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Online Learning of Pure States is as Hard as Mixed States

Maxime Meyer, Soumik Adhikary, Naixu Guo, Patrick Rebentrost

TL;DR

This work studies online learning of quantum states, focusing on whether pure states remain easier to learn than general mixed states in adversarial online settings. By analyzing the sequential fat-shattering dimension, the authors prove that pure and mixed states share essentially the same complexity, leading to identical minimax regret scaling of $\Theta(\sqrt{nT})$ under the $L_1$ loss. They extend these results to the $\epsilon$-realizable setting and to smoothed online learning, deriving bounds that remain tight or near-tight under structured noise and restricted measurements, and connect the regret to the Rademacher complexity for the $L_2$ loss. The results provide a unified framework that bridges full tomography with PAC-like and smoothed online models, offering new insights into the fundamental limits of online quantum state learning and its practical implications for device certification and property estimation. Overall, the paper shows that the online learning hardness does not separate pure from mixed states in the general adversarial setting, with robust extensions to more realistic experimental scenarios.

Abstract

Quantum state tomography, the task of learning an unknown quantum state, is a fundamental problem in quantum information. In standard settings, the complexity of this problem depends significantly on the type of quantum state that one is trying to learn, with pure states being substantially easier to learn than general mixed states. A natural question is whether this separation holds for any quantum state learning setting. In this work, we consider the online learning framework and prove the surprising result that learning pure states in this setting is as hard as learning mixed states. More specifically, we show that both classes share almost the same sequential fat-shattering dimension, leading to identical regret scaling. We also generalize previous results on full quantum state tomography in the online setting to (i) the $ε$-realizable setting and (ii) learning the density matrix only partially, using smoothed analysis.

Online Learning of Pure States is as Hard as Mixed States

TL;DR

This work studies online learning of quantum states, focusing on whether pure states remain easier to learn than general mixed states in adversarial online settings. By analyzing the sequential fat-shattering dimension, the authors prove that pure and mixed states share essentially the same complexity, leading to identical minimax regret scaling of under the loss. They extend these results to the -realizable setting and to smoothed online learning, deriving bounds that remain tight or near-tight under structured noise and restricted measurements, and connect the regret to the Rademacher complexity for the loss. The results provide a unified framework that bridges full tomography with PAC-like and smoothed online models, offering new insights into the fundamental limits of online quantum state learning and its practical implications for device certification and property estimation. Overall, the paper shows that the online learning hardness does not separate pure from mixed states in the general adversarial setting, with robust extensions to more realistic experimental scenarios.

Abstract

Quantum state tomography, the task of learning an unknown quantum state, is a fundamental problem in quantum information. In standard settings, the complexity of this problem depends significantly on the type of quantum state that one is trying to learn, with pure states being substantially easier to learn than general mixed states. A natural question is whether this separation holds for any quantum state learning setting. In this work, we consider the online learning framework and prove the surprising result that learning pure states in this setting is as hard as learning mixed states. More specifically, we show that both classes share almost the same sequential fat-shattering dimension, leading to identical regret scaling. We also generalize previous results on full quantum state tomography in the online setting to (i) the -realizable setting and (ii) learning the density matrix only partially, using smoothed analysis.

Paper Structure

This paper contains 28 sections, 13 theorems, 47 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Works
  4. Preliminaries
  5. Quantum States and Measurements
  6. Online Learning of Quantum States
  7. Sequential fat-shattering dimension
  8. Lower Bounds for Online Learning of Quantum States
  9. Learning with respect to a single measurement
  10. Learning uniform superposition states
  11. Learning general states using Von Neumann measurements
  12. Tightness of sequential fat-shattering dimension of quantum states
  13. Online Learning of Pure States is as Hard as Mixed States
  14. Extensions of Online Learning of Quantum States
  15. The epsilon-realizable setting
  16. ...and 13 more sections

Key Result

Theorem 3.1

Let $E\in\mathrm{Herm}_{\mathbb{C}}(2^n), \operatorname{Spec}(E)\subset[0,1]$ be a fixed measurement, and $\mathcal{X}=\{E\}$ be the sample space. Let $\mathcal{H}_n=\{\operatorname{Tr}_\omega, \omega\in\mathcal{C}_n\}$ be the hypothesis class, where $\mathcal{C}_n$ is the set of all $n$-qubit quant

Figures (5)

  • Figure 1: Real-valued part of the halving tree $\mathbf{T}_h$, up to $\frac{1}{N}$ factor.
  • Figure 2: $\mathcal{X}$-valued part of the Von Neumann tree $\mathbf{T}_{vn}$.
  • Figure 3: Von Neumann Halving Tree (Horizontal)
  • Figure 4: Graph representation of $\mathcal{G}=(V,E)$
  • Figure 5: General form of the partial matrix described in \ref{['lemma_completion']}. The interrogation marks denote the unspecified entries in the matrix.

Theorems & Definitions (33)

  • Definition 2.1: Minimax regret
  • Definition 2.1: Minimax regret
  • Definition 2.2: Sequential fat-shattering dimension
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Lemma 3.5
  • Theorem 3.6
  • ...and 23 more