Unifying quantum measurement constructions via a relative-entropy minimum change principle

2026-08-04

AI Summary

This work proposes a unified framework for quantum measurement grounded in quantum information theory to generalize classical Bayesian inference. By leveraging the principle of minimum change based on quantum relative entropy, and through dual modeling of classical-to-quantum preparation and quantum-to-classical measurement channels, the authors derive a closed-form solution for optimal measurements using techniques from dual optimization, unconstrained Hermitian parameterization, and entropy-regularized semidefinite programming. The framework unifies several known measurement schemes—such as pretty good measurements and Fermi-Dirac thermal measurements—and introduces a novel family of softmin thermal measurements, revealing a statistical-mechanical analogy in quantum measurement. Key contributions include an explicit construction of optimal measurements, a proof of the additivity of the minimum-change principle, and empirical validation of the superior performance of Fermi-Dirac thermal measurements in quantum hypothesis testing.

Abstract

The minimum change principle provides an information-theoretic characterization of the Bayes reversal channel in classical probability theory and has recently been proposed as a framework for extending Bayes' rule to quantum information theory. Using quantum relative entropy, we investigate a minimum change principle for the setting of quantum statistical inference. Specifically, we consider a forward process based on a classical-to-quantum preparation channel and a reverse process based on a quantum-to-classical measurement channel. We establish a closed-form characterization of measurements that are optimal for this principle, and this optimal measurement can be found via a dual formulation involving a single unconstrained Hermitian variable. This perspective allows us to recover some notable measurements within the same framework, including pretty good measurements and Fermi-Dirac thermal measurements, and we use it to discover a novel family that we call softmin thermal measurements. We further show that softmin thermal measurements arise as optimal solutions to entropy-regularized semidefinite optimization problems, demonstrating that they play a role for measurements analogous to that of thermal states in statistical mechanics. Finally, we prove an additivity property for the relative-entropy minimum change principle and investigate the performance of Fermi-Dirac thermal measurements for quantum hypothesis testing.

Problem

• quantum measurement
• relative entropy
• minimum change principle
• quantum statistical inference
• Bayes reversal

Innovation

• minimum change principle
• quantum relative entropy
• softmin thermal measurements
• Bayes reversal
• entropy-regularized optimization