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Anytime-Valid Quantum Tomography via Confidence Sequences

Aldo Cumitini, Luca Barletta, Osvaldo Simeone

TL;DR

This work addresses sequential quantum state tomography by introducing Anytime-Valid QST (AV-QST), which attaches time-uniform confidence sets to point estimates using confidence sequences. AV-QST constructs uncertainty sets $\mathcal{C}_t^{\text{AV-QST}}(\alpha)$ via a likelihood-ratio statistic $R_t(\rho)$ anchored to a predictor $\hat{\rho}_{t-1}$, and proves that $\mathbb{P}(\forall t, \rho^* \in \mathcal{C}_t^{\text{AV-QST}}(\alpha)) \ge 1-\alpha$ through a test-martingale argument and Ville's inequality. The paper contrasts AV-QST with standard LR-QST and Bayesian QST, showing that AV-QST offers reliable time-uniform coverage and typically tighter uncertainty sets than LR-QST’s batch approach while avoiding prior-dependence and asymptotic limitations of Bayesian methods. Empirical results on two- and four-qubit systems demonstrate favorable performance in terms of miscoverage control and set size, supporting practical applicability for real-time quantum tomography and sequential decision-making.

Abstract

In this letter, we address the problem of developing quantum state tomography (QST) methods that remain valid at any time during a sequence of measurements. Specifically, the aim is to provide a rigorous quantification of the uncertainty associated with the current state estimate as data are acquired incrementally. To this end, the proposed framework augments existing QST techniques by associating current point estimates of the state with confidence sets that are guaranteed to contain the true quantum state with a user-defined probability. The methodology is grounded in recent statistical advances in anytime-valid confidence sequences. Numerical results confirm the theoretical coverage properties of the proposed anytime-valid QST.

Anytime-Valid Quantum Tomography via Confidence Sequences

TL;DR

This work addresses sequential quantum state tomography by introducing Anytime-Valid QST (AV-QST), which attaches time-uniform confidence sets to point estimates using confidence sequences. AV-QST constructs uncertainty sets via a likelihood-ratio statistic anchored to a predictor , and proves that through a test-martingale argument and Ville's inequality. The paper contrasts AV-QST with standard LR-QST and Bayesian QST, showing that AV-QST offers reliable time-uniform coverage and typically tighter uncertainty sets than LR-QST’s batch approach while avoiding prior-dependence and asymptotic limitations of Bayesian methods. Empirical results on two- and four-qubit systems demonstrate favorable performance in terms of miscoverage control and set size, supporting practical applicability for real-time quantum tomography and sequential decision-making.

Abstract

In this letter, we address the problem of developing quantum state tomography (QST) methods that remain valid at any time during a sequence of measurements. Specifically, the aim is to provide a rigorous quantification of the uncertainty associated with the current state estimate as data are acquired incrementally. To this end, the proposed framework augments existing QST techniques by associating current point estimates of the state with confidence sets that are guaranteed to contain the true quantum state with a user-defined probability. The methodology is grounded in recent statistical advances in anytime-valid confidence sequences. Numerical results confirm the theoretical coverage properties of the proposed anytime-valid QST.
Paper Structure (10 sections, 1 theorem, 14 equations, 3 figures)

This paper contains 10 sections, 1 theorem, 14 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Conventional Methods
  4. Frequentist Quantum State Tomography
  5. Bayesian Quantum State Tomography
  6. Anytime-Valid Quantum Tomography
  7. Confidence Sequences
  8. Theoretical Properties
  9. Numerical Results
  10. Conclusion

Key Result

Proposition 1

For any sequence of point estimates $\{\hat{\rho}_t\}_t$, the sequence of AV-QST uncertainty sets $\{\mathcal{C}_t^{\text{AV-QST}}(\alpha)\}_t$ in (eq:Ct) satisfies the anytime-valid coverage property (eq:prob).

Figures (3)

  • Figure 1: Confidence regions obtained as measurements are carried out on copies of a true state $\rho^*$ over discrete time $t=1,2,...$ using Bayesian credible regions (B-QST) Blume_Kohout_2010 and the proposed anytime-valid QST (AV-QST). Bayesian QST (B-QST) only provides asymptotic statistical guarantee (highlighted at $t=22$); while the proposed AV-QST ensures that the confidence region include the true state $\rho^*$ (red dot) with probability no smaller than $1-\alpha$uniformly over all time steps $t$.
  • Figure 2: Empirical miscoverage and normalized set size for B-QST Blume_Kohout_2010, LR-QST blume2012robust, and AV-QST (this paper) for the two-qubit setting. The lines represent median values and the shaded areas correspond to the interquartile range (25th-75th percentiles).
  • Figure 3: Empirical miscoverage and normalized set size for B-QST Blume_Kohout_2010, LR-QST blume2012robust, and AV-QST (this paper) for the four-qubit setting. The lines represent median values and the shaded areas correspond to the interquartile range (25th-75th percentiles).

Theorems & Definitions (2)

  • Proposition 1
  • proof