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Title: The variance of relative surprisal as single-shot quantifier
Authors: Boes, Paul
Ng, Nelly Huei Ying
Wilming, Henrik
Keywords: Science::Physics
Issue Date: 2022
Source: Boes, P., Ng, N. H. Y. & Wilming, H. (2022). The variance of relative surprisal as single-shot quantifier. PRX Quantum, 3(1), 010325-1-010325-31.
Journal: PRX Quantum
Abstract: The variance of (relative) surprisal, also known as varentropy, so far mostly plays a role in information theory as quantifying the leading order corrections to asymptotic i.i.d.~limits. Here, we comprehensively study the use of it to derive single-shot results in (quantum) information theory. We show that it gives genuine sufficient and necessary conditions for approximate state-transitions between pairs of quantum states in the single-shot setting, without the need for further optimization. We also clarify its relation to smoothed min- and max-entropies, and construct a monotone for resource theories using only the standard (relative) entropy and variance of (relative) surprisal. This immediately gives rise to enhanced lower bounds for entropy production in random processes. We establish certain properties of the variance of relative surprisal which will be useful for further investigations, such as uniform continuity and upper bounds on the violation of sub-additivity. Motivated by our results, we further derive a simple and physically appealing axiomatic single-shot characterization of (relative) entropy which we believe to be of independent interest. We illustrate our results with several applications, ranging from interconvertibility of ergodic states, over Landauer erasure to a bound on the necessary dimension of the catalyst for catalytic state transitions and Boltzmann's H-theorem.
ISSN: 2691-3399
DOI: 10.1103/PRXQuantum.3.010325
Schools: School of Physical and Mathematical Sciences 
Rights: © 2022 the Authors. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Fulltext Permission: open
Fulltext Availability: With Fulltext
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