Using and reusing coherence to realize quantum processes

TL;DR

This paper introduces a measure called the robustness of coherence for quantum channels, linking it to the resources needed for their implementation and demonstrating that even weakly coherent states can implement certain channels, with implications for quantum resource theories.

Contribution

It defines the robustness of coherence for quantum channels and establishes its operational significance in channel implementation and resource quantification.

Findings

Robustness of coherence quantifies minimal resource states for channels.

Logarithm of robustness measures implementation cost and asymptotic costs.

Any pure coherent state in dimension >2 can implement some coherent unitary channel.

Abstract

Coherent superposition is a key feature of quantum mechanics that underlies the advantage of quantum technologies over their classical counterparts. Recently, coherence has been recast as a resource theory in an attempt to identify and quantify it in an operationally well-defined manner. Here we study how the coherence present in a state can be used to implement a quantum channel via incoherent operations and, in turn, to assess its degree of coherence. We introduce the robustness of coherence of a quantum channel---which reduces to the homonymous measure for states when computed on constant-output channels---and prove that: i) it quantifies the minimal rank of a maximally coherent state required to implement the channel; ii) its logarithm quantifies the amortized cost of implementing the channel provided some coherence is recovered at the output; iii) its logarithm also quantifies the zero-error asymptotic cost of implementation of many independent copies of a channel. We also consider the generalized problem of imperfect implementation with arbitrary resource states. Using the robustness of coherence, we find that in general a quantum channel can be implemented without employing a maximally coherent resource state. In fact, we prove that \textit{every} pure coherent state in dimension larger than $2$, however weakly so, turns out to be a valuable resource to implement \textit{some} coherent unitary channel. We illustrate our findings for the case of single-qubit unitary channels.