The measured relative entropies of quantum states and channels find operational significance in quantum information theory as achievable error rates in hypothesis testing tasks. They are of interest in the near term, as they correspond to hybrid quantum--classical strategies with technological requirements far less challenging to implement than required by the most general strategies allowed by quantum mechanics. In this paper, we prove that these measured relative entropies can be calculated efficiently by means of semi-definite programming, by making use of variational formulas for the measured relative entropies of states and semi-definite representations of the weighted geometric mean and the operator connection of the logarithm. Not only do the semi-definite programs output the optimal values of the measured relative entropies of states and channels, but they also provide numerical characterizations of optimal strategies for achieving them, which is of significant practical interest for designing hypothesis testing protocols.

翻译：量子态与量子信道的测量相对熵在量子信息理论中具有操作意义，它们对应着假设检验任务中可达到的错误率。这些度量在当前阶段尤其受到关注，因为它们对应于混合量子-经典策略，其技术实现要求远低于量子力学所允许的最一般策略。本文证明，通过利用量子态测量相对熵的变分公式、加权几何平均的半定表示以及对数算子连接的半定表示，这些测量相对熵可通过半定规划高效计算。这些半定规划不仅能输出量子态与量子信道测量相对熵的最优值，还能提供实现这些最优值的策略的数值表征，这对于设计假设检验协议具有重要的实际意义。