Quantum Jensen-Shannon Divergences Between Infinite-Dimensional Positive Definite Operators

Abstract

AbstractThis work studies a parametrized family of symmetric divergences on the set of Hermitian positive definite matrices which are defined using the \(\alpha \)-Tsallis entropy \(\forall \alpha \in \mathbb {R}\). This family unifies in particular the Quantum Jensen-Shannon divergence, defined using the von Neumann entropy, and the Jensen-Bregman Log-Det divergence. The divergences, along with their metric properties, are then generalized to the setting of positive definite trace class operators on an infinite-dimensional Hilbert space \(\forall \alpha \in \mathbb {R}\). In the setting of reproducing kernel Hilbert space (RKHS) covariance operators, all divergences admit closed form formulas in terms of the corresponding kernel Gram matrices.KeywordsQuantum Jensen-Shannon divergencesPositive definite operatorsTrace class operators