Quantum relative entropy for unravelings of master equations

This work explores connections between the quantum relative entropy of two faithful states $\rho,\sigma$ (i.e. full-rank density matrices) and the Kullback-Leibler divergences of classical measures $\mu,\nu$. Here, $\mu$ and $\nu$ are measures on the space of pure states, realizing $\rho$ and $\sigma$ respectively. The motivation for this result is to establish a notion of quantum relative entropy in the space of pure state distributions, which are the resulting objects of unravelings of the Lindblad equation, such as the stochastic Schr\"{o}dinger equation. Our results show that the measures that achieve the minimal KL divergence are those supported on a (possibly non-orthogonal) common basis between $\rho$ and $\sigma$. Using the classical and quantum data-processing inequalities, our notion of quantum relative entropy is shown to be equivalent to the Belavkin-Staszewski entropy on states, revealing new insights on this quantity. Furthermore, the common basis is used to provide a novel proof of contraction of the relative entropy under Lindblad flow and offers insights into results from large deviation theory.