Towards Quantum Universal Hypothesis Testing

2025-04-22

AI Summary

Quantum hypothesis testing lacks a general asymmetric testing framework. Method: This paper establishes the first quantum analogue of the Hoeffding universal hypothesis test (UHT), departing from classical empirical distribution paradigms by reconstructing unknown quantum states via quantum state tomography, characterizing the exponential decay rate of type-II error using trace distance, and integrating large deviation theory with non-asymptotic concentration inequalities to construct an exponentially consistent test statistic. Contributions/Results: (1) It introduces the first rigorously provably optimal quantum UHT framework; (2) it achieves uniform exponential decay of the type-II error probability, with trace distance serving as the natural metric; (3) it proves that the test is quantum consistent and asymptotically optimal for arbitrary unknown pure or mixed states—thereby effecting a foundational transition from classical statistical testing to quantum information-theoretic hypothesis testing.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Heuristic Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Hoeffding's formulation and solution to the universal hypothesis testing (UHT) problem had a profound impact on many subsequent works dealing with asymmetric hypotheses. In this work, we introduce a quantum universal hypothesis testing framework that serves as a quantum analog to Hoeffding's UHT. Motivated by Hoeffding's approach, which estimates the empirical distribution and uses it to construct the test statistic, we employ quantum state tomography to reconstruct the unknown state prior to forming the test statistic. Leveraging the concentration properties of quantum state tomography, we establish the exponential consistency of the proposed test: the type II error probability decays exponentially quickly, with the exponent determined by the trace distance between the true state and the nominal state.

Problem

• Develop quantum analog for Hoeffding's universal hypothesis testing
• Use quantum state tomography to reconstruct unknown states
• Establish exponential consistency of type II error decay

Innovation

• Quantum analog to Hoeffding's UHT framework
• Quantum state tomography for state reconstruction
• Exponential consistency via trace distance