Relative entropy is the standard measure of distinguishability in classical and quantum information theory. In the classical case, its loss under channels admits an exact chain rule, while in the quantum case only asymptotic, regularized chain rules are known. We establish new chain rules for quantum relative entropy that apply already in the single-copy regime. The first inequality can be naturally obtained via POVM decompositions, extending the point distributions in the classical chain rule to quantum ensemble partitions. The second gives a sufficient condition for the most natural extension of the classical result, which uses projectors as an analog for the classical point distributions. We additionally find a semiclassical chain rule where the point distributions are replaced with the projectors of the initial states, and, finally, we find a relation to previous works on strengthened data processing inequalities and recoverability. These results show that meaningful chain inequalities are possible already at the single-copy level, but they also highlight that tighter bounds remain to be found.

翻译：相对熵是经典与量子信息论中区分性度量的标准量。在经典情形下，其通过信道后的损失存在精确的链式法则，而量子情形下仅已知渐近正则化的链式法则。我们针对量子相对熵建立了单副本情形下即可适用的新链式法则。第一类不等式可通过POVM分解自然获得，将经典链式法则中的点分布扩展为量子系综划分。第二类给出了经典结果最自然延伸的充分条件，该延伸使用投影算子模拟经典点分布。此外，我们发现了以初始态投影算子替代点分布的半经典链式法则，最后找到了与先前关于强化数据处理不等式及可恢复性研究之间的联系。这些结果表明，有意义的链式不等式在单副本层面即可实现，但同时也揭示出更紧的界仍待探索。