Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence

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 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 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 .