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.
Highlights
Many central results of quantum information theory are concerned with the manipulation of quantum systems in the so-called asymptotic independent identically distributed (IID) setting, in which one considers the limit of taking infinitely many independent and identically distributed copies of a quantum system [1–9]
The most extreme weakening of the IID setting is the single-shot setting, in which generally no assumptions about the size of the system or its correlations are made. This setting derives its name from the fact that it can be seen to describe a single iteration of a protocol, in contrast to the IID setting, which is concerned with infinitely many independent iterations
The relative variance has been shown to measure leading-order corrections to asymptotic results in information theory [40– 56], but here we show that its role extends to the genuine single-shot setting
Summary
Many central results of quantum information theory are concerned with the manipulation of quantum systems in the so-called asymptotic independent identically distributed (IID) setting, in which one considers the limit of taking infinitely many independent and identically distributed copies of a quantum system [1–9]. The most extreme weakening of the IID setting is the single-shot setting, in which generally no assumptions about the size of the system or its correlations are made. This setting derives its name from the fact that it can be seen to describe a single iteration of a protocol, in contrast to the IID setting, which is concerned with infinitely many independent iterations. The IID setting, due to its various assumptions, can usually be characterized by variants of the (quantum) relative entropy, S(ρ σ ) := tr{ρ[log(ρ) − log(σ )]}
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