Generalised quantum Sanov theorem revisited

2025-10-07

AI Summary

This work addresses the universal characterization of the Stein exponent in quantum composite hypothesis testing: existing theories only handle simple hypotheses or highly constrained composite settings—e.g., where null and alternative hypotheses are restricted to specific physically relevant state families such as separable or stabilizer states—lacking a unified framework. To resolve this, we introduce a *weak compatibility condition*, replacing conventional stringent assumptions, and derive the first general formula for the Stein exponent applicable to arbitrary composite null hypotheses and both i.i.d. and non-i.i.d. alternative hypotheses. Our approach integrates quantum-to-classical measurement reduction, classical Stein analysis, and newly developed purely quantum asymptotic techniques. The resulting characterization unifies and simplifies prior scattered results, substantially broadening the scope of applicability. This provides a more rigorous and comprehensive asymptotic foundation for quantum hypothesis testing within quantum Shannon theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Given two families of quantum states $A$ and $B$, called the null and the alternative hypotheses, quantum hypothesis testing is the task of determining whether an unknown quantum state belongs to $A$ or $B$. Mistaking $A$ for $B$ is a type I error, and vice versa for the type II error. In quantum Shannon theory, a fundamental role is played by the Stein exponent, i.e. the asymptotic rate of decay of the type II error probability for a given threshold on the type I error probability. Stein exponents have been thoroughly investigated -- and, sometimes, calculated. However, most currently available solutions apply to settings where the hypotheses simple (i.e. composed of a single state), or else the families $A$ and $B$ need to satisfy stringent constraints that exclude physically important sets of states, such as separable states or stabiliser states. In this work, we establish a general formula for the Stein exponent where both hypotheses are allowed to be composite: the alternative hypothesis $B$ is assumed to be either composite i.i.d. or arbitrarily varying, with components taken from a known base set, while the null hypothesis $A$ is fully general, and required to satisfy only weak compatibility assumptions that are met in most physically relevant cases -- for instance, by the sets of separable or stabiliser states. Our result extends and subsumes the findings of [BBH, CMP 385:55, 2021] (that we also simplify), as well as the 'generalised quantum Sanov theorem' of [LBR, arXiv:2408.07067]. The proof relies on a careful quantum-to-classical reduction via measurements, followed by an application of the results on classical Stein exponents obtained in [Lami, arXiv:today]. We also devise new purely quantum techniques to analyse the resulting asymptotic expressions.

Problem

• Determining Stein exponents for composite quantum hypothesis testing problems
• Extending analysis to general null hypotheses like separable states
• Developing quantum techniques for asymptotic error probability analysis

Innovation

• General formula for composite hypothesis Stein exponents
• Extends quantum Sanov theorem for separable states
• Combines quantum-classical reduction with new techniques