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Open AccessArticle

Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence

School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, China
Author to whom correspondence should be addressed.
Entropy 2020, 22(3), 297;
Received: 18 January 2020 / Revised: 18 February 2020 / Accepted: 25 February 2020 / Published: 5 March 2020
(This article belongs to the Special Issue Quantum Information Processing)
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 1 -norm of coherence C 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 n-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 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 1 ( ρ ) for any state ρ . View Full-Text
Keywords: quantum coherence; measure; lower bound; upper bound quantum coherence; measure; lower bound; upper bound
MDPI and ACS Style

Zhang, C.; Guo, Z.; Cao, H. Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence. Entropy 2020, 22, 297.

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