Classifying coherence with a finite set of witnesses

TL;DR

This paper introduces a structured set of coherence witnesses for quantum states, providing necessary and sufficient conditions for detecting quantum coherence in finite-dimensional systems, and explicitly constructs a finite set that detects all coherent states.

Contribution

It defines a structured set of coherence witnesses and provides explicit finite sets capable of detecting all coherent states in quantum systems.

Findings

Necessary and sufficient conditions for coherence detection

Explicit finite set of witnesses for all coherent states

Detection of quantum coherence in finite-dimensional systems

Abstract

Coherence is a fundamental resource in quantum information processing, which can be certified by a coherence witness. In order to detect all the coherent states, we introduce a useful concept of coherence witness and structure the set of coherence witnesses $C^{d}_{[m,M]}$. We present necessary and sufficient conditions of detecting quantum coherence of $d-$dimensional quantum states based on $C^{d}_{[m,M]}$. Moreover, we show that each coherent state can be detected by one of the coherence witnesses in a finite set. The corresponding finite set of coherence witnesses is presented explicitly, which detects all the coherent states.