The resource theory of coherence for quantum channels

TL;DR

This paper introduces the quantum-incoherent relative entropy of coherence ($ ext{QI}$ REC) for quantum channels, establishing its properties, relationships with quantum discord, and its role as a measure of quantumness in the resource theory framework.

Contribution

It defines the $ ext{QI}$ REC for quantum channels using Choi-Jamiolkowsky isomorphism and explores its properties, including its relation to quantum discord and coherence-breaking channels.

Findings

$ ext{QI}$ REC is decreasing for divisible incoherent channels.

Quantum asymmetric discord is bounded above by $ ext{QI}$ REC.

For qubit channels, REC of channels equals REC of Choi states.

Abstract

We define the quantum-incoherent relative entropy of coherence ($\mathcal{QI}$ REC) of quantum channels in the framework of the resource theory by using the Choi-Jamiolkowsky isomorphism. Coherence-breaking channels are introduced as free operations and their corresponding Choi states as free states. We also show the relationship between the coherence of channel and the quantum discord and find that basis-dependent quantum asymmetric discord can never be more than the $\mathcal{QI}$ REC for any quantum channels. {Also}, we prove the $\mathcal{QI}$ REC is decreasing for any divisible quantum incoherent channel and we also claim it can be considered as the quantumness of quantum channels. Moreover, we demonstrate that for qubit channels, the relative entropy of coherence (REC) can be equivalent to the REC of their corresponding Choi states and the basis-dependent quantum symmetric discord can never exceed the coherence.