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：一组经典或量子态若存在一对经典或量子信道可将任一组映射至另一组，则称这两组等价。对于二分态（状态对），这通常与（经典或量子）Rényi散度及数据处理不等式密切相关：若某Rényi散度在信道作用于二分态后保持不变，则存在恢复信道将像映射回初始二分态。本文证明，对于经典二分态，Rényi散度相等本身已足以保证任一方向上存在信道，并讨论若干应用。我们推测最小量子Rényi散度相等在量子情形下具有充分性，并在特殊情况下给出证明。同时证明Petz量子Rényi散度与最大量子Rényi散度均不具备充分性。作为方法的副产品，我们得到经典、Petz量子及最大量子Rényi散度满足的无穷不等式列表，而最小量子Rényi散度不满足这些不等式。