R\'enyi divergences and binary state discrimination error exponents for fermionic quasi-free states

arXiv:2605.31379v1 Announce Type: cross Abstract: The trade-off relations between the two types of error probabilities in binary i.i.d. quantum state discrimination can be expressed by single-copy formulas in terms of the Petz-type and the sandwiched R\'enyi divergences of the two states representing the two hypotheses. In the non-i.i.d. setting, the error exponents can usually be expressed in terms of regularized R\'enyi divergences, which do not admit explicit formulas in general. Here, we consider a class of states, translation-invariant and gauge-invariant quasifree states on doubly infinite fermionic chains, and give explicit formulas for a wide range of regularized R\'enyi divergences between such states, including $(\alpha,z)$, log-Euclidean, maximal, measured, and the recently introduced integral R\'enyi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized R\'enyi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed $\alpha$, the regularized R\'enyi $(\alpha,z)$-divergences give different regularized values for different $z$ parameters in general. We also generalize a previous construction from [Bunth, Mar\'oti, Mosonyi, Zimbor\'as, Lett.~Math.~Phys.~113:(7), 2023] to the case of multiple modes per lattice site to obtain a large class of states exhibiting super-exponential decay of the discrimination error probabilities.