A set of classical or quantum states is equivalent to another one if there exists a pair of classical or quantum channels mapping either set to the other one. For dichotomies (pairs of states) this is closely connected to (classical or quantum) R\'enyi divergences (RD) and the data-processing inequality: If a RD remains unchanged when a channel is applied to the dichotomy, then there is a recovery channel mapping the image back to the initial dichotomy. Here, we prove for classical dichotomies that equality of the RDs alone is already sufficient for the existence of a channel in any of the two directions and discuss some applications. We conjecture that equality of the minimal quantum RDs is sufficient in the quantum case and prove it for special cases. We also show that neither the Petz quantum nor the maximal quantum RDs are sufficient. As a side-result of our techniques we obtain an infinite list of inequalities fulfilled by the classical, the Petz quantum, and the maximal quantum RDs. These inequalities are not true for the minimal quantum RDs.

翻译：一组经典或量子态与另一组等价，当且仅当存在一对经典或量子通道能将其中一组映射为另一组。对于二分态（状态对），这一性质与（经典或量子）Rényi散度（RD）及数据处理不等式密切相关：若施加通道于二分态后RD保持不变，则存在一个恢复通道将像映射回初始二分态。本文证明：对于经典二分态，仅需RD相等即可保证任一方向通道的存在性，并讨论若干应用。我们推测在量子情形下，最小量子RD相等即具有充分性，并在特例中予以证明。同时证明Petz量子RD与最大量子RD均不具备充分性。作为技术方法的副产品，我们获得了经典、Petz量子及最大量子RD所满足的无限不等式序列，而最小量子RD不满足这些不等式。