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

Hà Quang Minh

Abstract: This 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.