Decoherence is all around us. Every quantum system that interacts with the environment is doomed to decohere. While preserving quantum coherence is a major challenge faced in quantum technologies, the potential benefits for information processing are very promising since coherence can lead to various operational advantages, such as in quantum algorithms. Hence, much work has been devoted in recent years to quantify the coherence present in a system. In the present paper, we formulate the quantum resource theory of dynamical coherence. The underlying physical principle we follow is that the free dynamical objects are those that neither store nor output coherence. This leads us to identify classical channels as the free elements in this theory. Consequently, even the quantum identity channel is not free as all physical systems undergo decoherence and hence, the preservation of coherence should be considered a resource. The maximally coherent channel is then the quantum Fourier transform because of its abillity to preserve entanglement and generate maximal coherence from nothing. In our work, we introduce four different types of free superchannels (analogous to MIO, DIO, IO, and SIO) and discuss in detail two of them, namely, dephasing-covariant incoherent superchannels (DISC), maximally incoherent superchannels (MISC). The latter consists of all superchannels that do not generate non-classical channels from classical ones. We quantify dynamical coherence using channel-divergence-based monotones for $\mathrm{MISC}$ and $\mathrm{DISC}$. We show that some of these monotones have operational interpretations as the exact, the approximate, and the liberal coherence cost of a quantum channel. Moreover, we prove that the liberal asymptotic cost of a channel is equal to a new type of regularized relative entropy. Finally, we show that the conversion distance between two channels under $\mathrm{MISC}$ and $\mathrm{DISC}$ can be computed using a semidefinite program (SDP).

• Received 17 October 2019
• Accepted 13 May 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.023298

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Published by the American Physical Society