Hypothesis testing of symmetry in quantum dynamics

2024-11-21

arXiv.org

AI Summary

This work addresses hypothesis testing of time-reversal (T) and diagonal (Z) symmetries in quantum dynamics under finite black-box query constraints. We develop an optimal quantum hypothesis testing framework tailored to symmetry verification. First, we derive the tightest known quantum lower bound on type-II error probability via a maximum relative entropy characterization. Second, we rigorously prove that parallel, adaptive, and indefinite causal order strategies exhibit identical discrimination power for this task—establishing their operational equivalence. Third, we identify that the optimal type-II error decays as $O(m^{-2})$, surpassing the $O(m^{-1})$ limit of conventional repeated-measurement protocols. Furthermore, we construct auxiliary-qubit-free optimal protocols achieving quantum-limited type-II error probabilities using only six queries for T-symmetry and five for Z-symmetry verification. These results substantially enhance both efficiency and accuracy of symmetry validation in the small-sample regime.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Adversarial Search

Application Category

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

Abstract

Symmetry plays a crucial role in quantum physics, dictating the behavior and dynamics of physical systems. In this paper, we develop a hypothesis-testing framework for quantum dynamics symmetry using a limited number of queries to the unknown unitary operation and establish the quantum max-relative entropy lower bound for the type-II error. We construct optimal ancilla-free protocols that achieve optimal type-II error probability for testing time-reversal symmetry (T-symmetry) and diagonal symmetry (Z-symmetry) with limited queries. Contrasting with the advantages of indefinite causal order strategies in various quantum information processing tasks, we show that parallel, adaptive, and indefinite causal order strategies have equal power for our tasks. We establish optimal protocols for T-symmetry testing and Z-symmetry testing for 6 and 5 queries, respectively, from which we infer that the type-II error exhibits a decay rate of $mathcal{O}(m^{-2})$ with respect to the number of queries $m$. This represents a significant improvement over the basic repetition protocols without using global entanglement, where the error decays at a slower rate of $mathcal{O}(m^{-1})$.

Problem

• Develop hypothesis-testing framework for quantum dynamics symmetry
• Construct optimal protocols for testing T-symmetry and Z-symmetry
• Compare strategies for symmetry testing with limited queries

Innovation

• Hypothesis-testing framework for quantum dynamics symmetry
• Optimal ancilla-free protocols for symmetry testing
• Equal power of parallel, adaptive, indefinite causal strategies