A new estimation of the quantum Chernoff bound

TL;DR

The paper tackles tightening upper bounds in quantum state discrimination by expanding the class of functions used to bound the quantum Chernoff bound beyond matrix monotone/interpolation families. It introduces the extended class $\widetilde{\mathcal{F}}$ and proves a generalized Powers-Størmer-type inequality, yielding a bound $1-CH_{\widetilde{\mathcal{F}}\cup\mathcal{F}}(\rho,\sigma)\le \phi(\rho,\sigma)\le \sqrt{1-CH_{\widetilde{\mathcal{F}}\cup\mathcal{F}}(\rho,\sigma)^2}$ for quantum states $\rho,\sigma$, which potentially improves the standard Chernoff-based bounds. The work also develops matrix-perspective function inequalities, offers a characterization of matrix-decreasing functions via these perspectives, and provides an alternative proof approach for a key lemma using Hansen–Ha representations. Collectively, these results extend the toolkit for quantum hypothesis testing and may yield tighter, more general estimations of the quantum Chernoff bound in various operator settings.

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-Störmer type inequalities for perspective functions.

Figures (1)

• Figure : Graphs of $f(x)=W(x)$ (left) and $g(x)=x/W(x)$ (right)

Theorems & Definitions (16)

• Proposition 3.1
• proof
• Remark 3.2
• Theorem 3.3
• Theorem 3.4
• Lemma 3.5
• Theorem 4.1
• proof
• Remark 4.2
• Corollary 4.3