Fully quantum second-law--like statements from the theory of statistical comparisons

Abstract

In generalized resource theories, one aims to reformulate the problem of deciding whether a suitable transition (typically a transition that preserves the Gibbs state of the theory) between two given states $\rho$ and $\sigma$ exists or not, into the problem of checking whether a set of second-law--like inequalities hold or not. The aim of these preliminary notes is to show how the theory of statistical comparisons (in the sense of Blackwell, LeCam, and Torgersen) can be useful in such scenarios. In particular, we propose one construction, in which the second laws are formulated in terms of a suitable conditional min-entropy. Though a general, fully quantum result is also presented, stronger results are obtained for the case of qubits, and the case of $\sigma$ commuting with the Gibbs state.

arXiv comment11 pages of preliminary notes to be presented at QIT32, 25/26 May 2015, Osaka University

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• A complete and operational resource theory of measurement sharpness
• The resource theory under conditioned thermal operations
• A Compendious Review of Majorization-Based Resource Theories: Quantum Information and Quantum Thermodynamics