Resource theory of coherence: Beyond states

We generalize the recently proposed resource theory of coherence (or superposition) [T. Baumgratz, Phys. Rev. Lett. 113, 140401 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.140401; A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)PRLTAO0031-900710.1103/PhysRevLett.116.120404] to the setting where not just the free ("incoherent") resources, but also the manipulated objects, are quantum operations rather than states. In particular, we discuss an information theoretic notion of the 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, and one of rank d2 for "strictly incoherent" operations. 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 nonzero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence.