Preprint Article Version 1 This version is not peer-reviewed

# Symmetry-like Relation of Relative Entropy Measure of Quantum Coherence

Version 1 : Received: 30 January 2020 / Approved: 30 January 2020 / Online: 30 January 2020 (11:50:50 CET)

How to cite: Zhang, C.; Guo, Z.; Cao, H. Symmetry-like Relation of Relative Entropy Measure of Quantum Coherence. Preprints 2020, 2020010366 (doi: 10.20944/preprints202001.0366.v1). Zhang, C.; Guo, Z.; Cao, H. Symmetry-like Relation of Relative Entropy Measure of Quantum Coherence. Preprints 2020, 2020010366 (doi: 10.20944/preprints202001.0366.v1).

## Abstract

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. In this paper, we obtain a symmetry-like relation of relative entropy measure $C_r(\rho)$ of coherence for $n$-partite quantum states $\rho$, which gives lower and upper bounds for $C_r(\rho)$. Meanwhile, we discuss the conjecture about the validity of the inequality $C_r(\rho)\leq C_{\ell_1}(\rho)$ for any state $\rho$. We observe that every mixture $\eta$ of a state $\rho$ satisfying $C_r(\rho)\leq C_{\ell_1}(\rho)$ and any incoherent state $\sigma$ also satisfies the conjecture. We also note that if the von Neumann entropy is defined by the natural logarithm $\ln$ instead of $\log_2$, then the reduced relative entropy measure of coherence $\bar{C}_r(\rho)=-\rho_{\rm{diag}}\ln \rho_{\rm{diag}}+\rho\ln \rho$ satisfies the inequality ${\bar{C}}_r(\rho)\leq C_{\ell_1}(\rho)$ for any mixed state $\rho$.

## Subject Areas

quantum coherence; measure; lower bound; upper bound