Quantum R\'enyi divergences and the strong converse exponent of state discrimination in operator algebras

TL;DR

This paper extends the operational understanding of sandwiched R\'enyi divergences from finite-dimensional quantum states to normal states on injective von Neumann algebras, establishing their significance in state discrimination and their equivalence to measured divergences.

Contribution

It demonstrates the operational significance of sandwiched R\'enyi divergences in von Neumann algebras and extends their properties to nuclear C*-algebras, bridging finite and infinite-dimensional quantum settings.

Findings

Sandwiched R\'enyi divergences quantify error trade-offs in state discrimination.

These divergences coincide with regularized measured R\'enyi divergences in von Neumann algebras.

Operational interpretation extends to pairs of states on nuclear C*-algebras.

Abstract

The sandwiched R\'enyi $\alpha$-divergences of two finite-dimensional quantum states play a distinguished role among the many quantum versions of R\'enyi divergences as the tight quantifiers of the trade-off between the two error probabilities in the strong converse domain of state discrimination. In this paper we show the same for the sandwiched R\'enyi divergences of two normal states on an injective von Neumann algebra, thereby establishing the operational significance of these quantities. Moreover, we show that in this setting, again similarly to the finite-dimensional case, the sandwiched R\'enyi divergences coincide with the regularized measured R\'enyi divergences, another distinctive feature of the former quantities. Our main tool is an approximation theorem (martingale convergence) for the sandwiched R\'enyi divergences, which may be used for the extension of various further results from the finite-dimensional to the von Neumann algebra setting. We also initiate the study of the sandwiched R\'enyi divergences of pairs of states on a $C^*$-algebra, and show that the above operational interpretation, as well as the equality to the regularized measured R\'enyi divergence, holds more generally for pairs of states on a nuclear $C^*$-algebra.