Quantum Fluctuation-Response Inequality and Its Application in Quantum Hypothesis Testing

TL;DR

This paper introduces a quantum fluctuation-response inequality that bounds the mean difference of observables between quantum states using quantum relative entropy, leading to improved bounds in quantum hypothesis testing and applications in thermodynamics.

Contribution

The paper establishes a new quantum fluctuation-response inequality and applies it to derive stronger, nonasymptotic bounds for quantum hypothesis testing and other quantum information tasks.

Findings

Derived a bound linking observable differences to quantum relative entropy.

Established a novel error bound for quantum hypothesis testing based on sub-Gaussian properties.

Demonstrated applications in thermodynamic inference and quantum speed limits.

Abstract

We uncover the quantum fluctuation-response inequality, which, in the most general setting, establishes a bound for the mean difference of an observable at two different quantum states, in terms of the quantum relative entropy. When the spectrum of the observable is bounded, the sub-Gaussian property is used to further our result by explicitly linking the bound with the sub-Gaussian norm of the observable, based on which we derive a novel bound for the sum of statistical errors in quantum hypothesis testing. This error bound holds nonasymptotically and is stronger and more informative than that based on quantum Pinsker's inequality. We also show the versatility of our results by their applications in problems like thermodynamic inference and speed limit.