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：凸分裂是量子信息理论中的一项强大技术，用于证明众多信息处理协议（如量子态重分配和量子网络信道编码）的可实现性。在本工作中，我们建立了以迹距离为误差准则的凸分裂的单次误差指数和单次强逆定理。我们的结果表明，导出的误差指数（强逆指数）为正，当且仅当速率位于（外部）可达区域。这导致了各种任务中的新单次指数结果，例如量子窃听信道上的通信、密钥蒸馏、单向量子消息压缩、量子测量模拟以及发射端具有边信息的量子信道编码。我们还建立了凸分裂样本复杂度的近最优单次刻画，该刻画匹配了二阶渐近特性。这进而为许多量子信息理论任务提供了更强的单次分析。