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信道能将其中一组映射到另一组。对于二分态（态对），这密切关联于（经典或量子）Rényi散度（RD）及其数据处理不等式：若当某个信道作用于二分态时RD保持不变，则存在一个恢复信道将该映像映射回原始二分态。本文证明，对于经典二分态，仅凭RDs相等这一条件即可充分保证存在任意方向的信道，并讨论若干应用。我们推测在量子情形下，最小量子RDs相等亦具充分性，并在特殊情形下给出证明。同时证明Petz量子RD与最大量子RD均不满足充分性。作为技术方法的副产品，我们获得由经典、Petz量子及最大量子RDs满足的无穷不等式序列。这些不等式对于最小量子RDs不成立。