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.

翻译：我们研究了渐近状态下量子二分法精确与近似转换的问题，即对于大量n，是否存在量子信道ε将ρ₁^⊗n映射为ρ₂^⊗Rₙn（误差为εₙ，由迹距离度量），同时将σ₁^⊗n精确映射为σ₂^⊗Rₙn。我们推导了在微小偏差、中等偏差和大偏差误差区域以及零误差区域内，对于任意初始态对(ρ₁,σ₁)和可交换末态对(ρ₂,σ₂)，最优转换率Rₙ的二阶渐近表达式。我们还证明，当σ₁和σ₂由热吉布斯态给定时，前三类区域中的最优转换率可通过热力学操作实现。这使我们首次能够研究具有完全一般性初态（可能在不同能量本征空间之间存在相干性）的热力学态互变二阶渐近行为。因此，我们讨论了相干输入下热力学协议的最优性能，并描述了三种新颖的共振现象，这些现象可显著减少由有限大小效应引起的转换误差。此外，关于量子二分法的结果还可用于获得局部操作与经典通信下纯双粒子纠缠态之间的最优转换率（精确到二阶渐近项）。