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

Understand

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.