Title: Online Learning of Pure States is as Hard as Mixed States
Venue: NeurIPS 2025
Accepted: yes
Keywords: quantum state learning

tomography

pure and mixed states

online learning

sequential fat-shattering dimension

smoothed analysis
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 $\epsilon$-realizable setting and (ii) learning the density matrix only partially, using smoothed analysis.


Peer reviews:
comments: SUMMARY:
This paper focuses on the problem of online learning for pure and mixed quantum states. The authors establish the equivalence between these two settings by first proving a lower bound on the sfat dimension and then demonstrating the equivalence for $L_1$-loss regret using Rakhlin’s characterization.
Furthermore, they extend the results to a more practical setting called $\epsilon$-realizable and discuss the concept of smoothness to better understand the effects of adversariality.

STRENGTHS AND WEAKNESSES:
Strengths:

1.	This paper provides rigorous proofs for every result it presents.

2.	The illustration of the sfat-tree construction makes the concept easy to understand.

3.	Smoothness is quite an interesting topic to discuss, as the adversariality of nature might not always be so extreme.

Weaknesses:

1.	While the contribution is solid, the proof of the lower bound by constructing an sfat-tree is straightforward and somewhat unsurprising. It appears to lack technical novelty.

QUESTIONS:
1.	Is it correct that your work closes the sfat dimension of pure states? What would be required to close the offline version, the $\delta$-fat-shattering dimension?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 4
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nQuantum state tomography seeks to reconstruct an unknown quantum state $\\rho$ from measurements. In standard (offline) settings, learning \\emph{pure} states is substantially easier than learning general \\emph{mixed} states, with the sample complexity for pure states scaling as $\\widetilde{\\Theta}(N^2)$ versus $\\widetilde{\\Theta}(N^4)$ for mixed states, where $N = 2^n$ for $n$ qubits. However, these results do not account for adversarial or sequential measurement scenarios.\n\n\nIn the online learning framework, learning proceeds as a repeated game over $T$ rounds. In each round $t$, the adversary selects a measurement $E_t$, and the learner predicts $\\mathrm{Tr}(E_t \\omega_t)$ for some hypothesis state $\\omega_t$. The learner then incurs a loss (e.g., $\\ell_1$ or $\\ell_2$) with respect to the true value $\\mathrm{Tr}(E_t \\rho)$. The performance is measured by the \\emph{regret} denoted $V_T(\\mathcal{C}_n)$ \nwhere $\\mathcal{C}_n$ is the set of $n$-qubit quantum states.\n\nTheir main results are as follows:\n \\textbf{No separation in online learning:} The paper proves that, in the online learning framework, \\emph{learning pure states is as hard as learning mixed states}. Both classes have essentially the same \\emph{sequential fat-shattering dimension}, leading to identical regret scaling.\n    \n \\textbf{Regret scaling:} For both pure and mixed states, the minimax regret scales as\n$$\n    V_T = \\Theta(\\sqrt{nT})\n    $$\n    where $n$ is the number of qubits and $T$ is the number of rounds.\n    \\item \\textbf{Technical approach:} The authors provide new lower bounds on the sequential fat-shattering dimension for pure states, matching those for mixed states, by constructing information-extracting trees and using matrix completion arguments.\n  \nThey have some additional results as well\n In the \\(\\epsilon\\)-realizable setting (where the learner receives noisy feedback), the lower bound on regret remains unchanged and is independent of $\\epsilon$.\n In the \\emph{smoothed analysis} setting (where the adversary is restricted to be close to a fixed distribution), the regret upper bound improves to\n        $$\n        V_T = O\\left(\\sqrt{\\frac{nT \\log T}{\\sigma}}\\right)\n        $$\n        where $\\sigma$ quantifies the degree of adversariality.\n\n\\textbf{Contrast with other settings:} In some restricted online learning scenarios (e.g., bandit feedback), pure states can be learned exponentially faster than mixed states. However, in the general online learning setting considered here, this separation vanishes.\n\\end{itemize}\n\n\\textbf{Implications:}\n\nThe result is counterintuitive compared to classical learning, where restricting the hypothesis class (e.g., to low-rank matrices) typically reduces complexity.\nIn adversarial or highly dynamic experimental settings, learning pure quantum states cannot be made significantly easier than learning mixed states, despite the much lower sample complexity for pure states in standard tomography.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- I think the paper addresses a fairly natural question and makes a surprising contribution.\n- I like the idea of using smoothed analysis for quantum learning (as it has shown some interesting results classical recently)\n- Understanding pure state versus mixed state learning is a natural question\n\nWeakness:\n- It is not very clear to me at least from the introduction, what is the main technical challenge of their work? I do feel many of their results extend straightforward from classical techniques, but it'd be nice if they made clear what was the roadblocks in establishing these results. \n\n- I think from an ML perspective and quantum perspective the technical novelty is a tad limited (again the authors should correct me if wrong), but i still its a nice surprising result.\n\n- The bounds are non-optimal, did you make any efforts in tightening the upper and lower bounds? It seems like the bounds that are shown here directly follow from classical results (which are quantized in this paper).\n\nQUESTIONS:\nNone.\n\nLIMITATIONS:\nMentioned above.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work explores quantum state learning with online learning theory. The authors prove that pure and mixed states share nearly the same sequential fat-shattering dimension, which is a key measure in online learning that dictates the performance of a learning algorithm. As a result, the regret scales identically for both types of states. This is a significant finding because, in standard settings, pure states are known to be much easier to learn than mixed states. The paper also extends its findings to more practical scenarios, including the case where measurements are not perfectly accurate, and smoothed analysis.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The central claim—that online learning of pure quantum states is as hard as learning mixed states—is highly impactful. It challenges the common intuition derived from standard tomography, where a clear separation in learning complexity exists. This can inspire the following research, including the classical learning theory. \n2. The authors introduce a new proof for the sequential fat-shattering dimension by the matrix completion. It is interesting to see new techniques coming out from the intersection between quantum and learning.\n3. The authors also introduce several extensions, including a realizable setting, smoothed analysis, and L2 loss. These additions make this work comprehensive.\n\nWeakness:\n1. As also mentioned in the conclusion and limitation section, it is not clear whether the conclusion for the online learning setting is still true for the offline setting (difference between sequential fat-shattering dimension and fat-shattering dimension). If true, this can lead to an even stronger result.\n2. The main result focuses on the L1 loss case. Authors also have some discussions on the L2 loss case, but not perfect. It feels a lack of explanation why L1 loss is important, or what are the technical limitations that authors can not explore other loss.\n3. For the smoothed analysis part, it is not clear when such a smooth distribution could be possible in practice and why this generalization is important for quantum.\n\nQUESTIONS:\n1. The paper's central result is that the separation in learning difficulty between pure and mixed states vanishes in the online setting. What are the specific properties of the adversarial online framework, as opposed to standard tomography and other quantum online learning works, that cause this equivalence in hardness?\n2. What is the main advantage of your new proof strategy from previous results? Is it possible to adapt this strategy in the classical learning theory?\n3. For the smoothed analysis, what would be a practical, experimental situation where the adversary's measurements correspond to $sigma$-smooth rather than completely adversarial or completely random?\n\nLIMITATIONS:\nThe limitations of this work are included in the Strengths and Weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}