Quantum dichotomies and coherent thermodynamics beyond first-order asymptotics
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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 ρ_1^⊗ n into ρ_2^⊗ R_nn with an error ϵ_n (measured by trace distance) and σ_1^⊗ n into σ_2^⊗ R_n n exactly, for a large number n. We derive second-order asymptotic expressions for the optimal transformation rate R_n in the small, moderate, and large deviation error regimes, as well as the zero-error regime, for an arbitrary pair (ρ_1,σ_1) of initial states and a commuting pair (ρ_2,σ_2) of final states. We also prove that for σ_1 and σ_2 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.
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