Quantum dichotomies and coherent thermodynamics beyond first-order asymptotics

TL;DR

This paper develops second-order asymptotic formulas for quantum state transformations, including thermodynamic and entanglement conversions, revealing new resonance effects and extending understanding beyond first-order limits.

Contribution

It introduces second-order asymptotics for quantum dichotomy transformations, including thermodynamic and entanglement conversions, with new resonance phenomena and applicability to coherent states.

Findings

Derived second-order asymptotic transformation rates for quantum dichotomies.

Achieved optimal thermodynamic state interconversion rates with coherence.

Identified resonance phenomena reducing finite-size transformation errors.

Abstract

We address the problem of exact and approximate transformation of quantum dichotomies in the asymptotic regime, i.e., the existence of a quantum channel $\mathcal E$ mapping $\rho_1^{\otimes n}$ into $\rho_2^{\otimes R_nn}$ with an error $\epsilon_n$ (measured by trace distance) and $\sigma_1^{\otimes n}$ into $\sigma_2^{\otimes R_n n}$ 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 $(\rho_1,\sigma_1)$ of initial states and a commuting pair $(\rho_2,\sigma_2)$ of final states. We also prove that for $\sigma_1$ and $\sigma_2$ 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.