AI 中文总结
本文证明，在二进制猜测游戏中，任何加性可区分性度量若尊重最优测量揭示的赔率，则必为Umegaki相对熵，这源于量子非对易性。
AI 中文摘要
量子相对熵是物理学中的核心概念，它支配着通信的极限、热力学不可逆性和量子资源转换。然而，物理过程不能增加状态可区分性（数据处理不等式）的要求允许无限多种替代散度度量。这里我们证明，量子相对熵是由一个更尖锐的操作原理唯一选择的。我们通过二进制猜测游戏来评估可区分性，其中观察者使用最优测量来区分一对量子态。我们证明，任何尊重这些最优测量揭示的赔率的加性度量都必须与Umegaki相对熵一致。这种刚性是一种纯粹的量子现象。经典理论允许一个连续的散度度量族，包括Rényi散度，而量子非对易性消除了这种数学自由度。结果是精确的，既不需要无限多副本的热力学极限，也不需要对关联态的超可加性假设。它将量子相对熵确立为不仅是渐近量，而且是单次量子区分中唯一的加性可区分性度量。
英文摘要
Quantum relative entropy is a core concept in physics, governing the limits of communication, thermodynamic irreversibility and quantum resource conversion. However, the requirement that physical processes cannot increase state distinguishability, the data-processing inequality, permits an infinite family of alternative divergence measures. Here we show that quantum relative entropy is uniquely selected by a sharper operational principle. We evaluate distinguishability through binary guessing games, in which an observer discriminates between pairs of quantum states using the optimal measurement. We prove that any additive measure that respects the odds revealed by these optimal measurements must coincide with the Umegaki relative entropy. This rigidity is a purely quantum phenomenon. Whereas classical theory permits a continuous family of valid divergence measures, including Rényi divergences, quantum noncommutativity. collapses this mathematical freedom. The result is exact, requiring neither a thermodynamic limit of infinitely many copies nor super-additivity assumptions for correlated states. It establishes quantum relative entropy not merely as an asymptotic quantity, but as the unique additive distinguishability measure compatible with single-shot quantum discrimination.
CommentsRevised version: added an operational interpretation of Frenkel's integral formula in terms of binary guessing probabilities, together with a new figure and an appendix proof; also improved the terminology and historical attribution
