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Towards Quantum Universal Hypothesis Testing

Arick Grootveld, Haodong Yang, Biao Chen, Venkata Gandikota, Jason Pollack

TL;DR

This work introduces quantum universal hypothesis testing (QUHT), the quantum analogue of Hoeffding's UHT, and shows that a framework combining quantum state tomography with hypothesis testing achieves exponential consistency. The type II error exponent scales with the squared trace distance $||\rho-\sigma||_1^2$ between the true state and the nominal state, with constants depending on dimension and measurement strategy. Across one-sample and two-sample settings, and for pure-state, qubit, and qudit cases, the paper derives explicit exponents for independent Pauli measurements and for entangled measurements, with entanglement offering the strongest exponents. The discussion situates QUHT relative to classical UHT and quantum information tools, highlighting the use of tomography-based concentration results and the trade-offs between measurement designs, while outlining open questions on tightening constants and extending the framework.

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.

Towards Quantum Universal Hypothesis Testing

TL;DR

This work introduces quantum universal hypothesis testing (QUHT), the quantum analogue of Hoeffding's UHT, and shows that a framework combining quantum state tomography with hypothesis testing achieves exponential consistency. The type II error exponent scales with the squared trace distance between the true state and the nominal state, with constants depending on dimension and measurement strategy. Across one-sample and two-sample settings, and for pure-state, qubit, and qudit cases, the paper derives explicit exponents for independent Pauli measurements and for entangled measurements, with entanglement offering the strongest exponents. The discussion situates QUHT relative to classical UHT and quantum information tools, highlighting the use of tomography-based concentration results and the trade-offs between measurement designs, while outlining open questions on tightening constants and extending the framework.

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.

Paper Structure

This paper contains 19 sections, 15 theorems, 67 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Quantum Hypothesis Testing
  4. Quantum Universal Hypothesis Testing
  5. Quantum Two-Sample Test
  6. Results
  7. Pure State QUHT
  8. Qubit QUHT
  9. Qudit Independent Measurements
  10. Entangled Measurements
  11. Discussion
  12. Proofs
  13. One-Sample Generic Result
  14. Two-Sample Generic Result
  15. Proof of Theorem 1 (Symmetric Pure State)
  16. ...and 4 more sections

Key Result

Theorem 1

Take $\rho = \lvert\phi\rangle\langle\phi\rvert \in \mathcal{D}(\mathcal{H})$ a pure state density operator and $\sigma \in \mathcal{D}(\mathcal{H})$ a (potentially mixed) density operator. In the QUHT setting, given $\sigma^{\otimes m}$, there exists a decision rule $M_m$ with independent measureme

Figures (2)

  • Figure 1: QUHT One Sample Problem. The yellow circle denotes the confidence region around our estimate $\hat{\sigma}$, with this region shrinking exponentially as $n$ increases. When $\rho$ is within this confidence region, we accept $H_0$.
  • Figure 2: QUHT Two Sample Problem. Colored circles represent the estimated states of $\rho$ and $\sigma$, while the enclosing balls denote the confidence regions surrounding the estimates. Depending on the overlap between the confidence regions we may accept or reject $H_0$.

Theorems & Definitions (21)

  • Theorem 1: Symmetric Pure State
  • Lemma 1: Pauli Operator Concentration
  • Theorem 2: Qubit One-Sample
  • Theorem 3: Qubit Two-Sample
  • Lemma 2: Theorem 1 guctua2020fast
  • Theorem 4: Independent One-Sample
  • Theorem 5: Independent Two-Sample
  • Lemma 3: Section 2.3 haah2016sample
  • Theorem 6: Entangled One-Sample
  • Theorem 7: Entangled Two-Sample
  • ...and 11 more