A Generalized Geometric Measurement Of Quantum Discord Exact Treatment

A generalization of the geometric measure of quantum discord is introduced in this article, based on Hellinger distance, which has virtue of computability and independence of local measurements. In addition our definition also does not suffer from the recently raised critiques about quantum discord. Importantly the exact result can be obtained for bipartite pure states with arbitrary levels, which is completely determined by the Schmidt decomposition of the states. For bipartite mixed states the exact result can also be found for X𝑋Xitalic_X type case, of which the sub matrices are spanned by the Schmidt-typed states. Furthermore this definition has a natural generalization into multipartite states. As for symmetric case, permutational or translational invariance, we shown that it can be evaluated exactly by supposing that the ”nearest” completely classical state shares the same symmetry of the state. In addition we show that our definition can also be used to mark the appearance of quantum phase transition in many-body systems.

quantum discord, geometric measure, quantum phase transition

pacs: 03.65.Ud, 03.67.Mn, 64.60.-i, 75.10.Pq

Quantum systems can exhibit non-classical correlation by a different way from quantum entanglement, which is known as quantum discord (QD). The quantum discord characterizes the minimal perturbation induced by single-party von Neumann measurement qd . And thus there exists non-entangled state with non-zero QD. Importantly QD is now shown as a resource to speed up the quantum information processing. For instance the determined quantum computation with one qubit dqc1 and quantum metrology with noised states qm have been demonstrated an advantage over classical computation even without entanglement. Moreover QD has also inspired great attention in other diverse contexts review .

However QD is so hard to determine exactly because of the optimization in the definition that there exist very few exact results, even for the simplest two-qubit case luo . ( Recently an exact evaluation of QD is proposed in Ref ali10 . However it is pointed out in huang13 that this approach is not exactly correct.) It has been shown that the computation of quantum discord is NP-complete huang14 ; the running time of the computation of QD is increased exponentially with the dimension of the Hilbert space. Thus one has to find another way to measure QD. With respect of this fact, geometric measure of QD is introduced by Dakić and the coauthors, which is defined as the square form of the shortest distance between arbitrary state ρ𝜌\rhoitalic_ρ and zero-discord state χ𝜒\chiitalic_χ in Hilbert space dakic . By this feature the optimization in the definition of geometric discord (GD) can be reduce greatly, and more a tight lower bound of GD can be determined for arbitrary states luo10 . However the square form of GD is not monotonic under local operations, of which the value can increase by local operations dakic ; pz . This deficit raises the question whether GD and furthermore QD can unambiguously manifest the non-classical correlation in quantum states pz ; gheorghiu14 . In order to overcome this problem, many generalization of GD have been proposed. For instance a rescaled GD is defined by rescaling the density operator with its norm in tufarelli . Furthermore the so-called Schatten p𝑝pitalic_p-norm is introduced to qualify the distance pnorm , instead 2222-norm in Ref. dakic . In addition the Bures distance is also introduced bures . However it is difficult genially by these generalization to find the analytical expression for GD since their involved evaluation and optimization of the eigenvalue and corresponding eigenvectors in the expressions. Recently an interesting generalization is proposed by introducing the Hellinger distance chang13 ; luo04 ; dajka11 . This definition has a simple structure and can be evaluated readily. Moreover it is also monotonically nonincreasing (contractivity) under local operations luo04 ; dajka11 .

In addition it is interesting how to generalize this definition into multipartite case. A direct way is to introduce the three-tangle as a generalization of QD for tripartite case 3tangle and its generalization for four-qubit case bai13 , as have done in studying tripartite entanglement. This approach has virtue of clear discrimination of bipartite and multipartite quantum correlation. However it is difficult to determine analytically and furthermore to generalize into many-body case. Another way is to find the minimum of the QD between arbitrary single party and the others ggd , termed as global QD. However it do not consider the other possible bipartite correlation, e.g. (n,N-n)𝑛𝑁𝑛(n,N-n)( italic_n , italic_N - italic_n ) division of the system with total N𝑁Nitalic_N parties, and thus is not a comprehensive measurement of QD. Additionally the exact treatment of global QD is still difficult since one has to find optimal single party von Neumann measurements for all parties. Recently a geometric generalization of global QD is proposed by finding the shortest distance from the zero global QD state xu12 . However the author adopt the 2222-norm of distance, which suffers from the critique of non-contractivity under local operations pz .

With respect of these facts, we present an alternative approaching to QD by a generalization of Hellinger distance chang13 ; luo04 ; dajka11 , of which satisfies the requirements of a good measure of quantum correlation luo04 ; dajka11 . Furthermore in order to avoid the critique in Ref. gheorghiu14 , the shortest distance is defined instead as from the completely classical state ll08 , which hence becomes independent on the local measurements. By this generalization QD can be exactly evaluated for arbitrary bipartite pure states and for some special mixed states, as shown in this article. Moreover this generalization can be readily applied to multipartite case. As for symmetric multipartite states the exact results can be founded. This article is divided into several sections. The definition is presented in Section II, and a general expression for this geometric QD is also presented. Then Section III presents the exact results for bipartite pure states and for a special type of mixed states. In Section IV we apply the definition for multipartite states and the exact result can be obtained for symmetric states. In addition we also show the ability of marking the quantum phase transition in many-body systems. Conclusion and discussion are presented in final section.

II definition and technical preparation

We first present the definition.

Given arbitrary state ρ𝜌\rhoitalic_ρ and completely classical state σ𝜎\sigmaitalic_σ, we define geometric measure of QD

DH=12minσ∥ρ-σ∥2,superscript𝐷𝐻12subscript𝜎superscriptnorm𝜌𝜎2\displaystyle D^H=\frac12\min_\sigma\parallel\sqrt\rho-\sqrt\sigma% \parallel^2,italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_min start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∥ square-root start_ARG italic_ρ end_ARG - square-root start_ARG italic_σ end_ARG ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (1) where ∥⋅∥fragmentsparallel-tonormal-⋅parallel-to\parallel\cdot\parallel∥ ⋅ ∥ is the Hilbert-Schmidt norm and the superscript means the Hellinger distance.

For qubit system, the completely classical state σ𝜎\sigmaitalic_σ is the probabilistic mixture of a special computational basis -⟩⊗Nsuperscriptketkettensor-productabsent𝑁\^\otimes N start_POSTSUPERSCRIPT ⊗ italic_N end_POSTSUPERSCRIPT, the single-party states |+⟩=cosθ2|1⟩+exp(-iϕ)sinθ2|0⟩ket𝜃2ket1𝑖italic-ϕ𝜃2ket0|+\rangle=\cos\frac\theta2|1\rangle+\exp(-i\phi)\sin\frac\theta2|0\rangle| + ⟩ = roman_cos divide start_ARG italic_θ end_ARG start_ARG 2 end_ARG | 1 ⟩ + roman_exp ( - italic_i italic_ϕ ) roman_sin divide start_ARG italic_θ end_ARG start_ARG 2 end_ARG | 0 ⟩ and |-⟩=exp(iϕ)sinθ2|1⟩-cosθ2|0⟩ket𝑖italic-ϕ𝜃2ket1𝜃2ket0|-\rangle=\exp(i\phi)\sin\frac\theta2|1\rangle-\cos\frac\theta2|0\rangle| - ⟩ = roman_exp ( italic_i italic_ϕ ) roman_sin divide start_ARG italic_θ end_ARG start_ARG 2 end_ARG | 1 ⟩ - roman_cos divide start_ARG italic_θ end_ARG start_ARG 2 end_ARG | 0 ⟩, in which θ∈[0,π)𝜃0𝜋\theta\in[0,\pi)italic_θ ∈ [ 0 , italic_π ) and ϕ∈[0,2π)italic-ϕ02𝜋\phi\in[0,2\pi)italic_ϕ ∈ [ 0 , 2 italic_π ). As for multi-level case, the similar definition can be constructed readily, which is not shown here.

Now we are ready to find a general expression of DH(ρ)superscript𝐷𝐻𝜌D^H(\rho)italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT ( italic_ρ ). First the definition Eq. (1) can be reduced to

DH=1-maxσTr[ρσ].superscript𝐷𝐻1subscript𝜎Trdelimited-[]𝜌𝜎\displaystyle D^H=1-\max_\sigma\textTr\left[\sqrt\rho\sqrt\sigma% \right].italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - roman_max start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT Tr [ square-root start_ARG italic_ρ end_ARG square-root start_ARG italic_σ end_ARG ] . (2) Hence the evaluation of DH(ρ)superscript𝐷𝐻𝜌D^H(\rho)italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT ( italic_ρ ) is reduced to find the maximal overlap of ρ𝜌\sqrt\rhosquare-root start_ARG italic_ρ end_ARG and σ𝜎\sqrt\sigmasquare-root start_ARG italic_σ end_ARG. Second σ𝜎\sigmaitalic_σ can be written generally as the probabilistic mixture of computational basis

σ=∑npn|σn⟩⟨σn|.𝜎subscript𝑛subscript𝑝𝑛ketsubscript𝜎𝑛brasubscript𝜎𝑛\displaystyle\sigma=\sum_np_n|\sigma_n\rangle\langle\sigma_n|.italic_σ = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ ⟨ italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | . (3) ketsubscript𝜎𝑛\\sigma_n\rangle\ italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ denotes the general computational basis state, which, for instance of qu-bit case, is composed from the single-party basis states|+⟩ket|+\rangle| + ⟩ and |-⟩ket|-\rangle| - ⟩. In follow we will present the general expressions for pure and mixed state cases respectively. For convenience we set N𝑁Nitalic_N as the total number of the computational basis states and pN=1-∑n=1N-1pnsubscript𝑝𝑁1superscriptsubscript𝑛1𝑁1subscript𝑝𝑛p_N=1-\sum_n=1^N-1p_nitalic_p start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = 1 - ∑ start_POSTSUBSCRIPT italic_n = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N - 1 end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT.

-Pure state- For a pure state |ψ⟩ket𝜓|\psi\rangle| italic_ψ ⟩ , one gets

DH=1-maxpn,σn∑n|⟨ψ|σn⟩|2pn.superscript𝐷𝐻1subscriptsubscript𝑝𝑛subscript𝜎𝑛subscript𝑛superscriptinner-product𝜓subscript𝜎𝑛2subscript𝑝𝑛\displaystyle D^H=1-\max_\p_n,\sigma_n\\sum_n\left|\langle\psi|% \sigma_n\rangle\right|^2\sqrtp_n.italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - roman_max start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT square-root start_ARG italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG . (4) The extremal value points happen when

∂DH∂pi=0⇒superscript𝐷𝐻subscript𝑝𝑖0⇒absent\displaystyle\frac\partial D^H\partial p_i=0\Rightarrowdivide start_ARG ∂ italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = 0 ⇒

|⟨ψ|σi⟩|2pi=|⟨ψ|σN⟩|2pN,i=1,2,⋯,N-1.formulae-sequencesuperscriptinner-product𝜓subscript𝜎𝑖2subscript𝑝𝑖superscriptinner-product𝜓subscript𝜎𝑁2subscript𝑝𝑁𝑖12⋯𝑁1\displaystyle\frac\langle\psi\sqrtp_i% =\frac\sigma_N\rangle\right\sqrtp_N,i=1,2,% \cdots,N-1.divide start_ARG | ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG end_ARG = divide start_ARG | ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG italic_p start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_ARG end_ARG , italic_i = 1 , 2 , ⋯ , italic_N - 1 . (5) It is not difficult to find ∂2DH(ρ)∂pi∂pj>0superscript2superscript𝐷𝐻𝜌subscript𝑝𝑖subscript𝑝𝑗0\frac\partial^2D^H(\rho)\partial p_i\partial p_j>0divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT ( italic_ρ ) end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∂ italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG >0. Then DH(ρ)superscript𝐷𝐻𝜌D^H(\rho)italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT ( italic_ρ ) has the minimal value when

pi=|⟨ψ|σi⟩|4∑n|⟨ψ|σn⟩|4,i=1,2,⋯,N.formulae-sequencesubscript𝑝𝑖superscriptinner-product𝜓subscript𝜎𝑖4subscript𝑛superscriptinner-product𝜓subscript𝜎𝑛4𝑖12⋯𝑁\displaystyle p_i=\frac\langle\psi\langle\psi,i=1,2,\cdots,N.italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG | ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG , italic_i = 1 , 2 , ⋯ , italic_N . (6) Consequently DHsuperscript𝐷𝐻D^Hitalic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT reduces to

DH=1-maxσn∑n|⟨ψ|σn⟩|4.superscript𝐷𝐻1subscriptsubscript𝜎𝑛subscript𝑛superscriptinner-product𝜓subscript𝜎𝑛4\displaystyle D^H=1-\max_\\sigma_n\\sqrt\sum_n\left.italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - roman_max start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT square-root start_ARG ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG . (7)

-Mixed state- As for the spectrum decomposition ρ=∑kλk|ϕk⟩⟨ϕk|𝜌subscript𝑘subscript𝜆𝑘ketsubscriptitalic-ϕ𝑘brasubscriptitalic-ϕ𝑘\rho=\sum_k\lambda_k|\phi_k\rangle\langle\phi_k|italic_ρ = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩ ⟨ italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |, Eq.(2) can be written as

DH=1-maxpn,σn∑n,kλkpn|⟨ϕk|σn⟩|2.superscript𝐷𝐻1subscriptsubscript𝑝𝑛subscript𝜎𝑛subscript𝑛𝑘subscript𝜆𝑘subscript𝑝𝑛superscriptinner-productsubscriptitalic-ϕ𝑘subscript𝜎𝑛2\displaystyle D^H=1-\max_\p_n,\sigma_n\\sum_n,k\sqrt\lambda_k% \sqrtp_n\left|\langle\phi_k|\sigma_n\rangle\right|^2.italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - roman_max start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n , italic_k end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG square-root start_ARG italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | ⟨ italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . (8) As for pisubscript𝑝𝑖p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, the extremal value points happen when

∂DH∂pi=0superscript𝐷𝐻subscript𝑝𝑖0\displaystyle\frac\partial D^H\partial p_i=0divide start_ARG ∂ italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = 0

⇒∑kλk|⟨ϕk|σi⟩|2pi=∑kλk|⟨ϕk|σN⟩|2pN,⇒absentsubscript𝑘subscript𝜆𝑘superscriptinner-productsubscriptitalic-ϕ𝑘subscript𝜎𝑖2subscript𝑝𝑖subscript𝑘subscript𝜆𝑘superscriptinner-productsubscriptitalic-ϕ𝑘subscript𝜎𝑁2subscript𝑝𝑁\displaystyle\Rightarrow\frac\sum_k\sqrt\lambda_k\left\sqrtp_i=\frac\sigma_N\rangle\right\sqrtp_N,⇒ divide start_ARG ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG | ⟨ italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG end_ARG = divide start_ARG ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG | ⟨ italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_σ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG italic_p start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_ARG end_ARG ,

i=1,2,⋯,N-1.𝑖12⋯𝑁1\displaystyle i=1,2,\cdots,N-1.italic_i = 1 , 2 , ⋯ , italic_N - 1 . (9) Directly ∂2DH∂pi∂pj>0superscript2superscript𝐷𝐻subscript𝑝𝑖subscript𝑝𝑗0\frac\partial^2D^H\partial p_i\partial p_j>0divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∂ italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG >0. Then DHsuperscript𝐷𝐻D^Hitalic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT reduces to

DH=1-maxσn∑n(∑kλk|⟨ϕk|σn⟩|2)2,superscript𝐷𝐻1subscriptsubscript𝜎𝑛subscript𝑛superscriptsubscript𝑘subscript𝜆𝑘superscriptinner-productsubscriptitalic-ϕ𝑘subscript𝜎𝑛22\displaystyle D^H=1-\max_\\sigma_n\\sqrt\sigma_n\rangle\right,italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - roman_max start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT square-root start_ARG ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG | ⟨ italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , (10) when

pi=(∑kλk|⟨ϕk|σi⟩|2)2∑n(∑kλk|⟨ϕk|σn⟩|2)2,i=1,2,⋯,N.formulae-sequencesubscript𝑝𝑖superscriptsubscript𝑘subscript𝜆𝑘superscriptinner-productsubscriptitalic-ϕ𝑘subscript𝜎𝑖22subscript𝑛superscriptsubscript𝑘subscript𝜆𝑘superscriptinner-productsubscriptitalic-ϕ𝑘subscript𝜎𝑛22𝑖12⋯𝑁\displaystyle p_i=\frac\left(\sum_k\sqrt\lambda_k\left\sqrt^2\right)^2,i=% 1,2,\cdots,N.italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG ( ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG | ⟨ italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT square-root start_ARG italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG | ⟨ italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG , italic_i = 1 , 2 , ⋯ , italic_N . (11)

With these preparations, we are ready to evaluate DH(ρ)superscript𝐷𝐻𝜌D^H(\rho)italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT ( italic_ρ ) explicitly. It should be pointed out that these expressions above is suitable for arbitrary multi-level case. However for simplicity the following discussions would focus only on qubit case since the extensive interest on quantum information procession. The extension into multi-level case is direct.

III bipartite state: exact treatment

-Pure state- From Eq. (7) the determination of DH(ρ)superscript𝐷𝐻𝜌D^H(\rho)italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT ( italic_ρ ) reduces to find the maximal overlap |⟨ψ|σn⟩|inner-product𝜓subscript𝜎𝑛\left|\langle\psi|\sigma_n\rangle\right|| ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ |. It is known that there exists Schmidt decomposition form for any bipartite pure state, which mathematically corresponds to the minimal expansion of a pure state. The corresponding Schmidt states then construct the simplest subspace, in which the state is a vector. Hence in order to find the maximal overlap between |ψ⟩ket𝜓|\psi\rangle| italic_ψ ⟩ and σ𝜎\sigmaitalic_σ, σ𝜎\sigmaitalic_σ is necessary to be a vector or the mixed combination of the vectors in this subspace. With respect of classicality of σnsubscript𝜎𝑛\sigma_nitalic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT and that the superposition of Schmidt states cannot be separable, the only reasonable choice of σ𝜎\sigmaitalic_σ is to be the mixed combination of Schmidt states.

As an example, we try to find DHsuperscript𝐷𝐻D^Hitalic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT for

|ψ⟩ABsubscriptket𝜓𝐴𝐵\displaystyle|\psi\rangle_AB| italic_ψ ⟩ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT =\displaystyle== 12|1⟩A(12|1⟩B+32|0⟩B)+limit-from12subscriptket1𝐴12subscriptket1𝐵32subscriptket0𝐵\displaystyle\frac1\sqrt2|1\rangle_A\left(\frac12|1\rangle_B+% \frac\sqrt32|0\rangle_B\right)+divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG | 1 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 1 ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT + divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG | 0 ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) + (12)

12|0⟩A(32|1⟩B+12|0⟩B),12subscriptket0𝐴32subscriptket1𝐵12subscriptket0𝐵\displaystyle\frac1\sqrt2|0\rangle_A\left(\frac\sqrt32|1\rangle_% B+\frac12|0\rangle_B\right),divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG | 0 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG | 1 ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | 0 ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) , of which the Schmidt form is

|ψ⟩ABsubscriptket𝜓𝐴𝐵\displaystyle|\psi\rangle_AB| italic_ψ ⟩ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT =\displaystyle== 2+32|1⟩A+|0⟩A2|1⟩B+|0⟩B2232subscriptket1𝐴subscriptket0𝐴2subscriptket1𝐵subscriptket0𝐵2\displaystyle\frac\sqrt2+\sqrt32\frac% \sqrt2\frac0\rangle_B\sqrt2divide start_ARG square-root start_ARG 2 + square-root start_ARG 3 end_ARG end_ARG end_ARG start_ARG 2 end_ARG divide start_ARG | 1 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT + | 0 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG divide start_ARG | 1 ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT + | 0 ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG (13)

+\displaystyle++ 2-32|1⟩A-|0⟩A2|1⟩B-|0⟩B2.232subscriptket1𝐴subscriptket0𝐴2subscriptket1𝐵subscriptket0𝐵2\displaystyle\frac\sqrt2-\sqrt32\frac1\rangle_A-% \sqrt2\frac0\rangle_B\sqrt2.divide start_ARG square-root start_ARG 2 - square-root start_ARG 3 end_ARG end_ARG end_ARG start_ARG 2 end_ARG divide start_ARG | 1 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT - | 0 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG divide start_ARG | 1 ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT - | 0 ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG . By explicit calculation, we obtain

∑n|⟨ψ|σn⟩AB|4fragmentssubscript𝑛|subscriptfragments⟨ψ|subscript𝜎𝑛⟩𝐴𝐵superscript|4\displaystyle\sum_n\left|_AB\langle\psi\right|^4∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_FLOATSUBSCRIPT italic_A italic_B end_FLOATSUBSCRIPT ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT (14)

=\displaystyle== (14-c0+c1+c2)2+(14+c0+c1-c2)2superscript14subscript𝑐0subscript𝑐1subscript𝑐22superscript14subscript𝑐0subscript𝑐1subscript𝑐22\displaystyle(\tfrac14-c_0+c_1+c_2)^2+(\tfrac14+c_0+c_1-c_% 2)^2( divide start_ARG 1 end_ARG start_ARG 4 end_ARG - italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG + italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

+(14+c0-c1+c2)2+(14-c0-c1-c2)2superscript14subscript𝑐0subscript𝑐1subscript𝑐22superscript14subscript𝑐0subscript𝑐1subscript𝑐22\displaystyle+(\tfrac14+c_0-c_1+c_2)^2+(\tfrac14-c_0-c_1-c% _2)^2+ ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG + italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG - italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

=\displaystyle== 4(116+c02+c12+c22)4116subscriptsuperscript𝑐20subscriptsuperscript𝑐21subscriptsuperscript𝑐22\displaystyle 4\left(\tfrac116+c^2_0+c^2_1+c^2_2\right)4 ( divide start_ARG 1 end_ARG start_ARG 16 end_ARG + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) in which,

c0subscript𝑐0\displaystyle c_0italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT =\displaystyle== 18cosθ1cosθ2fragments18fragmentssubscript𝜃1subscript𝜃2\displaystyle\tfrac18\left\\cos\theta_1\cos\theta_2\right.divide start_ARG 1 end_ARG start_ARG 8 end_ARG roman_cos italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT

-\displaystyle-- 12sinθ1sinθ2[cos(ϕ1+ϕ2)+3cos(ϕ1-ϕ2)]fragments12subscript𝜃1subscript𝜃2fragments[fragments(subscriptitalic-ϕ1subscriptitalic-ϕ2)3fragments(subscriptitalic-ϕ1subscriptitalic-ϕ2)]\displaystyle\left.\tfrac12\sin\theta_1\sin\theta_2\left[\cos(\phi_1% +\phi_2)+3\cos(\phi_1-\phi_2)\right]\right\divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_sin italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ roman_cos ( italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + 3 roman_cos ( italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ]

c1subscript𝑐1\displaystyle c_1italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =\displaystyle== 38sinθ1cosϕ138subscript𝜃1subscriptitalic-ϕ1\displaystyle\tfrac\sqrt38\sin\theta_1\cos\phi_1divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 8 end_ARG roman_sin italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT

c2subscript𝑐2\displaystyle c_2italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT =\displaystyle== 38sinθ2cosϕ238subscript𝜃2subscriptitalic-ϕ2\displaystyle\tfrac\sqrt38\sin\theta_2\cos\phi_2divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 8 end_ARG roman_sin italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_cos italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (15) It is obvious that ci(i=0,1,2)subscript𝑐𝑖𝑖012c_i(i=0,1,2)italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_i = 0 , 1 , 2 ) is symmetric under transformation θ1(2)↔π-θ1(2)↔subscript𝜃12𝜋subscript𝜃12\theta_1(2)\leftrightarrow\pi-\theta_1(2)italic_θ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT ↔ italic_π - italic_θ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT. Consequently as for cisubscript𝑐𝑖c_iitalic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT’s the extremal value of the overlap occurs only when θ1(2)=0,π,π/2subscript𝜃120𝜋𝜋2\theta_1(2)=0,\pi,\pi/2italic_θ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT = 0 , italic_π , italic_π / 2.

When θ1=θ2=0,πformulae-sequencesubscript𝜃1subscript𝜃20𝜋\theta_1=\theta_2=0,\piitalic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 , italic_π, c0=1/8subscript𝑐018c_0=1/8italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 / 8 and c1=c2=0subscript𝑐1subscript𝑐20c_1=c_2=0italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0. Then max∑n|⟨ψ|σn⟩AB|4=5/16fragmentssubscript𝑛|subscriptfragments⟨ψ|subscript𝜎𝑛⟩𝐴𝐵superscript|4516\max\sum_n\left|_AB\langle\psi\right|^4=5/16roman_max ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_FLOATSUBSCRIPT italic_A italic_B end_FLOATSUBSCRIPT ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT = 5 / 16. When one of θ1(2)subscript𝜃12\theta_1(2)italic_θ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT equals to 00 or π𝜋\piitalic_π, the other π/2𝜋2\pi/2italic_π / 2, c0=0subscript𝑐00c_0=0italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0 and one of c1(2)subscript𝑐12c_1(2)italic_c start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT is also vanishing . Then

∑n|⟨ψ|σn⟩AB|4fragmentssubscript𝑛|subscriptfragments⟨ψ|subscript𝜎𝑛⟩𝐴𝐵superscript|4\displaystyle\sum_n\left|_AB\langle\psi\right|^4∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_FLOATSUBSCRIPT italic_A italic_B end_FLOATSUBSCRIPT ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT (16)

=\displaystyle== 2(14+38cosϕ1(2))2+2(14-38cosϕ1(2))22superscript1438subscriptitalic-ϕ1222superscript1438subscriptitalic-ϕ122\displaystyle 2(\tfrac14+\tfrac\sqrt38\cos\phi_1(2))^2+2(\tfrac% 14-\tfrac\sqrt38\cos\phi_1(2))^22 ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG + divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 8 end_ARG roman_cos italic_ϕ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG - divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 8 end_ARG roman_cos italic_ϕ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

=\displaystyle== 14+316cosϕ1(2)2,14316superscriptsubscriptitalic-ϕ122\displaystyle\tfrac14+\tfrac316\cos\phi_1(2)^2,divide start_ARG 1 end_ARG start_ARG 4 end_ARG + divide start_ARG 3 end_ARG start_ARG 16 end_ARG roman_cos italic_ϕ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , of which the maximum occurs when cosϕ1(2)=±1subscriptitalic-ϕ12plus-or-minus1\cos\phi_1(2)=\pm 1roman_cos italic_ϕ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT = ± 1. Moreover we note

max∑n|⟨ψ|σn⟩AB|4fragmentssubscript𝑛|subscriptfragments⟨ψ|subscript𝜎𝑛⟩𝐴𝐵superscript|4\displaystyle\max\sum_n\left|\sigma_n\rangle\right|^4roman_max ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_FLOATSUBSCRIPT italic_A italic_B end_FLOATSUBSCRIPT ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT (17)

=\displaystyle== 12(2+34)2+12(2-34)2=71612superscript234212superscript2342716\displaystyle\tfrac12\left(\tfrac2+\sqrt34\right)^2+\tfrac12% \left(\tfrac2-\sqrt34\right)^2=\tfrac716divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG 2 + square-root start_ARG 3 end_ARG end_ARG start_ARG 4 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG 2 - square-root start_ARG 3 end_ARG end_ARG start_ARG 4 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG 7 end_ARG start_ARG 16 end_ARG

Another case is when θ1=θ2=π/2subscript𝜃1subscript𝜃2𝜋2\theta_1=\theta_2=\pi/2italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_π / 2. ∑n|⟨ψ|σn⟩AB|4fragmentssubscript𝑛|subscriptfragments⟨ψ|subscript𝜎𝑛⟩𝐴𝐵superscript|4\sum_n\left|\sigma_n\rangle\right|^4∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_FLOATSUBSCRIPT italic_A italic_B end_FLOATSUBSCRIPT ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT then is a function of ϕ1subscriptitalic-ϕ1\phi_1italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and ϕ2subscriptitalic-ϕ2\phi_2italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. By a plotting versus ϕ1subscriptitalic-ϕ1\phi_1italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and ϕ2subscriptitalic-ϕ2\phi_2italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, one can readily find the maximum when ϕ1(2)=0,π,2πsubscriptitalic-ϕ120𝜋2𝜋\phi_1(2)=0,\pi,2\piitalic_ϕ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT = 0 , italic_π , 2 italic_π. Then one has

max∑n|⟨ψ|σn⟩AB|4=(2+34)2+(2-34)2=78,fragmentssubscript𝑛|subscriptfragments⟨ψ|subscript𝜎𝑛⟩𝐴𝐵superscript|4superscriptfragments(234)2superscriptfragments(234)278,\displaystyle\max\sum_n\left|_AB\langle\psi\right|^% 4=\left(\tfrac2+\sqrt34\right)^2+\left(\tfrac2-\sqrt34\right)^% 2=\tfrac78,roman_max ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_FLOATSUBSCRIPT italic_A italic_B end_FLOATSUBSCRIPT ⟨ italic_ψ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT = ( divide start_ARG 2 + square-root start_ARG 3 end_ARG end_ARG start_ARG 4 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( divide start_ARG 2 - square-root start_ARG 3 end_ARG end_ARG start_ARG 4 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG 7 end_ARG start_ARG 8 end_ARG , (18) which clearly is the sum of the fourth power of the Schmidt coefficients.

Then one can obtain that the maximal overlap is 7/8787/87 / 8, which happens when the ”nearest” σ𝜎\sigmaitalic_σ has the form, obtained by setting θ1=θ2=π/2subscript𝜃1subscript𝜃2𝜋2\theta_1=\theta_2=\pi/2italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_π / 2 and ϕ1=ϕ2=0subscriptitalic-ϕ1subscriptitalic-ϕ20\phi_1=\phi_2=0italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0,

σABsubscript𝜎𝐴𝐵\displaystyle\sigma_ABitalic_σ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT =\displaystyle== 7+4314|1x⟩A⟨1x|A⊗|1x⟩B⟨1x|Btensor-product74314subscriptketsubscript1𝑥𝐴subscriptbrasubscript1𝑥𝐴subscriptketsubscript1𝑥𝐵subscriptbrasubscript1𝑥𝐵\displaystyle\tfrac7+4\sqrt314|1_x\rangle_A\langle 1_x|_A\otimes% |1_x\rangle_B\langle 1_x|_Bdivide start_ARG 7 + 4 square-root start_ARG 3 end_ARG end_ARG start_ARG 14 end_ARG | 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟨ 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ⟨ 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT (19)

+7-4314|0x⟩A⟨0x|A⊗|0x⟩B⟨0x|B,tensor-product74314subscriptketsubscript0𝑥𝐴subscriptbrasubscript0𝑥𝐴subscriptketsubscript0𝑥𝐵subscriptbrasubscript0𝑥𝐵\displaystyle+\tfrac7-4\sqrt314|0_x\rangle_A\langle 0_x|_A% \otimes|0_x\rangle_B\langle 0_x|_B,+ divide start_ARG 7 - 4 square-root start_ARG 3 end_ARG end_ARG start_ARG 14 end_ARG | 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟨ 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ⟨ 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , in which |1x⟩=12(|1⟩+|0⟩)ketsubscript1𝑥12ket1ket0|1_x\rangle=\tfrac1\sqrt2(|1\rangle+|0\rangle)| 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 1 ⟩ + | 0 ⟩ ) and |0x⟩=12(|1⟩-|0⟩)ketsubscript0𝑥12ket1ket0|0_x\rangle=\tfrac1\sqrt2(|1\rangle-|0\rangle)| 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 1 ⟩ - | 0 ⟩ ). The pisubscript𝑝𝑖p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT can be determined easily by Eq. (6). So DH=1-78superscript𝐷𝐻178D^H=1-\sqrt\tfrac78italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - square-root start_ARG divide start_ARG 7 end_ARG start_ARG 8 end_ARG end_ARG.

Then we obtain the first conclusion

Conclusion 1

For arbitrary pure bipartite state, which has Schmidt decomposition

|ψ⟩AB=∑ncn|n⟩A|n~⟩B,subscriptket𝜓𝐴𝐵subscript𝑛subscript𝑐𝑛subscriptket𝑛𝐴subscriptket~𝑛𝐵\displaystyle|\psi\rangle_AB=\sum_nc_n|n\rangle_A|\tilden\rangle_B,| italic_ψ ⟩ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_n ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | over~ start_ARG italic_n end_ARG ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , the ”nearest” completely classical state σ𝜎\sigmaitalic_σ can be written as

σ=1c∑n|cn|4|n⟩A⟨n|A⊗|n~⟩B⟨n~|B,𝜎1𝑐subscript𝑛tensor-productsuperscriptsubscript𝑐𝑛4subscriptket𝑛𝐴subscriptbra𝑛𝐴subscriptket~𝑛𝐵subscriptbra~𝑛𝐵\displaystyle\sigma=\frac1c\sum_n\left|c_n\right|^4|n\rangle_A% \langle n|_A\otimes|\tilden\rangle_B\langle\tilden|_B,italic_σ = divide start_ARG 1 end_ARG start_ARG italic_c end_ARG ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT | italic_n ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟨ italic_n | start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | over~ start_ARG italic_n end_ARG ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ⟨ over~ start_ARG italic_n end_ARG | start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , (20) in which c=∑n|cn|4𝑐subscript𝑛superscriptsubscript𝑐𝑛4c=\sum_n\left|c_n\right|^4italic_c = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_c start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT. Then

DH=1-csuperscript𝐷𝐻1𝑐\displaystyle D^H=1-\sqrtcitalic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - square-root start_ARG italic_c end_ARG (21)

-Mixed state- For ρ=∑kλk|ϕk⟩⟨ϕk|𝜌subscript𝑘subscript𝜆𝑘ketsubscriptitalic-ϕ𝑘brasubscriptitalic-ϕ𝑘\rho=\sum_k\lambda_k|\phi_k\rangle\langle\phi_k|italic_ρ = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩ ⟨ italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT |, in general these pure states |ϕk⟩ketsubscriptitalic-ϕ𝑘|\phi_k\rangle| italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩ do not share the same Schmidt basis. Thus we cannot find a general method to determine DHsuperscript𝐷𝐻D^Hitalic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT. However an exceptional case is that the density matrix shows ”X” form

ρX=(ρ110⋯⋯⋯0ρ1n0ρ220⋯0ρ2(n-1)0⋮⋱⋮⋮⋱⋮0ρ(n-1)20⋯0ρ(n-1)(n-1)0ρn10⋯⋯⋯0ρnn),subscript𝜌𝑋subscript𝜌110⋯⋯⋯0subscript𝜌1𝑛0subscript𝜌220⋯0subscript𝜌2𝑛10⋮missing-subexpression⋱missing-subexpressionmissing-subexpressionmissing-subexpression⋮⋮missing-subexpressionmissing-subexpressionmissing-subexpression⋱missing-subexpression⋮0subscript𝜌𝑛120⋯0subscript𝜌𝑛1𝑛10subscript𝜌𝑛10⋯⋯⋯0subscript𝜌𝑛𝑛\displaystyle\rho_X=\left(\beginarray[]ccccccc\rho_11&0&\cdots&\cdots&% \cdots&0&\rho_1n\\ 0&\rho_22&0&\cdots&0&\rho_2(n-1)&0\\ \vdots&&\ddots&&\beginrotate90.0$\ddots$\endrotate&&\vdots\\ \vdots&&\beginrotate90.0$\ddots$\endrotate&&\ddots&&\vdots\\ 0&\rho_(n-1)2&0&\cdots&0&\rho_(n-1)(n-1)&0\\ \rho_n1&0&\cdots&\cdots&\cdots&0&\rho_nn\endarray\right),italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL ⋯ end_CELL start_CELL ⋯ end_CELL start_CELL ⋯ end_CELL start_CELL 0 end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 1 italic_n end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL ⋯ end_CELL start_CELL 0 end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 2 ( italic_n - 1 ) end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋱ end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL ⋱ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT ( italic_n - 1 ) 2 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL ⋯ end_CELL start_CELL 0 end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT ( italic_n - 1 ) ( italic_n - 1 ) end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT italic_n 1 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL ⋯ end_CELL start_CELL ⋯ end_CELL start_CELL ⋯ end_CELL start_CELL 0 end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT italic_n italic_n end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) , (28) which actually is the direct sum of sub-matrices. It should be pointed out that the basis of Eq. (28) is not necessarily the computational basis. The only restriction is that the basis states for any sub-matrix are orthogonormlized as Schmidt form. Thus the eigenstates of sub-matrix are Schmidt-typed in their own form. More importantly since the basis states for different sub-matrices have no overlap, then the ”nearest” σ𝜎\sigmaitalic_σ is necessarily a mixed combination of the basis states in all sub-matrices.

As an example, we try to find DHsuperscript𝐷𝐻D^Hitalic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT for Werner state

ρW=1-r4I+r|ψ+⟩⟨ψ+|,subscript𝜌𝑊1𝑟4𝐼𝑟ketsuperscript𝜓brasuperscript𝜓\displaystyle\rho_W=\tfrac1-r4I+r|\psi^+\rangle\langle\psi^+|,italic_ρ start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT = divide start_ARG 1 - italic_r end_ARG start_ARG 4 end_ARG italic_I + italic_r | italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ ⟨ italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | , (29) in which r∈[0,1]𝑟01r\in[0,1]italic_r ∈ [ 0 , 1 ] and |ψ+⟩=12(|10⟩+|01⟩)ketsuperscript𝜓12ket10ket01|\psi^+\rangle=\tfrac1\sqrt2(|10\rangle+|01\rangle)| italic_ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 10 ⟩ + | 01 ⟩ ). It is obvious that ρWsubscript𝜌𝑊\rho_Witalic_ρ start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT has a ”X” form on the basis 11⟩,ket11ket10ket01ket00\left\11\rangle, 11 ⟩ ,

ρW=14(1-r00001+r2r002r1+r00001-r),subscript𝜌𝑊141𝑟00001𝑟2𝑟002𝑟1𝑟00001𝑟\displaystyle\rho_W=\frac14\left(\beginarray[]cccc1-r&0&0&0\\ 0&1+r&2r&0\\ 0&2r&1+r&0\\ 0&0&0&1-r\endarray\right),italic_ρ start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( start_ARRAY start_ROW start_CELL 1 - italic_r end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 + italic_r end_CELL start_CELL 2 italic_r end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 2 italic_r end_CELL start_CELL 1 + italic_r end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 1 - italic_r end_CELL end_ROW end_ARRAY ) , (34) that generally is direct sum of two sub-matrices

ρ1=14(1-r001-r);ρ2=14(1+r2r2r1+r),formulae-sequencesubscript𝜌1141𝑟001𝑟subscript𝜌2141𝑟2𝑟2𝑟1𝑟\displaystyle\rho_1=\tfrac14\left(\beginarray[]cc1-r&0\\ 0&1-r\endarray\right);\rho_2=\tfrac14\left(\beginarray[]cc1+r&2r\\ 2r&1+r\endarray\right),italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( start_ARRAY start_ROW start_CELL 1 - italic_r end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 - italic_r end_CELL end_ROW end_ARRAY ) ; italic_ρ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( start_ARRAY start_ROW start_CELL 1 + italic_r end_CELL start_CELL 2 italic_r end_CELL end_ROW start_ROW start_CELL 2 italic_r end_CELL start_CELL 1 + italic_r end_CELL end_ROW end_ARRAY ) , (39) defined on the basis ket11ket00\left\00\rangle\right\ and ket10ket01\left\01\rangle\right\ 10 ⟩ , respectively. Then there are four eigenstates

|1⟩ket1\displaystyle|1\rangle| 1 ⟩ =\displaystyle== |11⟩;|2⟩=|00⟩ket11ket2ket00\displaystyle|11\rangle;|2\rangle=|00\rangle| 11 ⟩ ; | 2 ⟩ = | 00 ⟩

By Eq.(10), one obtains

maxTr[ρWσ]=2maxc02+c12Trdelimited-[]subscript𝜌𝑊𝜎2superscriptsubscript𝑐02superscriptsubscript𝑐12\displaystyle\max\textTr\left[\sqrt\rho_W\sqrt\sigma\right]=2\max\sqrt% c_0^2+c_1^2roman_max Tr [ square-root start_ARG italic_ρ start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT end_ARG square-root start_ARG italic_σ end_ARG ] = 2 roman_max square-root start_ARG italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG (41) in which c0=(31-r+1+3r)/8subscript𝑐031𝑟13𝑟8c_0=\left(3\sqrt1-r+\sqrt1+3r\right)/8italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( 3 square-root start_ARG 1 - italic_r end_ARG + square-root start_ARG 1 + 3 italic_r end_ARG ) / 8, c1=cosΩ(1-r-1+3r)/8subscript𝑐1Ω1𝑟13𝑟8c_1=\cos\Omega\left(\sqrt1-r-\sqrt1+3r\right)/8italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = roman_cos roman_Ω ( square-root start_ARG 1 - italic_r end_ARG - square-root start_ARG 1 + 3 italic_r end_ARG ) / 8 and cosΩ=cosθ1cosθ2-sinθ1sinθ2cos(ϕ1-ϕ2)Ωsubscript𝜃1subscript𝜃2subscript𝜃1subscript𝜃2subscriptitalic-ϕ1subscriptitalic-ϕ2\cos\Omega=\cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2\cos(\phi_% 1-\phi_2)roman_cos roman_Ω = roman_cos italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - roman_sin italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_cos ( italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ). Then one has DH=1-123-r+(1-r)(1+3r)superscript𝐷𝐻1123𝑟1𝑟13𝑟D^H=1-\tfrac12\sqrt3-r+\sqrt(1-r)(1+3r)italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - divide start_ARG 1 end_ARG start_ARG 2 end_ARG square-root start_ARG 3 - italic_r + square-root start_ARG ( 1 - italic_r ) ( 1 + 3 italic_r ) end_ARG end_ARG when cosΩ=±1Ωplus-or-minus1\cos\Omega=\pm 1roman_cos roman_Ω = ± 1, which can occur, for example if θ1=θ2=0subscript𝜃1subscript𝜃20\theta_1=\theta_2=0italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 and ϕ1=ϕ2subscriptitalic-ϕ1subscriptitalic-ϕ2\phi_1=\phi_2italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The corresponding ”nearest” σ𝜎\sigmaitalic_σ has the form

σ𝜎\displaystyle\sigmaitalic_σ =\displaystyle== 121+r+(1-r)(1+3r)3-r+(1-r)(1+3r)(|10⟩⟨10|+|01⟩⟨01|)121𝑟1𝑟13𝑟3𝑟1𝑟13𝑟ket10bra10ket01bra01\displaystyle\tfrac12\tfrac1+r+\sqrt(1-r)(1+3r)3-r+\sqrt(1-r)(1+3r)% \left(|10\rangle\langle 10|+|01\rangle\langle 01|\right)divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG 1 + italic_r + square-root start_ARG ( 1 - italic_r ) ( 1 + 3 italic_r ) end_ARG end_ARG start_ARG 3 - italic_r + square-root start_ARG ( 1 - italic_r ) ( 1 + 3 italic_r ) end_ARG end_ARG ( | 10 ⟩ ⟨ 10 | + | 01 ⟩ ⟨ 01 | ) (42)

+1-r3-r+(1-r)(1+3r)(|11⟩⟨11|+|00⟩⟨00|),1𝑟3𝑟1𝑟13𝑟ket11bra11ket00bra00\displaystyle+\tfrac1-r3-r+\sqrt(1-r)(1+3r)\left(|11\rangle\langle 11|+|% 00\rangle\langle 00|\right),+ divide start_ARG 1 - italic_r end_ARG start_ARG 3 - italic_r + square-root start_ARG ( 1 - italic_r ) ( 1 + 3 italic_r ) end_ARG end_ARG ( | 11 ⟩ ⟨ 11 | + | 00 ⟩ ⟨ 00 | ) , which just is a mixedness of the Schmidt basis of the two sub-matrices. It should be pointed out the the choice of values of θ1(2)subscript𝜃12\theta_1(2)italic_θ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT and ϕ1(2)subscriptitalic-ϕ12\phi_1(2)italic_ϕ start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT is not unique.

Although the simplicity of the example, we can obtain the second conclusion in this article

Conclusion 2

p For a X𝑋Xitalic_X-type density operator Eq. (28) the ”nearest” completely classical state σ𝜎\sigmaitalic_σ is necessarily the mixed combination of the basis of the density matrix. The probability is determined by the eigenstates of ρXsubscript𝜌𝑋\rho_Xitalic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT, as shown in Eq.(11).

IV Multipartite state: Symmetric case

The definition Eq.(1) can be applied for multipartite states just by generalizing σ𝜎\sigmaitalic_σ into multipartite case. However since the absence of generalized Schmidt decomposition in this case, we focus on the symmetric states instead, which is invariant under the permutation of arbitrary two single-party states or under cyclic translation of single-party states. We will show by exemplifications in qubit case that the ”nearest” σ𝜎\sigmaitalic_σ necessarily displays the same symmetry of the state. Moreover σ𝜎\sigmaitalic_σ can be readily determined by supposing θi=θ0subscript𝜃𝑖subscript𝜃0\theta_i=\theta_0italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and ϕi=ϕ0subscriptitalic-ϕ𝑖subscriptitalic-ϕ0\phi_i=\phi_0italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT for alli=1,2,3⋯N𝑖123⋯𝑁i=1,2,3\cdots Nitalic_i = 1 , 2 , 3 ⋯ italic_N.

IV.1 3-qubit case

∑n|⟨GHZ|σn⟩|4subscript𝑛superscriptinner-productGHZsubscript𝜎𝑛4\displaystyle\sum_n\left|\langle\textGHZ|\sigma_n\rangle\right|^4∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | ⟨ GHZ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT (43)

=\displaystyle== 18(1+cos2θ1cos2θ2+cos2θ1cos2θ3+cos2θ2cos2θ3fragments18fragments(1superscript2subscript𝜃1superscript2subscript𝜃2superscript2subscript𝜃1superscript2subscript𝜃3superscript2subscript𝜃2superscript2subscript𝜃3\displaystyle\frac18\left(1+\cos^2\theta_1\cos^2\theta_2+\cos^2% \theta_1\cos^2\theta_3+\cos^2\theta_2\cos^2\theta_3\right.divide start_ARG 1 end_ARG start_ARG 8 end_ARG ( 1 + roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT

+cos2(ϕ1+ϕ2+ϕ3)∏i=13sin2θi),fragmentssuperscript2fragments(subscriptitalic-ϕ1subscriptitalic-ϕ2subscriptitalic-ϕ3)superscriptsubscriptproduct𝑖13superscript2subscript𝜃𝑖),\displaystyle\left.+\cos^2(\phi_1+\phi_2+\phi_3)\prod_i=1^3\sin^2% \theta_i\right),+ roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , of which the maximal value is determined by relations

∂∑n|⟨GHZ|σn⟩|4∂ϕi=0⇒sin(ϕ1+ϕ2+ϕ3)=0subscript𝑛superscriptinner-productGHZsubscript𝜎𝑛4subscriptitalic-ϕ𝑖0⇒subscriptitalic-ϕ1subscriptitalic-ϕ2subscriptitalic-ϕ30\displaystyle\frac\partial\sum_n\left\partial\phi_i=0\Rightarrow\sin(\phi_1+\phi_2+\phi_3)=0divide start_ARG ∂ ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | ⟨ GHZ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = 0 ⇒ roman_sin ( italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = 0

∂∑n|⟨GHZ|σn⟩|4∂θi=0⇒sin2θi=0.subscript𝑛superscriptinner-productGHZsubscript𝜎𝑛4subscript𝜃𝑖0⇒2subscript𝜃𝑖0\displaystyle\frac^4\partial\theta_i=0\Rightarrow\sin 2\theta_i=0.divide start_ARG ∂ ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | ⟨ GHZ | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = 0 ⇒ roman_sin 2 italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 . (44) There are many choices for ϕisubscriptitalic-ϕ𝑖\phi_iitalic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and θisubscript𝜃𝑖\theta_iitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. One can choose ϕ1=ϕ2=ϕ3=0subscriptitalic-ϕ1subscriptitalic-ϕ2subscriptitalic-ϕ30\phi_1=\phi_2=\phi_3=0italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 and θ1=θ2=θ3=0subscript𝜃1subscript𝜃2subscript𝜃30\theta_1=\theta_2=\theta_3=0italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0. Then DH=1-12superscript𝐷𝐻112D^H=1-\tfrac1\sqrt2italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG and σ=12(|1⟩⟨1|⊗3+|0⟩⟨0|⊗3)𝜎12ket1superscriptbra1tensor-productabsent3ket0superscriptbra0tensor-productabsent3\sigma=\tfrac12\left(|1\rangle\langle 1|^\otimes 3+|0\rangle\langle 0|^% \otimes 3\right)italic_σ = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( | 1 ⟩ ⟨ 1 | start_POSTSUPERSCRIPT ⊗ 3 end_POSTSUPERSCRIPT + | 0 ⟩ ⟨ 0 | start_POSTSUPERSCRIPT ⊗ 3 end_POSTSUPERSCRIPT ). Except that θi=π/2subscript𝜃𝑖𝜋2\theta_i=\pi/2italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_π / 2 for arbitrary party i=1,2,3𝑖123i=1,2,3italic_i = 1 , 2 , 3, one can find the same result for the other choices of ϕisubscriptitalic-ϕ𝑖\phi_iitalic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and θisubscript𝜃𝑖\theta_iitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. It is also obvious that the ”nearest” σ𝜎\sigmaitalic_σ shares the same symmetry with GHZ state, which is also invariant by permutation.

∑n|⟨W|σn⟩|4subscript𝑛superscriptinner-product𝑊subscript𝜎𝑛4\displaystyle\sum_n\left|\langle W|\sigma_n\rangle\right|^4∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | ⟨ italic_W | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT (45)

=\displaystyle== 172(9+a2+b2+c2+d2+∑i=13cos2θi),1729superscript𝑎2superscript𝑏2superscript𝑐2superscript𝑑2superscriptsubscript𝑖13superscript2subscript𝜃𝑖\displaystyle\tfrac172\left(9+a^2+b^2+c^2+d^2+\sum_i=1^3\cos^% 2\theta_i\right),divide start_ARG 1 end_ARG start_ARG 72 end_ARG ( 9 + italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , in which

a𝑎\displaystyle aitalic_a =\displaystyle== 3cosθ1cosθ2cosθ3-2cosδϕ1sinθ1sinθ2cosθ33subscript𝜃1subscript𝜃2subscript𝜃32𝛿subscriptitalic-ϕ1subscript𝜃1subscript𝜃2subscript𝜃3\displaystyle 3\cos\theta_1\cos\theta_2\cos\theta_3-2\cos\delta\phi_1% \sin\theta_1\sin\theta_2\cos\theta_33 roman_cos italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 2 roman_cos italic_δ italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT

-2cosδϕ2sinθ1cosθ2sinθ32𝛿subscriptitalic-ϕ2subscript𝜃1subscript𝜃2subscript𝜃3\displaystyle-2\cos\delta\phi_2\sin\theta_1\cos\theta_2\sin\theta_3- 2 roman_cos italic_δ italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT

-2cosδϕ3cosθ1sinθ2sinθ3,2𝛿subscriptitalic-ϕ3subscript𝜃1subscript𝜃2subscript𝜃3\displaystyle-2\cos\delta\phi_3\cos\theta_1\sin\theta_2\sin\theta_3,- 2 roman_cos italic_δ italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,

b𝑏\displaystyle bitalic_b =\displaystyle== cosθ1cosθ2-2cosδϕ1sinθ1sinθ2,subscript𝜃1subscript𝜃22𝛿subscriptitalic-ϕ1subscript𝜃1subscript𝜃2\displaystyle\cos\theta_1\cos\theta_2-2\cos\delta\phi_1\sin\theta_1% \sin\theta_2,roman_cos italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - 2 roman_cos italic_δ italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ,

c𝑐\displaystyle citalic_c =\displaystyle== cosθ1cosθ3-2cosδϕ2sinθ1sinθ3,subscript𝜃1subscript𝜃32𝛿subscriptitalic-ϕ2subscript𝜃1subscript𝜃3\displaystyle\cos\theta_1\cos\theta_3-2\cos\delta\phi_2\sin\theta_1% \sin\theta_3,roman_cos italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 2 roman_cos italic_δ italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,

d𝑑\displaystyle ditalic_d =\displaystyle== cosθ2cosθ3-2cosδϕ3sinθ2sinθ3,subscript𝜃2subscript𝜃32𝛿subscriptitalic-ϕ3subscript𝜃2subscript𝜃3\displaystyle\cos\theta_2\cos\theta_3-2\cos\delta\phi_3\sin\theta_2% \sin\theta_3,roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - 2 roman_cos italic_δ italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,

δϕ1𝛿subscriptitalic-ϕ1\displaystyle\delta\phi_1italic_δ italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =\displaystyle== ϕ1-ϕ2;δϕ2=ϕ1-ϕ3;δϕ3=ϕ2-ϕ3.formulae-sequencesubscriptitalic-ϕ1subscriptitalic-ϕ2𝛿subscriptitalic-ϕ2subscriptitalic-ϕ1subscriptitalic-ϕ3𝛿subscriptitalic-ϕ3subscriptitalic-ϕ2subscriptitalic-ϕ3\displaystyle\phi_1-\phi_2;\delta\phi_2=\phi_1-\phi_3;\delta\phi_3% =\phi_2-\phi_3.italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; italic_δ italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ; italic_δ italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT . (46) It is not difficult to find that the extremal points appear when sinδi=0subscript𝛿𝑖0\sin\delta_i=0roman_sin italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0. Furthermore one notes that a,b,c𝑎𝑏𝑐a,b,citalic_a , italic_b , italic_c and d𝑑ditalic_d are invariant under transformation θi↔π-θi↔subscript𝜃𝑖𝜋subscript𝜃𝑖\theta_i\leftrightarrow\pi-\theta_iitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ↔ italic_π - italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Thus the extremal points occurs when θisubscript𝜃𝑖\theta_iitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT equals to 0,π0𝜋0,\pi0 , italic_π or π/2𝜋2\pi/2italic_π / 2 and an the meanwhile δisubscript𝛿𝑖\delta_iitalic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is 00 or π𝜋\piitalic_π. There are many choices for θisubscript𝜃𝑖\theta_iitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT’s and δisubscript𝛿𝑖\delta_iitalic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT’s. Our calculation shows that except that θisubscript𝜃𝑖\theta_iitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT equals to π/2𝜋2\pi/2italic_π / 2 for any i=1,2,3𝑖123i=1,2,3italic_i = 1 , 2 , 3, one can always obtain the extremal value for other possible choices of θisubscript𝜃𝑖\theta_iitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Then one can find the choice that θ1=θ2=θ3=0subscript𝜃1subscript𝜃2subscript𝜃30\theta_1=\theta_2=\theta_3=0italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 and ϕ1=ϕ2=ϕ3=0subscriptitalic-ϕ1subscriptitalic-ϕ2subscriptitalic-ϕ30\phi_1=\phi_2=\phi_3=0italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0. Thus DH=1-13superscript𝐷𝐻113D^H=1-\tfrac1\sqrt3italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - divide start_ARG 1 end_ARG start_ARG square-root start_ARG 3 end_ARG end_ARG and

σ𝜎\displaystyle\sigmaitalic_σ =\displaystyle== 13(|1⟩⟨1|⊗|0⟩⟨0|⊗|0⟩⟨0|+|0⟩⟨0|⊗|1⟩⟨1|⊗|0⟩⟨0|fragments13fragments(|1⟩fragments⟨1|tensor-product|0⟩fragments⟨0|tensor-product|0⟩fragments⟨0||0⟩fragments⟨0|tensor-product|1⟩fragments⟨1|tensor-product|0⟩fragments⟨0|\displaystyle\tfrac13\left(|1\rangle\langle 1|\otimes|0\rangle\langle 0|% \otimes|0\rangle\langle 0|+|0\rangle\langle 0|\otimes|1\rangle\langle 1|% \otimes|0\rangle\langle 0|\right.divide start_ARG 1 end_ARG start_ARG 3 end_ARG ( | 1 ⟩ ⟨ 1 | ⊗ | 0 ⟩ ⟨ 0 | ⊗ | 0 ⟩ ⟨ 0 | + | 0 ⟩ ⟨ 0 | ⊗ | 1 ⟩ ⟨ 1 | ⊗ | 0 ⟩ ⟨ 0 | (47)

+|0⟩⟨0|⊗|0⟩⟨0|⊗|1⟩⟨1|),fragments|0⟩⟨0|tensor-product|0⟩⟨0|tensor-product|1⟩⟨1|),\displaystyle\left.+|0\rangle\langle 0|\otimes|0\rangle\langle 0|\otimes|1% \rangle\langle 1|\right),+ | 0 ⟩ ⟨ 0 | ⊗ | 0 ⟩ ⟨ 0 | ⊗ | 1 ⟩ ⟨ 1 | ) , which obviously is also permutationally invariant.

IV.2 4-qubit case

As for 4-qubit states, there exist another symmetry besides of permutational invariance, named as translation symmetry. The definition of translation of state is similar to that in solid systems. However the difference is that it is defined for single-party state in Hilbert space, instead of single particle in real configuration cui . We will display by two exemplifications that the ”nearest” σ𝜎\sigmaitalic_σ for the state of translational invariance is necessary also translational invariant. And DHsuperscript𝐷𝐻D^Hitalic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT can also be determined readily by setting the parameters have the same value respectively, as shown in 3-qubit case.

-|𝐺𝐻𝑍1⟩4subscriptketsubscript𝐺𝐻𝑍14|\textGHZ_1\rangle_4| GHZ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT state-, which is defined as |GHZ1⟩4=12(|1010⟩+|0101⟩)subscriptketsubscriptGHZ1412ket1010ket0101|\textGHZ_1\rangle_4=\tfrac1\sqrt2(|1010\rangle+|0101\rangle)| GHZ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 1010 ⟩ + | 0101 ⟩ ). It is obvious that the state is actually constructed by cyclic permutation of 1010101010101010, which is named as cyclic unit. It is not difficult to find

∑n|⟨GHZ1|σn⟩4|4fragmentssubscript𝑛|subscriptfragments⟨subscriptGHZ1|subscript𝜎𝑛⟩4superscript|4\displaystyle\sum_n\left|_4\langle\textGHZ_1|\sigma_n\rangle\right% |^4∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_FLOATSUBSCRIPT 4 end_FLOATSUBSCRIPT ⟨ GHZ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT (48)

=\displaystyle== 116(1+a2+b2+c2+d2+e2+f2+g2),1161superscript𝑎2superscript𝑏2superscript𝑐2superscript𝑑2superscript𝑒2superscript𝑓2superscript𝑔2\displaystyle\tfrac116\left(1+a^2+b^2+c^2+d^2+e^2+f^2+g^2% \right),divide start_ARG 1 end_ARG start_ARG 16 end_ARG ( 1 + italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_f start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_g start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , in which

a𝑎\displaystyle aitalic_a =\displaystyle== ∏i=14cosθi+cos(ϕ1-ϕ2+ϕ3-ϕ4)∏i=14sinθi,superscriptsubscriptproduct𝑖14subscript𝜃𝑖subscriptitalic-ϕ1subscriptitalic-ϕ2subscriptitalic-ϕ3subscriptitalic-ϕ4superscriptsubscriptproduct𝑖14subscript𝜃𝑖\displaystyle\prod_i=1^4\cos\theta_i+\cos(\phi_1-\phi_2+\phi_3-% \phi_4)\prod_i=1^4\sin\theta_i,∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + roman_cos ( italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_ϕ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_ϕ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT roman_sin italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,

b𝑏\displaystyle bitalic_b =\displaystyle== cosθ1cosθ2;c=cosθ1cosθ3;d=cosθ1cosθ4,formulae-sequencesubscript𝜃1subscript𝜃2𝑐subscript𝜃1subscript𝜃3𝑑subscript𝜃1subscript𝜃4\displaystyle\cos\theta_1\cos\theta_2;c=\cos\theta_1\cos\theta_3;d=% \cos\theta_1\cos\theta_4,roman_cos italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ; italic_c = roman_cos italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ; italic_d = roman_cos italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ,

e𝑒\displaystyle eitalic_e =\displaystyle== cosθ2cosθ3;f=cosθ2cosθ4;g=cosθ3cosθ4.formulae-sequencesubscript𝜃2subscript𝜃3𝑓subscript𝜃2subscript𝜃4𝑔subscript𝜃3subscript𝜃4\displaystyle\cos\theta_2\cos\theta_3;f=\cos\theta_2\cos\theta_4;g=% \cos\theta_3\cos\theta_4.roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ; italic_f = roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ; italic_g = roman_cos italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_cos italic_θ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT . It is not difficult to find that the overlap has maximal value 1/2121/21 / 2 and then DH=1-1/2superscript𝐷𝐻112D^H=1-1/\sqrt2italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 - 1 / square-root start_ARG 2 end_ARG when cosθi=±1(i=1,2,3,4)subscript𝜃𝑖plus-or-minus1𝑖1234\cos\theta_i=\pm 1(i=1,2,3,4)roman_cos italic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ± 1 ( italic_i = 1 , 2 , 3 , 4 ). ϕisubscriptitalic-ϕ𝑖\phi_iitalic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT’s can be any values and are set to be zero for simplicity. Consequently the ”nearest” σ=12(|1010⟩⟨1010|+|0101⟩⟨0101|)𝜎12ket1010bra1010ket0101bra0101\sigma=\tfrac12(|1010\rangle\langle 1010|+|0101\rangle\langle 0101|)italic_σ = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( | 1010 ⟩ ⟨ 1010 | + | 0101 ⟩ ⟨ 0101 | ), which is also translational invariant with the same cyclic unit to that of |GHZ1⟩4subscriptketsubscriptGHZ14|\textGHZ_1\rangle_4| GHZ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT.

-|W2⟩4statesubscriptketsubscript𝑊24𝑠𝑡𝑎𝑡𝑒|W_2\rangle_4state| italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT italic_s italic_t italic_a italic_t italic_e-, which is defined as

|W2⟩4=12(|1100⟩+|0110⟩+|0011⟩+|1001⟩).subscriptketsubscript𝑊2412ket1100ket0110ket0011ket1001\displaystyle|W_2\rangle_4=\tfrac12\left(|1100\rangle+|0110\rangle+|00% 11\rangle+|1001\rangle\right).| italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( | 1100 ⟩ + | 0110 ⟩ + | 0011 ⟩ + | 1001 ⟩ ) . (49) The state is actually constructed by cyclic unit 1100110011001100. Moreover it is bi-seperable since |W2⟩4=12(|10⟩+|01⟩)13⊗12(|10⟩+|01⟩)24subscriptketsubscript𝑊24tensor-product12subscriptket10ket011312subscriptket10ket0124|W_2\rangle_4=\tfrac1\sqrt2\left(|10\rangle+|01\rangle\right)_13% \otimes\tfrac1\sqrt2\left(|10\rangle+|01\rangle\right)_24| italic_W start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 10 ⟩ + | 01 ⟩ ) start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT ⊗ divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 10 ⟩ + | 01 ⟩ ) start_POSTSUBSCRIPT 24 end_POSTSUBSCRIPT. Thus the ”nearest” σ𝜎\sigmaitalic_σ can also be constructed by two parts, i.e. σ=σ13⊗σ24𝜎tensor-productsubscript𝜎13subscript𝜎24\sigma=\sigma_13\otimes\sigma_24italic_σ = italic_σ start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 24 end_POSTSUBSCRIPT, in which σ13subscript𝜎13\sigma_13italic_σ start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT and σ24subscript𝜎24\sigma_24italic_σ start_POSTSUBSCRIPT 24 end_POSTSUBSCRIPT are the ”nearest” completely classical states for 12(|10⟩+|01⟩)1312subscriptket10ket0113\tfrac1\sqrt2\left(|10\rangle+|01\rangle\right)_13divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 10 ⟩ + | 01 ⟩ ) start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT and 12(|10⟩+|01⟩)2412subscriptket10ket0124\tfrac1\sqrt2\left(|10\rangle+|01\rangle\right)_24divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 10 ⟩ + | 01 ⟩ ) start_POSTSUBSCRIPT 24 end_POSTSUBSCRIPT respectively. By Conclusion 1, one can obtain DH=1/2superscript𝐷𝐻12D^H=1/2italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT = 1 / 2 and

σ𝜎\displaystyle\sigmaitalic_σ =\displaystyle== 12(|10⟩⟨10|+|01⟩⟨01|)13⊗12(|10⟩⟨10|+|01⟩⟨01|)24tensor-product12subscriptket10bra10ket01bra011312subscriptket10bra10ket01bra0124\displaystyle\tfrac12\left(|10\rangle\langle 10|+|01\rangle\langle 01|% \right)_13\otimes\tfrac12\left(|10\rangle\langle 10|+|01\rangle\langle 0% 1|\right)_24divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( | 10 ⟩ ⟨ 10 | + | 01 ⟩ ⟨ 01 | ) start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT ⊗ divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( | 10 ⟩ ⟨ 10 | + | 01 ⟩ ⟨ 01 | ) start_POSTSUBSCRIPT 24 end_POSTSUBSCRIPT (50)

=\displaystyle== 14(|1100⟩⟨1100|+|0110⟩⟨0110|+|0011⟩⟨0011|fragments14fragments(|1100⟩fragments⟨1100||0110⟩fragments⟨0110||0011⟩fragments⟨0011|\displaystyle\tfrac14\left(|1100\rangle\langle 1100|+|0110\rangle\langle 0% 110|+|0011\rangle\langle 0011|\right.divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( | 1100 ⟩ ⟨ 1100 | + | 0110 ⟩ ⟨ 0110 | + | 0011 ⟩ ⟨ 0011 |

+|1001⟩⟨1001|),fragments|1001⟩⟨1001|),\displaystyle\left.+|1001\rangle\langle 1001|\right),+ | 1001 ⟩ ⟨ 1001 | ) , which is obviously translationally invariant.

IV.3 A short discussion

By the previous exemplifications, we can obtain the third conclusion

Conclusion 3

For multipartite state with permutational or translational invariance, the ”nearest” σ𝜎\sigmaitalic_σ necessarily has the same invariance, which can be determined by setting θisubscript𝜃𝑖\theta_iitalic_θ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in σ𝜎\sigmaitalic_σ to be the same, so do for ϕisubscriptitalic-ϕ𝑖\phi_iitalic_ϕ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT.

It should be pointed out that the form of σ𝜎\sigmaitalic_σ is not necessary to comply with the superposition terms in the states. As an example, we try to find the σ𝜎\sigmaitalic_σ for Dicke state |4,2⟩=16∑perm.|1100⟩ket4216subscriptperm.ket1100|4,2\rangle=\tfrac1\sqrt6\sum_\textperm.|1100\rangle| 4 , 2 ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 6 end_ARG end_ARG ∑ start_POSTSUBSCRIPT perm. end_POSTSUBSCRIPT | 1100 ⟩, which is the equally weighted sums of all permutations of computational basis states with two qubits being |1⟩ket1|1\rangle| 1 ⟩ and two qubits being |0⟩ket0|0\rangle| 0 ⟩. By explicit calculation, one obtain

∑n|⟨4,2|σn⟩|4subscript𝑛superscriptinner-product42subscript𝜎𝑛4\displaystyle\sum_n\left|\langle 4,2|\sigma_n\rangle\right|^4∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | ⟨ 4 , 2 | italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟩ | start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT (51)

=\displaystyle== 217-96cos(2θ)+108cos(4θ)+27cos(8θ)1536,217962𝜃1084𝜃278𝜃1536\displaystyle\frac217-96\cos(2\theta)+108\cos(4\theta)+27\cos(8\theta)1536,divide start_ARG 217 - 96 roman_cos ( 2 italic_θ ) + 108 roman_cos ( 4 italic_θ ) + 27 roman_cos ( 8 italic_θ ) end_ARG start_ARG 1536 end_ARG , which obviously has maximal value when θ=π/2𝜃𝜋2\theta=\pi/2italic_θ = italic_π / 2. Then DH≈0.46superscript𝐷𝐻0.46D^H\approx 0.46italic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT ≈ 0.46 and

σ𝜎\displaystyle\sigmaitalic_σ =\displaystyle== 0.482(|1x⟩⟨1x|⊗4+|0x⟩⟨0x|⊗4)0.482ketsubscript1𝑥superscriptbrasubscript1𝑥tensor-productabsent4ketsubscript0𝑥superscriptbrasubscript0𝑥tensor-productabsent4\displaystyle 0.482\left(|1_x\rangle\langle 1_x|^\otimes 4+|0_x\rangle% \langle 0_x|^\otimes 4\right)0.482 ( | 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ ⟨ 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT ⊗ 4 end_POSTSUPERSCRIPT + | 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ ⟨ 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT ⊗ 4 end_POSTSUPERSCRIPT ) (52)

+0.006(∑perm|1x⟩⟨1x|⊗2|0x⟩⟨0x|⊗2),0.006subscriptpermketsubscript1𝑥superscriptbrasubscript1𝑥tensor-productabsent2ketsubscript0𝑥superscriptbrasubscript0𝑥tensor-productabsent2\displaystyle+0.006\left(\sum_\textperm|1_x\rangle\langle 1_x|^% \otimes 2|0_x\rangle\langle 0_x|^\otimes 2\right),+ 0.006 ( ∑ start_POSTSUBSCRIPT perm end_POSTSUBSCRIPT | 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ ⟨ 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT ⊗ 2 end_POSTSUPERSCRIPT | 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ ⟨ 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT ⊗ 2 end_POSTSUPERSCRIPT ) , in which |1x⟩=12(|1⟩+|0⟩)ketsubscript1𝑥12ket1ket0|1_x\rangle=\tfrac1\sqrt2(|1\rangle+|0\rangle)| 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 1 ⟩ + | 0 ⟩ ) and |0x⟩=12(|1⟩-|0⟩)ketsubscript0𝑥12ket1ket0|0_x\rangle=\tfrac1\sqrt2(|1\rangle-|0\rangle)| 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 1 ⟩ - | 0 ⟩ ). The second term is the equally weighted sums of all permutations of the density operators with two qubits being |1x⟩ketsubscript1𝑥|1_x\rangle| 1 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩ and the other two being |0x⟩ketsubscript0𝑥|0_x\rangle| 0 start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⟩. It is obvious that σ𝜎\sigmaitalic_σ displays more complex form than that of |4,2⟩ket42|4,2\rangle| 4 , 2 ⟩.

As for the mixed case, the similar conclusion can be obtained since its eigenstates are also symmetric. A simple example is the discussion for Werner state ρWsubscript𝜌𝑊\rho_Witalic_ρ start_POSTSUBSCRIPT italic_W end_POSTSUBSCRIPT in Sec. Discord server III.

V DHsuperscript𝐷𝐻D^Hitalic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT and quantum phase transition in many-body system

In this section, we show that DHsuperscript𝐷𝐻D^Hitalic_D start_POSTSUPERSCRIPT italic_H end_POSTSUPERSCRIPT can also mark the quantum phase transition in many-body systems. Discord server For clarity and simplicity, this discussion focuses on two popular models, Lipkin-Meshkov-Glick (LMG) lmg and Dicke models dicke , of which the ground states can be determined analytically and additionally both are permutationally invariant.

V.1 Lipkin-Meshkov-Glick model

The LMG model describes a set of spin-half particles coupled to all others with an interaction independent of the position and the nature of the elements. The Hamiltonian can be written as

H=-λN(Sx2+γSy2)-hzSz,𝐻𝜆𝑁subscriptsuperscript𝑆2𝑥𝛾subscriptsuperscript𝑆2𝑦subscriptℎ𝑧subscript𝑆𝑧H=-\frac\lambdaN(S^2_x+\gamma S^2_y)-h_zS_z,italic_H = - divide start_ARG italic_λ end_ARG start_ARG italic_N end_ARG ( italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT + italic_γ italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) - italic_h start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT , (53) in which Sα=∑i=1Nσαi/2(α=x,y,z)subscript𝑆𝛼superscriptsubscript𝑖1𝑁subscriptsuperscript𝜎𝑖𝛼2𝛼𝑥𝑦𝑧S_\alpha=\sum_i=1^N\sigma^i_\alpha/2(\alpha=x,y,z)italic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT / 2 ( italic_α = italic_x , italic_y , italic_z ) and the σαsubscript𝜎𝛼\sigma_\alphaitalic_σ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT denotes the Pauli operator, and N𝑁Nitalic_N is the total particle number in this system. The prefactor 1/N1𝑁1/N1 / italic_N is essential to ensure the convergence of the free energy per spin in the thermodynamic limit. It is known that there is a second-order transition at h=hz/|λ|=1ℎsubscriptℎ𝑧𝜆1h=h_z/|\lambda|=1italic_h = italic_h start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT / | italic_λ | = 1 for the ferromagnetic case (λ>0𝜆0\lambda>0italic_λ >0) and a first-order one at h=0ℎ0h=0italic_h = 0 for the antiferromagnetic case (λ<0𝜆0\lambda<0italic_λ <0) botet ; vidal . The following discussion is divided into two parts by γ=1𝛾1\gamma=1italic_γ = 1 or not.

-γ=1𝛾1\gamma=1italic_γ = 1- In this case the model can be solved exactly, of which the eigenstate is |N/2,n⟩ket𝑁2𝑛|N/2,n\rangle| italic_N / 2 , italic_n ⟩, in which n𝑛nitalic_n denotes the quantum number of angular moment Szsubscript𝑆𝑧S_zitalic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, and the corresponding eigenenergy is En=-λ2(N2+1)+λNn2-hzn(ℏ=1)subscript𝐸𝑛𝜆2𝑁21𝜆𝑁superscript𝑛2subscriptℎ𝑧𝑛Planck-constant-over-2-pi1E_n=-\tfrac\lambda2(\tfracN2+1)+\tfrac\lambdaNn^2-h_zn(\hbar% =1)italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = - divide start_ARG italic_λ end_ARG start_ARG 2 end_ARG ( divide start_ARG italic_N end_ARG start_ARG 2 end_ARG + 1 ) + divide start_ARG italic_λ end_ARG start_ARG italic_N end_ARG italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_h start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT italic_n ( roman_ℏ = 1 ). For λ>0𝜆0\lambda>0italic_λ >0, the minimal value of Ensubscript𝐸𝑛E_nitalic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT appears when n=[hzλN2]𝑛delimited-[]subscriptℎ𝑧𝜆𝑁2n=\left[\tfrach_z\lambda\tfracN2\right]italic_n = [ divide start_ARG italic_h start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG start_ARG italic_λ end_ARG divide start_ARG italic_N end_ARG start_ARG 2 end_ARG ]. Then the ground state is |N/2,N/2⟩ket𝑁2𝑁2|N/2,N/2\rangle| italic_N / 2 , italic_N / 2 ⟩ for hz/λ>1subscriptℎ𝑧𝜆1h_z/\lambda>1italic_h start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT / italic_λ >1. As for hz/λ<1subscriptℎ𝑧𝜆1h_z/\lambda<1italic_h start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT / italic_λ <1, the ground state is |N/2,[hzλN2]⟩ket𝑁2delimited-[]subscriptℎ𝑧𝜆𝑁2|N/2,\left[\tfrach_z\lambda\tfracN2\right]\rangle| italic_N / 2 , [ divide start_ARG italic_h start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG start_ARG italic_λ end_ARG divide start_ARG italic_N end_ARG start_ARG 2 end_ARG ] ⟩, which actually is a Dicke state

|N,m⟩=1CmN∑perm|1⋯1⏟m0⋯0⏟N-m⟩,ket𝑁𝑚1superscriptsubscript𝐶𝑚𝑁subscriptpermketsubscript⏟1⋯1𝑚subscript⏟0⋯0𝑁𝑚\displaystyle|N,m\rangle=\tfrac1\sqrt^NC_m\sum_\textperm|% \underbrace1\cdots 1_m\underbrace0\cdots 0_N-m\rangle,| italic_N , italic_m ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG start_FLOATSUPERSCRIPT italic_N end_FLOATSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_ARG end_ARG ∑ start_POSTSUBSCRIPT perm end_POSTSUBSCRIPT | under⏟ start_ARG 1 ⋯ 1 end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT under⏟ start_ARG 0 ⋯ 0 end_ARG start_POSTSUBSCRIPT italic_N - italic_m end_POSTSUBSCRIPT ⟩ , (54) in which m=N2+[hzλN2]𝑚𝑁2delimited-[]subscriptℎ𝑧𝜆𝑁2m=\tfracN2+\left[\tfrach_z\lambda\tfracN2\right]italic_m = divide start_ARG italic_N end_ARG start_ARG 2 end_ARG + [ divide start_ARG italic_h start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG start_ARG italic_λ end_ARG divide start_ARG italic_N end_ARG start_ARG 2 end_ARG ]. Thus one has

DH=0,hz/λ>1;>0,0<hz λ<1.,superscript𝐷𝐻cases0subscriptℎ𝑧𝜆1absent00subscriptℎ𝑧𝜆1\displaystyle d^h="\left\\beginarray[]cc0,&h_z/\lambda">

1;\\ >0,&0

<italic_d start_postsuperscript italic_h end_postsuperscript="N/2,N/2⟩ket𝑁2𝑁2</body"></italic_d>

</hz>

A Generalized Geometric Measurement Of Quantum Discord Exact Treatment

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