝑏bra00\displaystyle\tilde\mathcalE_D(\sum_iK^i_b\rho_abK^i\dagger_b% \otimes\keti_b_1\brai)=D(\sigma^\star_aa^\primea^\prime_1a^% \prime_2bb_1b^\primeb^\prime_2)\geq D(\ket0,0_M_aM^\prime_% a\bra0,0\otimes\sigma^\star_aa^\primea^\prime_1a^\prime_2bb_% 1b^\primeb^\prime_2\otimes\ket0,0_M_bM^\prime_b\bra0,0)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ) = italic_D ( italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ≥ italic_D ( | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ⊗ italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | )



=\displaystyle== D[𝒰Ca1′Ma∘𝒰Cb1Mb(|0,0⟩MaMa′⟨0,0|⊗σaa′a1′a2′bb1b′b2′⋆⊗|0,0⟩MbMb′⟨0,0|)]≥ℰ~D(∑i|i⟩Ma⟨i|⊗KbiρabKbi†⊗|i⟩Mb⟨i|),𝐷delimited-[]subscriptsuperscript𝒰subscriptsuperscript𝑎′1subscript𝑀𝑎𝐶subscriptsuperscript𝒰subscript𝑏1subscript𝑀𝑏𝐶tensor-productsubscriptket00subscript𝑀𝑎subscriptsuperscript𝑀′𝑎bra00subscriptsuperscript𝜎⋆𝑎superscript𝑎′subscriptsuperscript𝑎′1subscriptsuperscript𝑎′2𝑏subscript𝑏1superscript𝑏′subscriptsuperscript𝑏′2subscriptket00subscript𝑀𝑏subscriptsuperscript𝑀′𝑏bra00subscript~ℰ𝐷subscript𝑖tensor-producttensor-productsubscriptket𝑖subscript𝑀𝑎bra𝑖subscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏subscriptket𝑖subscript𝑀𝑏bra𝑖\displaystyle D[\mathcalU^a^\prime_1M_a_C\circ\mathcalU^b_1M_% b_C(\ket0,0_M_aM^\prime_a\bra0,0\otimes\sigma^\star_aa^% \primea^\prime_1a^\prime_2bb_1b^\primeb^\prime_2\otimes\ket% 0,0_M_bM^\prime_b\bra0,0)]\geq\tilde\mathcalE_D(\sum_i\ket% i_M_a\brai\otimes K^i_b\rho_abK^i\dagger_b\otimes\keti_M_% b\brai),italic_D [ caligraphic_U start_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ∘ caligraphic_U start_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ⊗ italic_σ start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ | start_ARG 0 , 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 , 0 end_ARG | ) ] ≥ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ) , where the chain of inequalities follows from the definition, properties 2, 3, 4, and the last inequality follows from the fact that Eq.(C2) is a CSE of the last state.



Remark 3.



The proof is similar with Supplemental Material in Tan and Jeong (2018), where they constructed a set of entanglement quantifications with coherence measures in the sense of “symmetric extensions”, and proved their contractions under classical communications.



Property 7 (Contraction under LOCC). ℰ~D(⋅)subscript~ℰ𝐷⋅\tilde\mathcalE_D(\cdot)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ⋅ ) is nonincreasing under LOCC operations.



In fact, any LOCC operation consists of a series of process, i.e, a local quantum operation, classical communication, and followed by another local operations. We will prove that ℰ~Dsubscript~ℰ𝐷\tilde\mathcalE_Dover~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT is contractive under each step.



Firstly, suppose that Bob, represented by the subsystem b, will communicate classical information to Alice, represented by subsystem a. By Naimark’s theorem, Bob’s local quantum operation can be realized by adding ancillary subsystems b1b2subscript𝑏1subscript𝑏2b_1b_2italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in an initial pure state |0⟩b1⟨0|⊗|0⟩b2⟨0|tensor-productsubscriptket0subscript𝑏1bra0subscriptket0subscript𝑏2bra0\ket0_b_1\bra0\otimes\ket0_b_2\bra0| start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG |, followed by an unitary operation on bb1b2𝑏subscript𝑏1subscript𝑏2bb_1b_2italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Let subsystem b1subscript𝑏1b_1italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT contains all the classical information after the unitary performed and b2subscript𝑏2b_2italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is traced out.



Based on above discussion, one has



ℰ~D(ρab)subscript~ℰ𝐷subscript𝜌𝑎𝑏\displaystyle\tilde\mathcalE_D(\rho_ab)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≥ℰ~D(ρab⊗|0⟩b1⟨0|⊗|0⟩b2⟨0|)absentsubscript~ℰ𝐷tensor-producttensor-productsubscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0subscriptket0subscript𝑏2bra0\displaystyle\geq\tilde\mathcalE_D(\rho_ab\otimes\ket0_b_1\bra0% \otimes\ket0_b_2\bra0)≥ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | )



=\displaystyle== ℰ~D(Ubb1b2ρab⊗|0⟩b1⟨0|⊗|0⟩b2⟨0|Ubb1b2†)subscript~ℰ𝐷tensor-producttensor-productsubscript𝑈𝑏subscript𝑏1subscript𝑏2subscript𝜌𝑎𝑏subscriptket0subscript𝑏1bra0subscriptket0subscript𝑏2bra0subscriptsuperscript𝑈†𝑏subscript𝑏1subscript𝑏2\displaystyle\tilde\mathcalE_D(U_bb_1b_2\rho_ab\otimes\ket0_b% _1\bra0\otimes\ket0_b_2\bra0U^\dagger_bb_1b_2)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_U start_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ⊗ | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )



≥\displaystyle\geq≥ ℰ~D(∑iKbiρabKbi†⊗|i⟩b1⟨i|),subscript~ℰ𝐷subscript𝑖tensor-productsubscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏subscriptket𝑖subscript𝑏1bra𝑖\displaystyle\tilde\mathcalE_D(\sum_iK^i_b\rho_abK^i\dagger_b% \otimes\keti_b_1\brai),over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ) , where the chain of inequalities follows from Property 2,3,4, and the observation that the state in the penultimate parenthesis is an extension of the last one.



Secondly, property 6 guarantees that classical communication does not increase ℰ~Dsubscript~ℰ𝐷\tilde\mathcalE_Dover~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT, that is,



ℰ~D(∑iKbiρabKbi†⊗|i⟩b1⟨i|)subscript~ℰ𝐷subscript𝑖tensor-productsubscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏subscriptket𝑖subscript𝑏1bra𝑖\displaystyle\tilde\mathcalE_D(\sum_iK^i_b\rho_abK^i\dagger_b% \otimes\keti_b_1\brai)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | )



≥\displaystyle\geq≥ ℰ~D(∑i|i⟩a1⟨i|⊗KbiρabKbi†⊗|i⟩b1⟨i|),subscript~ℰ𝐷subscript𝑖tensor-producttensor-productsubscriptket𝑖subscript𝑎1bra𝑖subscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏subscriptket𝑖subscript𝑏1bra𝑖\displaystyle\tilde\mathcalE_D(\sum_i\keti_a_1\brai\otimes K^% i_b\rho_abK^i\dagger_b\otimes\keti_b_1\brai),over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ) , where registers Masubscript𝑀𝑎M_aitalic_M start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT and Mbsubscript𝑀𝑏M_bitalic_M start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT store classical information in party a and b.



In the end, similar to the first step, the local operation that Alice performed based on the classical information received from Bob also will not increase ℰ~Dsubscript~ℰ𝐷\tilde\mathcalE_Dover~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT. In fact, her local operation can be realized by adding ancillary subsystems a2subscript𝑎2a_2italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in some initial pure state |0⟩⟨0|ket0bra0\ket0\bra0| start_ARG 0 end_ARG ⟩ ⟨ start_ARG 0 end_ARG |, followed by an unitary operation on aa1𝑎subscript𝑎1aa_1italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and subsystem a1subscript𝑎1a_1italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT traced out at last.



In conclusion, one has that



ℰ~D(ρab)≥ℰ~D(∑iKbiρabKbi†⊗|i⟩b1⟨i|)subscript~ℰ𝐷subscript𝜌𝑎𝑏subscript~ℰ𝐷subscript𝑖tensor-productsubscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏subscriptket𝑖subscript𝑏1bra𝑖\displaystyle\tilde\mathcalE_D(\rho_ab)\geq\tilde\mathcalE_D(% \sum_iK^i_b\rho_abK^i\dagger_b\otimes\keti_b_1\brai)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) ≥ over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | )



≥\displaystyle\geq≥ ℰ~D(∑i|i⟩a1⟨i|⊗KbiρabKbi†⊗|i⟩b1⟨i|)subscript~ℰ𝐷subscript𝑖tensor-producttensor-productsubscriptket𝑖subscript𝑎1bra𝑖subscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏subscriptket𝑖subscript𝑏1bra𝑖\displaystyle\tilde\mathcalE_D(\sum_i\keti_a_1\brai\otimes K^% i_b\rho_abK^i\dagger_b\otimes\keti_b_1\brai)over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | )



=\displaystyle== ℰ~D[Uaa1a2(|0⟩a2⟨0|⊗fragmentssubscript~ℰ𝐷fragments[subscript𝑈𝑎subscript𝑎1subscript𝑎2fragments(subscriptket0subscript𝑎2bra0tensor-product\displaystyle\tilde\mathcalE_D[U_aa_1a_2(\ket0_a_2\bra0\otimesover~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT [ italic_U start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG 0 end_ARG | ⊗



∑i|i⟩a1⟨i|⊗KbiρabKbi†⊗|i⟩b1⟨i|)Uaa1a2†]fragmentsfragmentssubscript𝑖subscriptket𝑖subscript𝑎1bra𝑖tensor-productsubscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏tensor-productsubscriptket𝑖subscript𝑏1bra𝑖)subscriptsuperscript𝑈†𝑎subscript𝑎1subscript𝑎2]\displaystyle\sum_i\keti_a_1\brai\otimes K^i_b\rho_abK^i% \dagger_b\otimes\keti_b_1\brai)U^\dagger_aa_1a_2]∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ) italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ]



≥\displaystyle\geq≥ ℰ~D(∑i,j|i⟩a1⟨i|⊗Kai,jKbiρabKbi†Kai,j†⊗|i⟩b1⟨i|).subscript~ℰ𝐷subscript𝑖𝑗tensor-producttensor-productsubscriptket𝑖subscript𝑎1bra𝑖subscriptsuperscript𝐾𝑖𝑗𝑎subscriptsuperscript𝐾𝑖𝑏subscript𝜌𝑎𝑏subscriptsuperscript𝐾𝑖†𝑏subscriptsuperscript𝐾𝑖𝑗†𝑎subscriptket𝑖subscript𝑏1bra𝑖\displaystyle\tilde\mathcalE_D(\sum_i,j\keti_a_1\brai\otimes K% ^i,j_aK^i_b\rho_abK^i\dagger_bK^i,j\dagger_a\otimes\keti_% b_1\brai).over~ start_ARG caligraphic_E end_ARG start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ⊗ italic_K start_POSTSUPERSCRIPT italic_i , italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_K start_POSTSUPERSCRIPT italic_i , italic_j † end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⊗ | start_ARG italic_i end_ARG ⟩ start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⟨ start_ARG italic_i end_ARG | ) . The last line implies that when Alice performs an operation conditioned on the classical received from Bob, the measure also does not increase, which completes the proof.