The strong converse exponent of discriminating infinite-dimensional quantum states

Abstract

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.