Quantum ff-divergences via Nussbaum-Szko{\l}a Distributions with applications to Petz-R\'enyi and von Neumann Relative Entropy

We prove that the quantum $f$-divergence of two states (density operators on a Hilbert space of finite or infinite dimensions) is same as the classical $f$-divergence of the corresponding Nussbaum-Szko{\l}a distributions. This provides a general framework to study most of the known quantum entropic quantities using the corresponding classical entities. In this spirit, we study the Petz-R\'enyi and von Neumann relative entropy and prove that these quantum entropies are equal to the corresponding classical counterparts of the Nussbaum-Szko{\l}a distributions. This is a generalization of a finite dimensional result that was proved by Nussbaum and Szko{\l}a [Ann. Statist. 37, 2009, 2] to the infinite dimensions. We apply classical results about R\'enyi and Kullback-Leibler divergences to obtain new results and new proofs for some known results about the quantum relative entropies. Most important among these are (i) a quantum Pinsker type inequality in the infinite dimensions, and (ii) necessary and sufficient conditions for the finiteness of the Petz-R\'enyi $\alpha$-relative entropy for any order $\alpha \in [0, \infty]$. Furthermore, we construct an example to show that the information theoretic definition of the von Neumann relative entropy is different from Araki's definition of relative entropy when the dimension of the Hilbert space is infinite. This discrepancy can be bridged using the notion of the distribution of an unbounded positive selfadjoint operator with respect to a positive compact operator and Haagerup's extension of the trace. Our results are valid in both finite and infinite dimensions and hence can be applied to continuous variable systems as well.Comment: Major Revision. New results about quantum $f$-divergences are added. New formula for Petz-R\'enyi and von Neumann relative entropy are obtained using the idea of distribution of an observable. We thank Mark Wilde and Mil\'an Mosonyi for providing useful comments and relevant references for the earlier version. 38 page

Petz–Rényi relative entropy of thermal states and their displacements

In this letter, we obtain the precise range of the values of the parameter α such that Petz–Rényi α-relative entropy Dα(ρ||σ) of two faithful displaced thermal states is finite. More precisely, we prove that, given two displaced thermal states ρ and σ with inverse temperature parameters r1,r2,…,rn and s1,s2,…,sn, respectively, 01. Along the way, we also prove a special case of a conjecture of Seshadreesan et al. (J Math Phys 59(7):072204, 2018. https://doi.org/10.1063/1.5007167).Multidisciplinary University Research Initiative12 month embargo; first published 17 April 2024This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]

Real normal operators and Williamson’s normal form

A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose (adjoint). Astructure theorem for invertible skew-symmetric operators, which is analogous to the finite-dimensional situation, is also proved using elementary techniques. The second result is used to establish the main theorem of this article, which is a generalization of Williamson’s normal form for bounded positive operators on infinite-dimensional separable Hilbert spaces. This has applications in the study of infinite mode Gaussian states