From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions
Abstract
We prove a variety of new and refined uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinitedimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new meanconstrained Fanotype inequality and the notion of maximal coupling of random variables. We then employ this classical result to derive the first tight energyconstrained continuity bound for the von Neumann entropy of states of infinitedimensional quantum systems, when the Hamiltonian is the number operator, which is arguably the most relevant Hamiltonian in the study of infinitedimensional quantum systems in the context of quantum information theory. The above scheme works only for Shannon and von Neumann entropies. Hence, to deal with more general entropies, e.g. $\alpha$Rényi and $\alpha$Tsallis entropies, with $\alpha \in (0,1)$, for which continuity bounds are known only for finitedimensional systems, we develop a novel approximation scheme which relies on recent results on operator Hölder continuous functions and the equivalence of all Schatten norms in special spectral subspaces of the Hamiltonian. This approach is, as we show, motivated by continuity bounds for $\alpha$Rényi and $\alpha$Tsallis entropies of random variables that follow from the Hölder continuity of the entropy functionals. Bounds for $\alpha>1$ are provided, too. Finally, we settle an open problem on related approximation questions posed in the recent works by Shirokov on the socalled Finitedimensional Approximation (FA) property.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 DOI:
 10.48550/arXiv.2104.02019
 arXiv:
 arXiv:2104.02019
 Bibcode:
 2021arXiv210402019B
 Keywords:

 Quantum Physics;
 Computer Science  Information Theory;
 Mathematical Physics
 EPrint:
 comments welcome