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 [Formula: see text] and the other is the [Formula: see text]-norm of coherence [Formula: see text]. In this paper, we obtain a symmetry-like relation of relative entropy measure [Formula: see text] of coherence for an n-partite quantum states [Formula: see text] , which gives lower and upper bounds for [Formula: see text]. As application of our inequalities, we conclude that when each reduced states [Formula: see text] is pure, [Formula: see text] is incoherent if and only if the reduced states [Formula: see text] and [Formula: see text] are all incoherent. Meanwhile, we discuss the conjecture that [Formula: see text] for any state [Formula: see text] , 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 [Formula: see text] of a state [Formula: see text] satisfying the conjecture with any incoherent state [Formula: see text] also satisfies the conjecture. We also observe that when the von Neumann entropy is defined by the natural logarithm ln instead of [Formula: see text] , the reduced relative entropy measure of coherence [Formula: see text] satisfies the inequality [Formula: see text] for any state [Formula: see text].