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 Rényi 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 Rényi 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 Rényi divergences between such states, including $(α,z)$, log-Euclidean, maximal, measured, and the recently introduced integral Rényi 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 Rényi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed $α$, the regularized Rényi $(α,z)$-divergences give different regularized values for different $z$ parameters in general. We also generalize a previous construction from [Bunth, Maróti, Mosonyi, Zimborás, 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.

翻译：二元独立同分布量子态判别中两类误差概率之间的权衡关系可通过Petz型及夹层型Rényi散度（基于代表两个假设的态）的单副本公式表达。在非独立同分布框架下，误差指数通常可用正则化Rényi散度表示，但此类散度一般无显式公式。本文针对双无穷费米子链上平移不变且规范不变的拟自由态，给出了这类态间一系列正则化Rényi散度的显式公式，涵盖$(α,z)$型、对数欧几里得型、最大型、测量型以及近期引入的积分型Rényi散度。研究表明：当每个格点仅有单一模式时，系统渐近趋于经典情形，所有不同类型的正则化Rényi散度均相等；而当每个格点存在多模式时，非对易性在正则化过程中持续存在，且对固定$α$，正则化Rényi $(α,z)$-散度通常在不同$z$参数下给出不同的正则化值。此外，我们将[Bunth, Maróti, Mosonyi, Zimborás, Lett.~Math.~Phys.~113:(7), 2023]的先前构造推广至每个格点具有多模式的情形，从而获得一大类展现判别误差概率超指数衰减的态。