Convex splitting is a powerful technique in quantum information theory used in proving the achievability of numerous information-processing protocols such as quantum state redistribution and quantum network channel coding. In this work, we establish a one-shot error exponent and a one-shot strong converse for convex splitting with trace distance as an error criterion. Our results show that the derived error exponent (strong converse exponent) is positive if and only if the rate is in (outside) the achievable region. This leads to new one-shot exponent results in various tasks such as communication over quantum wiretap channels, secret key distillation, one-way quantum message compression, quantum measurement simulation, and quantum channel coding with side information at the transmitter. We also establish a near-optimal one-shot characterization of the sample complexity for convex splitting, which yields matched second-order asymptotics. This then leads to stronger one-shot analysis in many quantum information-theoretic tasks.

翻译：凸分割是量子信息论中的一项强大技术，常用于证明量子态重分配、量子网络信道编码等众多信息处理协议的可实现性。本文以迹距离为误差判据，建立了凸分割的单次误差指数和单次强逆界。结果表明：导出的误差指数（强逆指数）在速率处于（超出）可达区域时严格为正（为负）。这一成果为量子窃听信道通信、密钥蒸馏、单向量子消息压缩、量子测量模拟以及发射端含边信息的量子信道编码等任务提供了新的单次指数结果。我们还建立了凸分割样本复杂度的近最优单次刻画，获得了匹配的二阶渐近特性，进而推动了众多量子信息理论任务中更强的单次分析。