A new estimation of the quantum Chernoff bound

TL;DR

This paper introduces a new class of functions to improve the estimation of the quantum Chernoff bound, enhancing bounds on error probabilities in quantum state discrimination.

Contribution

It proposes a novel class of functions satisfying key inequalities, leading to a new estimation method for the quantum Chernoff bound and related matrix inequalities.

Findings

Derived a new estimation of the quantum Chernoff bound.

Characterized matrix decreasing functions and established matrix inequalities.

Extended perspective functions with Powers-Störmer type inequalities.

Abstract

Relating to finding possible upper bounds for the probability of error for discriminating between two quantum states, it is well-known that \begin{align*} \mathrm{tr}(A+B) - \mathrm{tr}|A-B|\leq 2\, \mathrm{tr}\big(f(A)g(B)\big) \end{align*} holds for every positive-valued matrix monotone function $f$, where $g(x)=x/f(x)$, and all positive definite matrices $A$ and $B$. In this paper, we introduce a new class of functions that satisfy the above inequality. As a consequence, we derive a novel estimation of the quantum Chernoff bound. Additionally, we characterize matrix decreasing functions and establish matrix Powers-St\"ormer type inequalities for perspective functions.