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 Schrödinger 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.