We address the problem of exact and approximate transformation of quantum
dichotomies in the asymptotic regime, i.e., the existence of a quantum channel
E mapping Ο1βnβ into Ο2βRnβnβ with an
error Ο΅nβ (measured by trace distance) and Ο1βnβ into
Ο2βRnβnβ exactly, for a large number n. We derive
second-order asymptotic expressions for the optimal transformation rate Rnβ
in the small, moderate, and large deviation error regimes, as well as the
zero-error regime, for an arbitrary pair (Ο1β,Ο1β) of initial states
and a commuting pair (Ο2β,Ο2β) of final states. We also prove that
for Ο1β and Ο2β given by thermal Gibbs states, the derived
optimal transformation rates in the first three regimes can be attained by
thermal operations. This allows us, for the first time, to study the
second-order asymptotics of thermodynamic state interconversion with fully
general initial states that may have coherence between different energy
eigenspaces. Thus, we discuss the optimal performance of thermodynamic
protocols with coherent inputs and describe three novel resonance phenomena
allowing one to significantly reduce transformation errors induced by
finite-size effects. What is more, our result on quantum dichotomies can also
be used to obtain, up to second-order asymptotic terms, optimal conversion
rates between pure bipartite entangled states under local operations and
classical communication.Comment: 51 pages, 6 figures, comments welcom