Work

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

Year: 2023

Type: article

Abstract: We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional 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 mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions,e.g.$\alpha $-Rényi and$\alpha $-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.

Source: IEEE Transactions on Information Theory

DOI: 10.1109/tit.2023.3248228

Authors: Simon Becker, Nilanjana Datta, Michael G. Jabbour

Institutions: ETH Zurich, University of Cambridge, Technical University of Denmark

Language: English

Topic: Quantum Information and Cryptography

Subfield: Artificial Intelligence

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: University of Cambridge, Carlsbergfondet, Engineering and Physical Sciences Research Council

Awards: G1413339353, EPSRC Centre for Doctoral Training in Analysis (Cambridge Centre for Analysis)