The variance of relative surprisal as single-shot quantifier

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.

v3: published version

References in corpus (4)

• Entanglement Spectrum and Entanglement Thermodynamics of Quantum Hall Bilayers at nu=1
• Quantum Relative Lorenz Curves
• Quantum state discrimination: a geometric approach
• Smoothed generalized free energies for thermodynamics

Cited by in corpus (7)

• Catalytic quantum teleportation and beyond
• Correlation in Catalysts Enables Arbitrary Manipulation of Quantum Coherence
• Fundamental Limits on Correlated Catalytic State Transformations
• Capacity of Entanglement for Non-local Hamiltonian
• Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture
• Probing RG flows, symmetry resolution and quench dynamics through the capacity of entanglement
• Fluctuation-dissipation relations for thermodynamic distillation processes