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

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 for all alpha is an element of 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 for all alpha is an element of 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.