Quantum relative entropy for unravelings of master equations

TL;DR

This paper introduces a new way to measure quantum relative entropy through classical measures on pure states, linking it to the Belavkin-Staszewski entropy and providing insights into Lindblad dynamics and large deviations.

Contribution

It establishes a quantum relative entropy framework based on pure state measures, connecting classical KL divergence with quantum states and proving contraction properties under Lindblad flow.

Findings

Measures minimizing KL divergence are supported on a common basis

Quantum relative entropy equals Belavkin-Staszewski entropy for faithful states

Provides a new proof of entropy contraction under Lindblad evolution

Abstract

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.