Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hypothesis testing of symmetry in quantum dynamics

Yu-Ao Chen, Chenghong Zhu, Keming He, Yingjian Liu, Xin Wang

TL;DR

This work formulates symmetry testing in quantum dynamics as an asymmetric hypothesis test against two symmetry families (time-reversal T-symmetry and diagonal Z-symmetry) using a fixed number of queries to an unknown unitary. It derives a quantum max-relative entropy lower bound on the type-II error and constructs optimal ancilla-free parallel protocols that achieve a decay of the type-II error as $\mathcal{O}(m^{-2})$, outperforming naive repetition that yields $\mathcal{O}(m^{-1})$ decay. The authors show that sequential, adaptive, and indefinite causal order strategies offer no advantage for these tasks under their framework, and provide a unified optimality condition linking zero-type-I-error results to the small-type-I-error regime. The results are computed via semidefinite programming using performance operators $\Omega^{(m)}_{\mu}$ and common-eigenstate constructions, with explicit protocols for $m=2,4,6$ in the T-symmetry case and $m=1$–$5$ in the Z-symmetry case. This framework advances practical quantum-dynamics symmetry testing and highlights fundamental limits in distinguishing symmetry-structured unitaries from Haar-random ones.

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 $\mathcal{O}(m^{-1})$.

Hypothesis testing of symmetry in quantum dynamics

TL;DR

This work formulates symmetry testing in quantum dynamics as an asymmetric hypothesis test against two symmetry families (time-reversal T-symmetry and diagonal Z-symmetry) using a fixed number of queries to an unknown unitary. It derives a quantum max-relative entropy lower bound on the type-II error and constructs optimal ancilla-free parallel protocols that achieve a decay of the type-II error as , outperforming naive repetition that yields decay. The authors show that sequential, adaptive, and indefinite causal order strategies offer no advantage for these tasks under their framework, and provide a unified optimality condition linking zero-type-I-error results to the small-type-I-error regime. The results are computed via semidefinite programming using performance operators and common-eigenstate constructions, with explicit protocols for in the T-symmetry case and – in the Z-symmetry case. This framework advances practical quantum-dynamics symmetry testing and highlights fundamental limits in distinguishing symmetry-structured unitaries from Haar-random ones.

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 with respect to the number of queries . This represents a significant improvement over the basic repetition protocols without using global entanglement, where the error decays at a slower rate of .

Paper Structure

This paper contains 11 sections, 16 theorems, 83 equations, 2 figures, 2 tables.

Table of Contents

  1. Preliminaries and basic setup
  2. Quantum comb and the Choi isomorphism
  3. Haar unitary integrals
  4. Identifying between symmetries of unitary
  5. Upper bound and protocol constructing
  6. Upper bound
  7. The optimal testing protocols
  8. Proofs of optimal testing protocols
  9. The native repetition testing protocols
  10. Lower bound
  11. Unified optimality condition of symmetry testing

Key Result

Theorem 1

There is an upper bound $\beta_{up}^{(m)}$ that $\beta_{0}^{(m)}\le\beta_{up}^{(m)}$, where $\beta_{0}^{(m)}$ is the minimal average type-II error for testing the unitary distributions $\mu_0$ and $\mu_1$ within $m$ calls without type-I error: or equivalently $|\psi\rangle$ is a common eigenstate of the $m$-th tensor powers of almost all unitaries in the support of $\mu_0$.

Figures (2)

  • Figure 1: The framework of symmetry distinguishing task via quantum comb. Given two unitary distributions $\mu_0$, $\mu_1$ of interest, the task is to perform operations on the given unitaries and decide which distribution it belongs.
  • Figure 2: Illustration of the protocol: To query the unknown unitary $U$$m$ times, the protocol for achieving the optimal type-II error probability is constructed using $U^{\otimes m}$ in parallel.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition S1: T-symmetry
  • Definition S2: Z-symmetry
  • Proposition S1: Performance operator
  • Lemma S2
  • Corollary S3
  • Theorem S4
  • Corollary S5
  • ...and 8 more