# Resource Theory of Coherence - Beyond States

## TL;DR

This paper extends the resource theory of quantum coherence from states to quantum operations, introducing coherence capacity and cost for channels, and revealing phenomena like bound coherence in operations.

## Contribution

It generalizes coherence resource theory to quantum channels, providing formulas for coherence capacity and cost, and discovering bound coherence in operations.

## Key findings

- Single-letter formula for coherence capacity of unitaries
- Maximally coherent states suffice for channel simulation with incoherent operations
- Existence of bound coherence in quantum operations

## Abstract

We generalize the recently proposed resource theory of coherence (or superposition) [Baumgratz, Cramer & Plenio, Phys. Rev. Lett. 113:140401; Winter & Yang, Phys. Rev. Lett. 116:120404] to the setting where not only the free ("incoherent") resources, but also the objects manipulated are quantum operations, rather than states. In particular, we discuss an information theoretic notion of coherence capacity of a quantum channel, and prove a single-letter formula for it in the case of unitaries. Then we move to the coherence cost of simulating a channel, and prove achievability results for unitaries and general channels acting on a $d$-dimensional system; we show that a maximally coherent state of rank $d$ is always sufficient as a resource if incoherent operations are allowed. We also show lower bounds on the simulation cost of channels that allow us to conclude that there exists bound coherence in operations, i.e. maps with non-zero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence.