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

Understand

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.