The sandwiched R\'enyi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched R\'enyi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical R\'enyi divergences for R\'enyi parameter $\alpha>1$. The known proof of this goes by showing that the sandwiched R\'enyi divergence coincides with the regularized measured R\'enyi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched R\'enyi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (even for states of a von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured R\'enyi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched R\'enyi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class. This is interesting from the purely mathematical point of view of extending the concept of R\'enyi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of R\'enyi $(\alpha,z)$-divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.
翻译:R\'enyi 的变异。 已知的证据是, R\'enyi 的变异与测得的 R\'enyi 的变异相吻合, 反过来又通过测得的混杂度差异化而得到证明。 因此, 混杂的 R\'enyi 的变异概念最近被扩展至一个无限的Hilbert 空间(即使对于 von Neumann algebra 的状态来说,这些变异也远没有与测得的R\'enyi 的变异性相类似的操作解释。 在本文中, 我们通过解答这两个变异性的问题, 在一个直线性平面的 Rilbert 空间的变异性运行者, 使用一个简单的直线性平面的变异性定义, 也从一个直线性平面的 Ryaltial 直径直径直径直径直至一个直线直径直径直径直径直径直的直径直径直径直径直径直径直径直径直径直径直径直达直径直径直径直径直径直径直径直径直的轨道直径直径直径直径直径直径直距。