Zhang Chengyang, Guo Zhihua, Cao Huaixin
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119, China.
Entropy (Basel). 2020 Mar 5;22(3):297. doi: 10.3390/e22030297.
Quantum coherence is an important physical resource in quantum information science, and also as one of the most fundamental and striking features in quantum physics. To quantify coherence, two proper measures were introduced in the literature, the one is the relative entropy of coherence C r ( ρ ) = S ( ρ diag ) - S ( ρ ) and the other is the ℓ 1 -norm of coherence C ℓ 1 ( ρ ) = ∑ i ≠ j | ρ i j | . In this paper, we obtain a symmetry-like relation of relative entropy measure C r ( ρ A 1 A 2 ⋯ A n ) of coherence for an -partite quantum states ρ A 1 A 2 ⋯ A n , which gives lower and upper bounds for C r ( ρ ) . As application of our inequalities, we conclude that when each reduced states ρ A i is pure, ρ A 1 ⋯ A n is incoherent if and only if the reduced states ρ A i and tr A i ρ A 1 ⋯ A n ( i = 1 , 2 , … , n ) are all incoherent. Meanwhile, we discuss the conjecture that C r ( ρ ) ≤ C ℓ 1 ( ρ ) for any state ρ , which was proved to be valid for any mixed qubit state and any pure state, and open for a general state. We observe that every mixture η of a state ρ satisfying the conjecture with any incoherent state σ also satisfies the conjecture. We also observe that when the von Neumann entropy is defined by the natural logarithm ln instead of log 2 , the reduced relative entropy measure of coherence C ¯ r ( ρ ) = - ρ diag ln ρ diag + ρ ln ρ satisfies the inequality C ¯ r ( ρ ) ≤ C ℓ 1 ( ρ ) for any state ρ .
量子相干是量子信息科学中一种重要的物理资源，也是量子物理学中最基本、最显著的特征之一。为了量化相干性，文献中引入了两种合适的度量，一种是相干性的相对熵(C_r(\rho)=S(\rho_{diag}) - S(\rho))，另一种是相干性的(\ell_1)范数(C_{\ell_1}(\rho)=\sum_{i\neq j}|\rho_{ij}|)。在本文中，我们得到了关于(n)部量子态(\rho_{A_1A_2\cdots A_n})的相干性相对熵度量(C_r(\rho_{A_1A_2\cdots A_n}))的一个类似对称的关系，它给出了(C_r(\rho))的上下界。作为我们不等式的应用，我们得出当每个约化态(\rho_{A_i})是纯态时，(\rho_{A_1\cdots A_n})是非相干的当且仅当约化态(\rho_{A_i})和(\text{tr}{A_i}\rho{A_1\cdots A_n}(i = 1, 2, \cdots, n))都是非相干的。同时，我们讨论了对于任意态(\rho)，(C_r(\rho)\leq C_{\ell_1}(\rho))的猜想，该猜想已被证明对于任意混合量子比特态和任意纯态是成立的，而对于一般态仍然是开放的。我们观察到，态(\rho)与任意非相干态(\sigma)的每个混合态(\eta)，如果(\rho)满足该猜想，那么(\eta)也满足该猜想。我们还观察到，当冯·诺依曼熵由自然对数(\ln)而不是(\log_2)定义时，相干性的约化相对熵度量(\overline{C}r(\rho)=-\rho{diag}\ln\rho_{diag}+\rho\ln\rho)对于任意态(\rho)满足不等式(\overline{C}r(\rho)\leq C{\ell_1}(\rho))。
