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：我们研究了渐近条件下量子二分法精确及近似转换的问题，即对于大量n，是否存在量子通道$\mathcal E$将$\rho_1^{\otimes n}$映射为$\rho_2^{\otimes R_n n}$（误差为$\epsilon_n$，以迹距离度量）并将$\sigma_1^{\otimes n}$精确映射为$\sigma_2^{\otimes R_n n}$。针对初始态任意对$(\rho_1,\sigma_1)$与最终态可交换对$(\rho_2,\sigma_2)$，我们推导了小偏差、中等偏差、大偏差误差区间以及零误差区间下最优转换速率$R_n$的二阶渐近表达式。我们还证明：当$\sigma_1$和$\sigma_2$由热吉布斯态给出时，在前三个区间内推导出的最优转换速率可通过热操作实现。这使我们首次能够研究具有完全一般性初始态（可能在不同能量本征空间之间存在相干性）的热力学态互转换的二阶渐近行为。因此，我们讨论了含相干输入的热力学协议的最优性能，并描述了三种新型共振现象，这些现象可显著降低由有限尺寸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域操作与经典通信条件下纯两体纠缠态之间转换速率的二阶渐近最优表达式。