Comparison Of Quantum Discord And Fully Entangled Fraction Of Two Classes Of Dd2 States

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The quantumness of a generic state is the resource of many applications in quantum information theory and it is interesting to survey the measures which are able to detect its trace in the properties of the state. In this work we study the quantum discord and fully entangled fraction of two classes of bipartite states and compare their behaviors. These classes are complements to the d⊗dtensor-product𝑑𝑑d\otimes ditalic_d ⊗ italic_d Werner and isotropic states, in the sense that each class possesses the same purification as the corresponding complemental class of states. Our results show that maximally entangled mixed states are also maximally discordant states, leading to a generalization of the well-known fact that all maximally entangled pure states have also maximum quantum discord. Moreover, it is shown that the separability-entanglement boundary of a Werner or isotropic state is manifested as an inflection point in the diagram of quantum discord of the corresponding complemental state. Gaming News



The exploitation of quantum states would result in the stronger ability in the processing of information with respect to the case of using classical systems Nielsen and Chuang (2000). The origin of the speed-up in quantum information processing is the quantum correlation of the states which itself originates from the strange properties of quantum mechanics. One of these properties is the superposition principle and can be emanated in the entanglement Horodecki et al. (2009). Entanglement plays a significant role in the theory of quantum information and quantum computation and was believed to be the only proper nominate for the quantum correlation, because of the efficiency of quantum algorithms in analogy to the classical ones, when they are applied alongside with entangled states. However, entanglement is not able to represent all of the quantumness of correlations, in the sense that there are some disentangled states which can be used in efficient quantum algorithms Knill and Laflamme (1998); Datta et al. (2005); Datta and Vidal (2007). Therefore, another strange property of quantum systems, i.e. the measurement, is considered which leads to a measure of quantum correlation called quantum discord Ollivier and Zurek (2001). Aside the advantages and disadvantages of any of these two concepts, considering both of them will give us stronger vision about the quantumness of system.



Because of the importance of the maximally entangled states in quantum information tasks, it may be interesting to study a measure which is able to determine the degree that the state is close to a maximally entangled state. Fully entangled fraction, as a measure for the degree of being close to the maximal entangled states, is defined for a state ρ𝜌\rhoitalic_ρ acting on d⊗dtensor-product𝑑𝑑d\otimes ditalic_d ⊗ italic_d Hilbert space as follows Bennett et al. (1996)



ℱ(ρ)=maxU,V⟨ψmax|(U⊗V)ρ(U†⊗V†)|ψmax⟩=maxV⟨ψmax|(𝕀⊗V)ρ(𝕀⊗V†)|ψmax⟩,ℱ𝜌subscript𝑈𝑉conditionalsuperscript𝜓tensor-product𝑈𝑉𝜌tensor-productsuperscript𝑈†superscript𝑉†superscript𝜓subscript𝑉conditionalsuperscript𝜓tensor-product𝕀𝑉𝜌tensor-product𝕀superscript𝑉†superscript𝜓\mathcalF(\rho)=\max_U,V\langle\psi^\max|(U\otimes V)\rho(U^\dagger% \otimes V^\dagger)|\psi^\max\rangle=\max_V\langle\psi^\max|(\mathbbI% \otimes V)\rho(\mathbbI\otimes V^\dagger)|\psi^\max\rangle,caligraphic_F ( italic_ρ ) = roman_max start_POSTSUBSCRIPT italic_U , italic_V end_POSTSUBSCRIPT ⟨ italic_ψ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT | ( italic_U ⊗ italic_V ) italic_ρ ( italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ⊗ italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) | italic_ψ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ⟩ = roman_max start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ⟨ italic_ψ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT | ( blackboard_I ⊗ italic_V ) italic_ρ ( blackboard_I ⊗ italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) | italic_ψ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ⟩ , (1) where |ψmax⟩=1d∑j=1d|jj⟩ketsuperscript𝜓1𝑑superscriptsubscript𝑗1𝑑ket𝑗𝑗|\psi^\max\rangle=\frac1\sqrtd\sum_j=1^d|jj\rangle| italic_ψ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_d end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | italic_j italic_j ⟩ is the maximally entangled pure state of the d⊗dtensor-product𝑑𝑑d\otimes ditalic_d ⊗ italic_d system, U𝑈Uitalic_U and V𝑉Vitalic_V are unitary operators acting on the marginal subsystems and 𝕀𝕀\mathbbIblackboard_I denotes the identity operator acting on the first subsystem. Although all of the maximally entangled states of the d⊗dtensor-product𝑑𝑑d\otimes ditalic_d ⊗ italic_d bipartite systems are pure, for a general d⊗d′tensor-product𝑑superscript𝑑′d\otimes d^\primeitalic_d ⊗ italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT system there are mixed states which are maximally entangled. Let ρ𝜌\rhoitalic_ρ be an arbitrary state, acting on d⊗d′tensor-product𝑑superscript𝑑′d\otimes d^\primeitalic_d ⊗ italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT Hilbert space with d≤d′𝑑superscript𝑑′d\leq d^\primeitalic_d ≤ italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and define K𝐾Kitalic_K via d′=Kd+rsuperscript𝑑′𝐾𝑑𝑟d^\prime=Kd+ritalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_K italic_d + italic_r where 0≤r<d0𝑟𝑑0\leq r0 ≤ italic_r <italic_d and k≥1𝐾1k\geq 1italic_k ≥ 1. a generalization of eq. (1) is then proposed as the maximum overlap state ρ𝜌\rhoitalic_ρ with k𝐾kitalic_k maximally entangled pure states follows zhao (2015)< p>















</d0𝑟𝑑0\leq>



ℱ(ρ)=maxU,V∑i=1K⟨ψimax|(U⊗V)ρ(U†⊗V†)|ψimax⟩=maxV∑i=1K⟨ψimax|(𝕀⊗V)ρ(𝕀⊗V†)|ψimax⟩,ℱ𝜌subscript𝑈𝑉superscriptsubscript𝑖1𝐾quantum-operator-productsubscriptsuperscript𝜓𝑖tensor-product𝑈𝑉𝜌tensor-productsuperscript𝑈†superscript𝑉†subscriptsuperscript𝜓𝑖subscript𝑉superscriptsubscript𝑖1𝐾quantum-operator-productsubscriptsuperscript𝜓𝑖tensor-product𝕀𝑉𝜌tensor-product𝕀superscript𝑉†subscriptsuperscript𝜓𝑖\mathcalF(\rho)=\max_U,V\sum_i=1^K\langle\psi^\max_i|(U\otimes V)% \rho(U^\dagger\otimes V^\dagger)|\psi^\max_i\rangle=\max_V\sum_i=1% ^K\langle\psi^\max_i|(\mathbbI\otimes V)\rho(\mathbbI\otimes V^% \dagger)|\psi^\max_i\rangle,caligraphic_F ( italic_ρ ) = roman_max start_POSTSUBSCRIPT italic_U , italic_V end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT ⟨ italic_ψ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ( italic_U ⊗ italic_V ) italic_ρ ( italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ⊗ italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) | italic_ψ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ = roman_max start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT ⟨ italic_ψ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ( blackboard_I ⊗ italic_V ) italic_ρ ( blackboard_I ⊗ italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) | italic_ψ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ , (2) where, in the last term, maximum is taken over all unitary operators acting on the second subsystem. Moreover, ψimax⟩i=1Ksuperscriptsubscriptketsubscriptsuperscript𝜓𝑖𝑖1𝐾\\psi^\max_i\rangle\_i=1^K start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT is the set of maximally entangled pure states of the d⊗d′tensor-product𝑑superscript𝑑′d\otimes d^\primeitalic_d ⊗ italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT system, defined by



|ψimax⟩=1d∑j=1d|j⟩⊗|j+(i-1)d⟩.ketsubscriptsuperscript𝜓𝑖1𝑑superscriptsubscript𝑗1𝑑tensor-productket𝑗ket𝑗𝑖1𝑑|\psi^\max_i\rangle=\frac1\sqrtd\sum_j=1^d|j\rangle\otimes|j+(i-% 1)d\rangle.| italic_ψ start_POSTSUPERSCRIPT roman_max end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_d end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | italic_j ⟩ ⊗ | italic_j + ( italic_i - 1 ) italic_d ⟩ . (3) It is shown that Kdd′≤ℱ(ρ)≤1𝐾𝑑superscript𝑑′ℱ𝜌1\fracKdd^\prime\leq\mathcalF(\rho)\leq 1divide start_ARG italic_K end_ARG start_ARG italic_d italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ≤ caligraphic_F ( italic_ρ ) ≤ 1 for all d⊗d′tensor-product𝑑superscript𝑑′d\otimes d^\primeitalic_d ⊗ italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT quantum states, while Kdd′≤ℱ(ρ)≤1d𝐾𝑑superscript𝑑′ℱ𝜌1𝑑\fracKdd^\prime\leq\mathcalF(\rho)\leq\frac1ddivide start_ARG italic_K end_ARG start_ARG italic_d italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ≤ caligraphic_F ( italic_ρ ) ≤ divide start_ARG 1 end_ARG start_ARG italic_d end_ARG, for all d⊗d′tensor-product𝑑superscript𝑑′d\otimes d^\primeitalic_d ⊗ italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT separable states Zhao (2015). Although fully entangled fraction cannot be regarded as a measure of entanglement in its own right, it may be used as a quantifier of entanglement applications. Historically, the first such usage was worked out by Horodecki et al. Horodecki et al. (1999), where the authors found a relationship between fully entangled fraction and the optimal fidelity of teleportation. In addition, in a two-qubit system, the fully entangled fraction can also be related to the fidelities of other important quantum information tasks such as dense coding, entanglement swapping and quantum cryptography in such a way as to provide an inclusive measure of these entanglement applications Grondalski et al. (2002). Additionally, it can be exploited as an index to characterize the non-local correlation Zhou and Guo (2000).



Quantum discord, introduced by Ollivier and Zurk in their seminal work Ollivier and Zurek (2001), is defined for a generic bipartite state ρACsuperscript𝜌𝐴𝐶\rho^ACitalic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT as follows



𝒟←(ρAC)=I(ρAC)-ℐ←(ρAC),superscript𝒟←superscript𝜌𝐴𝐶𝐼superscript𝜌𝐴𝐶superscriptℐ←superscript𝜌𝐴𝐶\mathcalD^\leftarrow(\rho^AC)=I(\rho^AC)-\mathcalI^\leftarrow(\rho% ^AC),caligraphic_D start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) - caligraphic_I start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) , (4) where I(ρAC)=S(ρA)+S(ρC)-S(ρAC)𝐼superscript𝜌𝐴𝐶𝑆superscript𝜌𝐴𝑆superscript𝜌𝐶𝑆superscript𝜌𝐴𝐶I(\rho^AC)=S(\rho^A)+S(\rho^C)-S(\rho^AC)italic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) is the mutual information of ρACsuperscript𝜌𝐴𝐶\rho^ACitalic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT, and



ℐ←(ρAC)=S(ρA)-∑ipiS(ρA|ΠiC),superscriptℐ←superscript𝜌𝐴𝐶𝑆superscript𝜌𝐴subscript𝑖subscript𝑝𝑖𝑆superscript𝜌conditional𝐴superscriptsubscriptΠ𝑖𝐶\mathcalI^\leftarrow(\rho^AC)=S(\rho^A)-\sum_ip_iS(\rho^\\Pi_% i^C\),caligraphic_I start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) - ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A | roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ) , (5) is the classical correlation between parts of ρACsuperscript𝜌𝐴𝐶\rho^ACitalic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT. Here ΠiC=superscriptsubscriptΠ𝑖𝐶ket𝑖bra𝑖\\Pi_i^C\=\left\i\rangle\langle i roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT = italic_i ⟩ ⟨ italic_i is the set of projection operators on the subsystem C𝐶Citalic_C, pi=Tr[(𝕀⊗ΠiC)ρAC(𝕀⊗ΠiC)]subscript𝑝𝑖Trdelimited-[]tensor-product𝕀superscriptsubscriptΠ𝑖𝐶superscript𝜌𝐴𝐶tensor-product𝕀superscriptsubscriptΠ𝑖𝐶p_i=\mathrmTr[(\mathbbI\otimes\Pi_i^C)\rho^AC(\mathbbI\otimes% \Pi_i^C)]italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_Tr [ ( blackboard_I ⊗ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ) italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ( blackboard_I ⊗ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ) ] is the probability of the i𝑖iitalic_i-th outcome for the subsystem C𝐶Citalic_C, and ρA|ΠiC=1piTrC[(𝕀⊗ΠiC)ρAC(𝕀⊗ΠiC)]superscript𝜌conditional𝐴superscriptsubscriptΠ𝑖𝐶1subscript𝑝𝑖subscriptTr𝐶delimited-[]tensor-product𝕀superscriptsubscriptΠ𝑖𝐶superscript𝜌𝐴𝐶tensor-product𝕀superscriptsubscriptΠ𝑖𝐶\rho^\\Pi_i^C\=\frac1p_i\mathrmTr_C[(\mathbbI\otimes% \Pi_i^C)\rho^AC(\mathbbI\otimes\Pi_i^C)]italic_ρ start_POSTSUPERSCRIPT italic_A | roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG roman_Tr start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT [ ( blackboard_I ⊗ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ) italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ( blackboard_I ⊗ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ) ] represents the post-measurement state. An independent work by Henderson and Vedral leads to a similar measure Henderson and Vedral (2001). Finding the quantum discord of an arbitrary state is still an open problem in quantum information theory and the analytical formula for the measure has been found just for a limited class of states Henderson and Vedral (2001); Ollivier and Zurek (2001); Luo (2008); Rulli and Sarandy (2011); Okrasa and Walczak (2011); Xu (2013).



Let |ψABC⟩ketsuperscript𝜓𝐴𝐵𝐶|\psi^ABC\rangle| italic_ψ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ⟩ be a pure state of the Hilbert space ℋA⊗ℋB⊗ℋCtensor-productsuperscriptℋ𝐴superscriptℋ𝐵superscriptℋ𝐶\mathcalH^A\otimes\mathcalH^B\otimes\mathcalH^Ccaligraphic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ caligraphic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⊗ caligraphic_H start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT. It is shown that the following relation can be found between entanglement of formation Bennett et al. (1996) of the bipartite state ρAB=TrC|ψABC⟩⟨ψABC|superscript𝜌𝐴𝐵subscriptTr𝐶ketsuperscript𝜓𝐴𝐵𝐶brasuperscript𝜓𝐴𝐵𝐶\rho^AB=\mathrmTr_Citalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT | italic_ψ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ⟩ ⟨ italic_ψ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT | and the so-called post-measurement mutual information of the bipartite state ρAC=TrB|ψABC⟩⟨ψABC|superscript𝜌𝐴𝐶subscriptTr𝐵ketsuperscript𝜓𝐴𝐵𝐶brasuperscript𝜓𝐴𝐵𝐶\rho^AC=\mathrmTr_Bitalic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT | italic_ψ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ⟩ ⟨ italic_ψ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT | Koashi and Winter (2004)



Ef(ρAB)+ℐ←(ρAC)=S(ρA),subscript𝐸𝑓superscript𝜌𝐴𝐵superscriptℐ←superscript𝜌𝐴𝐶𝑆superscript𝜌𝐴E_f(\rho^AB)+\mathcalI^\leftarrow(\rho^AC)=S(\rho^A),italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + caligraphic_I start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) , (6) where Ef(⋅)subscript𝐸𝑓⋅E_f(\cdot)italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ( ⋅ ) is the entanglement of formation of the state and ℐ←(⋅)superscriptℐ←⋅\mathcalI^\leftarrow(\cdot)caligraphic_I start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( ⋅ ) is the right post-measurement mutual information. The state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is called the C𝐶Citalic_C complement to ρACsuperscript𝜌𝐴𝐶\rho^ACitalic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT, and similarly, ρACsuperscript𝜌𝐴𝐶\rho^ACitalic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT is called the B𝐵Bitalic_B complement to ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT. From Eqs. (4) and (6) one infer that



𝒟←(ρAC)superscript𝒟←superscript𝜌𝐴𝐶\displaystyle\mathcalD^\leftarrow(\rho^AC)caligraphic_D start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) =\displaystyle== Ef(ρAB)+S(ρC)-S(ρAC)subscript𝐸𝑓superscript𝜌𝐴𝐵𝑆superscript𝜌𝐶𝑆superscript𝜌𝐴𝐶\displaystyle E_f(\rho^AB)+S(\rho^C)-S(\rho^AC)italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) (7)



=\displaystyle== Ef(ρAB)+S(ρAB)-S(ρB),subscript𝐸𝑓superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\displaystyle E_f(\rho^AB)+S(\rho^AB)-S(\rho^B),italic_E start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) , where the second line follows from S(ρC)=S(ρAB)𝑆superscript𝜌𝐶𝑆superscript𝜌𝐴𝐵S(\rho^C)=S(\rho^AB)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) and S(ρAC)=S(ρB)𝑆superscript𝜌𝐴𝐶𝑆superscript𝜌𝐵S(\rho^AC)=S(\rho^B)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ). This equation leads to an alternative way for calculating the quantum discord of a class of states, without the hard process of optimization which is a part of the definition of quantum discord. Evidently, the use of this method is limited to the states whose entanglement of formation of their complemental states is known. However, the strength of the mentioned method was shown in Ref. Shi et al. (2011) where the authors could find the quantum discord of some two-qubit rank-2 states.



The aim of this paper is to compare the fully entangled fraction and quantum discord of two classes of d⊗d2tensor-product𝑑superscript𝑑2d\otimes d^2italic_d ⊗ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT states. The states considered here are complement to the d⊗dtensor-product𝑑𝑑d\otimes ditalic_d ⊗ italic_d Werner and isotropic states. Using Eq. (7) we obtain a closed relation for the quantum discord of these states. We also use the notion of fully entangled fraction and find an exact relation for the first class of our d⊗d2tensor-product𝑑superscript𝑑2d\otimes d^2italic_d ⊗ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT states, i.e. states complement to the Werner states. For the second class of d⊗d2tensor-product𝑑superscript𝑑2d\otimes d^2italic_d ⊗ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT states, i.e. states complement to d⊗dtensor-product𝑑𝑑d\otimes ditalic_d ⊗ italic_d isotropic states, we provide a lower bound for the fully entangled fraction. A comparison of the measures shows that the quantum discord and fully entangled fraction of each class behave similarly. In particular, our results show that maximally entangled mixed states are also maximally discordant states. This generalizes the well known fact that all maximally entangled pure states have also maximum quantum discord. In addition, our results show that the separability-entanglement boundary of a Werner or isotropic state is manifested as an inflection point in the diagram of quantum discord of the corresponding complemental state. Therefore, further insight into the notion of separability-entanglement paradigm may be provided by investigating quantum discord of the complemental state.



The remainder of the paper is arranged as follows. In Section II, we present a class of d⊗d2tensor-product𝑑superscript𝑑2d\otimes d^2italic_d ⊗ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT states complement to d⊗dtensor-product𝑑𝑑d\otimes ditalic_d ⊗ italic_d Werner states and provide an exact solution for their fully entangled fraction and quantum discord. Section III is devoted to the d⊗d2tensor-product𝑑superscript𝑑2d\otimes d^2italic_d ⊗ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT states complement to d⊗dtensor-product𝑑𝑑d\otimes ditalic_d ⊗ italic_d isotropic states, and finding a lower bound for fully entangled fraction. In Section IV we provide a discussion by presenting some examples, i.e. the cases d=2,3𝑑23d=2,3italic_d = 2 , 3, and compare the fully entangled fraction and quantum discord of the mentioned states. We conclude the paper in Section V.



II The d⊗d2tensor-product𝑑superscript𝑑2d\otimes d^2italic_d ⊗ italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT states complement to the d⊗dtensor-product𝑑𝑑d\otimes ditalic_d ⊗ italic_d Werner states



A generic d⊗dtensor-product𝑑𝑑d\otimes ditalic_d ⊗ italic_d Werner state acting on ℋA⊗ℋBtensor-productsuperscriptℋ𝐴superscriptℋ𝐵\mathcalH^A\otimes\mathcalH^Bcaligraphic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ caligraphic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT is defined by



ρwAB=d-xd3-d𝕀d2+dx-1d3-dF,x∈[-1,1],formulae-sequencesuperscriptsubscript𝜌w𝐴𝐵𝑑𝑥superscript𝑑3𝑑subscript𝕀superscript𝑑2𝑑𝑥1superscript𝑑3𝑑𝐹𝑥11\rho_\mathrmw^AB=\fracd-xd^3-d\mathbbI_d^2+\fracdx-1d^3% -dF,\quad x\in[-1,1],italic_ρ start_POSTSUBSCRIPT roman_w end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = divide start_ARG italic_d - italic_x end_ARG start_ARG italic_d start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - italic_d end_ARG blackboard_I start_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + divide start_ARG italic_d italic_x - 1 end_ARG start_ARG italic_d start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - italic_d end_ARG italic_F , italic_x ∈ [ - 1 , 1 ] , (8) where F=∑k,l=1d|kl⟩⟨lk|𝐹superscriptsubscript𝑘𝑙1𝑑ket𝑘𝑙bra𝑙𝑘F=\sum_k,l=1^d|kl\rangle\langle lk|italic_F = ∑ start_POSTSUBSCRIPT italic_k , italic_l = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | italic_k italic_l ⟩ ⟨ italic_l italic_k |, and x=Tr(ρwABF)𝑥Trsuperscriptsubscript𝜌w𝐴𝐵𝐹x=\mathrmTr(\rho_\mathrmw^ABF)italic_x = roman_Tr ( italic_ρ start_POSTSUBSCRIPT roman_w end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT italic_F ). It is shown that Werner state is separable if and only if 0≤x≤10𝑥10\leq x\leq 10 ≤ italic_x ≤ 1 Vollbrecht and Werner (2001). The Werner state (8) has the following spectral decomposition



ρwAB=λ+P(+)+λ-P(-).superscriptsubscript𝜌w𝐴𝐵subscript𝜆superscript𝑃subscript𝜆superscript𝑃\rho_\mathrmw^AB=\lambda_+P^(+)+\lambda_-P^(-).italic_ρ start_POSTSUBSCRIPT roman_w end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = italic_λ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT + italic_λ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT italic_P start_POSTSUPERSCRIPT ( - ) end_POSTSUPERSCRIPT . (9) Here λ±=1±xd(d±1)subscript𝜆plus-or-minusplus-or-minus1𝑥𝑑plus-or-minus𝑑1\lambda_\pm=\frac1\pm xd(d\pm 1)italic_λ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = divide start_ARG 1 ± italic_x end_ARG start_ARG italic_d ( italic_d ± 1 ) end_ARG, and P(+)superscript𝑃P^(+)italic_P start_POSTSUPERSCRIPT ( + ) end_POSTSUPERSCRIPT and P(-)superscript𝑃P^(-)italic_P start_POSTSUPERSCRIPT ( - ) end_POSTSUPERSCRIPT are projection operators on the d+=d(d+1)/2subscript𝑑𝑑𝑑12d_+=d(d+1)/2italic_d start_POSTSUBSCRIPT + end_POSTSUBSCRIPT = italic_d ( italic_d + 1 ) / 2- and d-=d(d-1)/2subscript𝑑𝑑𝑑12d_-=d(d-1)/2italic_d start_POSTSUBSCRIPT - end_POSTSUBSCRIPT = italic_d ( italic_d - 1 ) / 2-dimensional symmetric and antisymmetric subspaces of ℋA⊗ℋBtensor-productsuperscriptℋ𝐴superscriptℋ𝐵\mathcalH^A\otimes\mathcalH^Bcaligraphic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ caligraphic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT, respectively, defined by



P(+)=∑k≤ld|λkl(+)⟩⟨λkl(+)|,P(-)=∑k<ld|λkl(-)⟩⟨λkl(-)|,formulae-sequencesuperscript𝑃superscriptsubscript𝑘𝑙𝑑ketsuperscriptsubscript𝜆𝑘𝑙brasuperscriptsubscript𝜆𝑘𝑙superscript𝑃superscriptsubscript𝑘𝑙𝑑ketsuperscriptsubscript𝜆𝑘𝑙brasuperscriptsubscript𝜆𝑘𝑙p^(+)=\sum_k\leq l^d|\lambda_kl^(+)\rangle\langle\lambda_kl^(+)|% ,\qquad p^(-)="\sum_" , (10)

where



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