The sandwiched rényi divergence and quantum positive evidence order in infinite-dimensional hilbert space

Abstract

The aim of this note is to extend some main results of the sandwiched Rényi divergence and quantum positive evidence order from finite- to the infinite-dimensional separable Hilbert spaces. We first define sandwiched Rényi divergence Dα (ρ||ρ) for α > 1 in separable Hilbert space and consider the positivity, monotonicity and limitation of Dα (ρ||σ) (α → ∞). Then we give some sufficient and necessary conditions of σ ⊑ ρ with respect to the quantum positive evidence order, where σ and ρ are quantum states.