The variance of relative surprisal as single-shot quantifier
- URL: http://arxiv.org/abs/2009.08391v3
- Date: Tue, 29 Mar 2022 12:51:46 GMT
- Title: The variance of relative surprisal as single-shot quantifier
- Authors: Paul Boes, Nelly H. Y. Ng, Henrik Wilming
- Abstract summary: We show that (relative) surprisal gives sufficient conditions for approximate state-transitions between pairs of quantum states in single-shot setting.
We further derive a simple and physically appealing axiomatic single-shot characterization of (relative) entropy.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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