Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence.

Chengyang Zhang,Zhihua Guo,Huaixin Cao

doi:10.3390/e22030297

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

Highlights

Quantum computing utilizes the superposition and entanglement of quantum states to operate and process information
Its most significant advantage lies in the parallelism of operations [1,2,3]
Quantum coherence arising from quantum superposition plays a central role in quantum mechanics and so becomes an important physical resource in quantum information and quantum computation [4]

Summary

Introduction

Quantum computing utilizes the superposition and entanglement of quantum states to operate and process information. In 2014, Baumgratz et al [15] proposed a framework to quantify coherence In their seminal work, conditions that a suitable measure of coherence should satisfy have been put forward, including nonnegativity, the monotonicity under incoherent completely positive and trace preserving operations, the monotonicity under selective incoherent operations on average and the convexity under mixing of states. Conditions that a suitable measure of coherence should satisfy have been put forward, including nonnegativity, the monotonicity under incoherent completely positive and trace preserving operations, the monotonicity under selective incoherent operations on average and the convexity under mixing of states By introducing such a rigorous theoretical framework, a mass of properties and operations of quantification of coherence were discussed.

Preliminaries

Lower and Upper Bounds for the Relative Entropy of Coherence

Conclusions