Having a distance measure between quantum states satisfying the right properties is of fundamental importance in all areas of quantum information. In this work, we present a systematic study of the geometric Rényi divergence (GRD), also known as the maximal Rényi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties, which are not satisfied by other quantum Rényi divergences. For example we prove a chain rule inequality that immediately implies the “amortization collapse” for the geometric Rényi divergence, addressing an open question by Berta et al. [Letters in Mathematical Physics 110:2277–2336, 2020, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric Rényi divergence, sharpening the previously best-known bounds based on the max-relative entropy while still keeping the new bounds single-letter and efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.

在量子态的所有领域中，具有满足正确性质的量子态之间的距离测量至关重要。在这项工作中，我们对Rényi几何散度进行了系统的研究（GRD），从量子信息理论的角度，也称为最大Rényi发散。我们表明，这种发散及其向通道的扩展，具有许多吸引人的结构特性，而其他量子Rényi发散却无法满足这些特性。例如，我们证明了一个链规则不等式，该不等式立即暗示了几何Rényi散度的“摊销崩溃”，这解决了Berta等人的一个未解决的问题。量子信道辨别领域中的[数学物理学报110：2277-2336，2020，等式（55）]。作为应用，我们探索各种信道容量问题，并根据几何Rényi散度构造新的信道信息度量，根据最大相对熵锐化以前最著名的范围，同时仍使新范围保持单个字母且可有效计算。对大量示例进行了研究，并且对于几乎所有情况而言，改进都是显而易见的。