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Machine learning the effects of many quantum measurements

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A generative neural network trained only on measurement outcomes reveals entanglement induced by measuring many qubits, with no model of the state preparation and no postselection on outcome strings.

desk verdict The negativity detection is real and worth taking seriously; the learnability-transition story at large theta is oversold and needs a scaling test. read the letter →

arxiv 2509.08890 v1 pith:QT2UAL7K submitted 2025-09-10 quant-ph cond-mat.stat-mech

Wanda Hou , Samuel J. Garratt , Norhan M. Eassa , Eliott Rosenberg , Pedram Roushan , Yi-Zhuang You , Ehud Altman This is my paper · ORCID
classification quant-phcond-mat.stat-mech PACS 03.65.Ud03.67.-a03.67.Lx
keywords measurement-inducedentanglementunsupervisedlearningneuralnetworksclassicalshadowsnegativityphasetransitionclusterstateslearnability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Many measurements create long-range entanglement between the qubits that were not measured, but the post-measurement state is different for each random outcome string, so the entanglement is normally invisible without postselecting on those outcomes. This paper shows that the entanglement can be certified from raw data alone: an unsupervised attention-based neural network maps observed outcome strings to estimates of the two-qubit post-measurement state, and cross-correlating those estimates with held-out classical shadows gives positive lower bounds on measurement-averaged entanglement negativity. The evidence includes one-dimensional cluster chains up to 34 qubits and 6×6 two-dimensional arrays near an intermediate measurement angle. In the 2D data, sweeping the measurement basis produces a sharp change in how much the network can learn, which the paper identifies as a learnability transition closely related to a measurement-induced phase transition. The reason to care: if measurement-induced effects can be decoded from unlabelled data, they become observable in systems whose preparation is not accurately known, including prospective quantum error correction and control settings.

What carries the argument

The load-bearing objects are the quantum-classical cross-correlation inequalities. Given any computational model ρ^C_m for the two-probe post-measurement state, the quantum Kullback-Leibler divergence ensures S^QC_m = −Tr[ρ_m log ρ^C_m] ≥ S_m, and the projector onto negative eigenvalues of the partial transpose ensures N^QC_m = −Tr[(ρ_m)^TA Π((ρ^C_m)^TA)] ≤ N_m. These bounds become experimentally accessible because weighted averages of classical shadows over repeats converge to the same quantities: the shadow noise has zero mean. The second ingredient is the unsupervised generative network: an attention-based transformer that takes the outcome string m as input and outputs a valid 4×4 densit

What would settle it

Run the same 2D experiment at large measurement angle with a much larger, better-trained network (or an exact classical simulation of the outcome-to-state map). If a scalable model then reproduces the gate-based negativity bound, the claimed learnability transition is an artifact of finite model capacity; if even an optimal learner fails while the gate-based model succeeds, the transition is intrinsic.

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Extended reading notes

Core claim

Central claim: measurement-induced entanglement can be revealed from measurement data alone—no model of the prepared state, no postselection on outcome strings. For any model m→ρ^C_m, cross-correlations give S^QC ≥ S_m and N^QC ≤ N_m, and classical-shadow averages realize these bounds because shadow noise averages to zero. An unsupervised attention-based network learns such a map from outcomes. Cross-correlating its predictions with held-out shadows yields positive negativity bounds in 1D cluster chains up to L=34 (matching a gate-based model) and in 2D 6×6 arrays near an intermediate angle. The network's sharp failure at larger angles is read as a learnability transition tied to the measure

Load-bearing premise

The central claim would collapse if the network's failure at large measurement angles were due to the fixed 20-epoch training budget or the chosen architecture rather than to an intrinsic difficulty of learning the measurement-to-state map.

Editorial extensions

If this is right

  • If a trained network can certify measurement-induced entanglement without a state-preparation model, then measurement-induced phase transitions can in principle be located in any experimental platform with repeated state preparation and single-shot readout.
  • Because the same cross-correlation bounds apply to coherent information and general observables, the scheme transfers directly to quantum error correction, where syndrome-to-logical-state maps could be learned rather than assumed.
  • The data show that the detectability window of the network (intermediate angles) is narrower than the entanglement window seen by the gate-based model, so a 'learnability transition' does not coincide exactly with the physical transition at finite size; this distinction matters for interpreting measurement-induced phase transition experiments.
  • In 1D, learned models match gate-based models for negativity bounds even though they ignore the preparation circuit, suggesting the method remains useful as systems grow and gate-level descriptions become unreliable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A sharper test the paper leaves implicit: retrain at large angles with larger capacity or longer schedules; if the entropy bound drops toward the gate-based value, the observed transition is partly a model-capacity artifact rather than an intrinsic property of the data.
  • The learned model's insensitivity to outcomes in the farthest row, combined with sensitivity to the adjacent row, suggests the trained network itself can be used as a diagnostic of the 'light cone' of measurement-induced correlations—an experimental tool not developed in the paper.
  • If the same unsupervised protocol is applied to monitored random circuits, the peak in KL-divergence reduction during training may serve as a generic finite-size order parameter for measurement-induced criticality, independent of any chosen entanglement measure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper reports experiments on Google's Sycamore and Willow processors in which post-measurement states of two probe qubits are characterized from the outcomes of many other measurements. Cluster states are prepared in 1D (L up to 34) and 2D (6x6) arrays; all non-probe qubits are measured, and the probe qubits are measured in random Pauli bases to generate classical shadows. A generative transformer NN is trained unsupervised, mapping outcome sets m to estimated probe states rho^C_m; cross-correlations with independent test shadows, via the inequalities S^SC >= S_m and N^SC <= N_m (Eqs. 1-2), yield bounds on average entropy and negativity. Main claims: (i) in 1D, the NN and tensor-network models give positive negativity bounds comparable to a gate-based model, up to L=34; (ii) in 2D, a peak in the detected negativity and in the amount of information learned appears at intermediate measurement-basis angle theta, while at large theta the NN produces near-maximally-mixed estimates and fails to detect MIE although the gate-based model still does; this failure is interpreted as a learnability transition related to the measurement-induced phase transition, observable without advance knowledge of the quantum state and without postselection.

Significance. The methodological core is sound and valuable. The cross-correlation bounds (Eqs. (1) and (2)) are derived self-contained in the SI and are rigorous for any model rho^C_m; the train/test split makes the ML evaluation non-circular; the code and data are released (Refs. [47,48]); and the 1D experiment gives clean evidence of MIE up to L=34 from unsupervised models. The 2D results provide a concrete finite-size demonstration that a data-driven model can certify positive negativity in a window of intermediate measurement angles, and the row-flip sensitivity analysis (SI Fig. 12A) is a falsifiable diagnostic of nonlocal dependence on outcomes. The principal weakness is the interpretive claim that the NN's failure at large theta is a physical learnability transition: the paper's own Fig. 3B shows that the gate-based model captures structure that the 20-epoch NN misses, so without a scaling study the observed transition may reflect model capacity and training budget rather than an information-theoretic MIPT signature. I therefore view the MIE detection claims as supported, and the transition narrative as requiring additional evidence or substantially weakened wording.

major comments (2)
  1. [Two dimensions; Figs. 3A-C, 4A] The learnability-transition claim rests on the NN's failure at large theta, but the evidence is one architecture trained for a fixed t=20 epochs. Fig. 3B shows the gate-based model yields S^QC well below 2 bits at theta/pi ~ 0.5, so the m -> rho_m structure is present in the data, and the text concedes the NN cannot approximate it. The peak in D_KL reduction (Fig. 3C) and the vanishing negativity bound (Fig. 4A) could then reflect model capacity or the 20-epoch budget rather than an intrinsic transition. Please report (a) training curves beyond 20 epochs (at least for theta/pi ~ 0.3-0.5) or saturation evidence; (b) a scaling test in model size/training-set size showing the ~2-bit plateau persists; (c) the Fig. 3C peak location relative to the gate-based finite-size crossing (SI Fig. 10A). Without these, the abstract's transition claim should be weakened to 'this NN under the fixed traini
  2. [Abstract vs SI Sec. I A] The abstract's 'without postselection' is internally inconsistent with the SI, which discards runs flagged by error-detection qubits and states 'This corresponds to post-selecting on error-free repeats of experiment.' This is not postselection on the exponentially many outcome strings m, so the method's advantage over Ref. [16] survives, but the wording should be 'without postselecting on measurement outcomes' and the discarded fraction should be stated. Also, the NN receives positional encodings and a 2D causal masking schedule; 'without advance knowledge of the quantum state' should be scoped to exclude knowledge of the measurement layout.
minor comments (4)
  1. [SI Sec. V B, Eq. (22)] Both rho_i1 and rho_i2 are defined with averages over R1; the second should average over R2.
  2. [Figs. 3-4] The axes and captions use 'mu' (mu/deg) while the text uses theta for the measurement basis; unify the notation, preferably with theta/pi as in the text.
  3. [Fig. 3C] The caption states the plotted quantity is S^QC_m(t) - S^QC_m(20); clarify that this equals D_KL(t) - D_KL(20) and describe how non-monotonic training would appear.
  4. [SI Sec. V B] For the classification estimates in Eq. (23), report the class sizes R1, R2 and whether depolarization was needed; the text says it was not, but the convergence to the post-measurement ensemble average should be stated more explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cross-correlation bounds are rederived in the SI, the neural network is trained and evaluated on independent data splits, and the learnability-transition claim is empirical rather than definitional.

full rationale

The derivation chain is self-contained. The central inequalities (Eqs. (1) and (2)) are not merely imported from Ref. [22] (by coauthors Garratt and Altman); the Supplemental Information (Sec. II) rederives them from the classical-shadow property and non-negativity of the quantum relative entropy, so the citation is not load-bearing. The neural-network models are trained on ~7.8e7 repeats and cross-correlated with a separate 1e6-repeat test set, so the reported positive negativity bounds and entropy bounds are out-of-sample evaluations; no parameter of the neural network is fitted to the claimed negativity. The only fitted parameter is the depolarizing strength epsilon=0.3 in the gate-based baseline, which is a comparison model, not the machine-learning prediction. The 'learnability transition' is an empirical observation (peak in D_KL reduction and failure at large theta) and is not derived from a uniqueness theorem or from self-citation; whether the large-theta failure persists with more capacity or training time is a scaling/correctness concern, not circularity. One internal inconsistency is worth noting: the abstract claims observation 'without postselection,' while the SI states that error-detection runs are discarded, 'This corresponds to post-selecting on error-free repeats of experiment.' This is an overstatement, but it is not a circular step. Overall, no load-bearing step reduces by construction to its own inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper's central claim is an experimental demonstration. It borrows the cross-correlation inequalities from Ref. [22] and the classical shadow method from Ref. [29]. The main tunable knob introduced in this paper is epsilon=0.3 for the gate-based baselines; the ML pipeline depends on hyperparameters such as 20 training epochs and the network architecture. No new physical entities are introduced.

free parameters (2)
  • Depolarizing strength epsilon in gate-based model = 0.3
    Chosen because it improves the entropy and negativity bounds for both 1D and 2D arrays (footnote 34). This is a hand-tuned parameter in the comparison model, not in the ML model, but it affects the gate-based baseline curves used to validate the ML results.
  • NN training epochs = 20
    The paper reports results after t=20 epochs; the 'learnability transition' conclusion depends on this fixed training budget. With more epochs or a different architecture, the network might learn the mapping at large theta, which would change the interpretation.
assumptions (5)
  • standard math Quantum relative entropy is non-negative and non-increasing under quantum channels (data processing inequality), giving S_QC >= S and I_QC <= I.
    Used in Supplemental Sec. II to derive the entropy upper bound and the coherent information lower bound from cross-correlations.
  • standard math The classical shadow construction with the random Pauli bases satisfies E[rho_S] = rho_m (Eq. 6).
    Invoked in Supplemental Sec. II; needed so that averages over shadows converge to the true post-measurement density matrix in the cross-correlation formulas.
  • domain assumption The cluster state preparation circuit is well described by the ideal unitaries plus a simple depolarizing channel; the gate-based model uses a pure-state simulation with a single depolarizing parameter epsilon.
    Used in Supplemental Sec. IV to compute the gate-based rho_C_m. If the real noise is not depolarizing, the epsilon-depolarized pure model may be a poor reference.
  • domain assumption Discarding runs with detected measurement errors does not bias the ensemble of post-measurement states in a way that changes the sign of the negativity bound.
    Used in Supplemental Sec. I A; the experiment postselects on error-free error-detection qubits. Correlated measurement errors could bias the detected entanglement.
  • domain assumption The trained neural network generalizes from the training repeats to the test repeats; the test data are independent and identically distributed.
    Needed to interpret cross-correlations on 'previously unseen' data as bounds on physical states. If the test set is not independent, e.g. due to calibration drift, the bounds can be biased.

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Cite this review

Pith. "Pith review of Machine learning the effects of many quantum measurements." pith.science (2026). https://pith.science/paper/QT2UAL7K

@misc{pith2026250908890,
  author       = {Pith},
  title        = {Pith review of: Machine learning the effects of many quantum measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QT2UAL7K}},
  note         = {Machine review of arXiv:2509.08890}
}
read the original abstract

Measurements are essential for the processing and protection of information in quantum computers. They can also induce long-range entanglement between unmeasured qubits. However, when post-measurement states depend on many non-deterministic measurement outcomes, there is a barrier to observing and using the entanglement induced by prior measurements. Here we demonstrate a new approach for detecting such measurement-induced entanglement. We create short-range entangled states of one- and two-dimensional arrays of qubits in a superconducting quantum processor, and aim to characterize the long-range entanglement induced between distant pairs of qubits when we measure all of the others. To do this we use unsupervised training of neural networks on observations to create computational models for post-measurement states and, by correlating these models with experimental data, we reveal measurement-induced entanglement. Our results additionally demonstrate a transition in the ability of a classical agent to accurately model the experimental data; this is closely related to a measurement-induced phase transition. We anticipate that our work can act as a basis for future experiments on quantum error correction and more general problems in quantum control.

Figures

Figures reproduced from arXiv: 2509.08890 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Measurement-induced entanglement in one [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Learning post-measurement states of two-dimensional cluster states of 6 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Measurement-induced mixed-state entanglement in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Experimental realization of the quantum circuit that creates one-dimensional cluster states, and the measurements [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Experiments on two-dimensional cluster states. The cluster state (on red and green qubits) is prepared by applying [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Transformer NN for predicting the probe qubit state from preparation measurements. The input is tokenized and [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Examples of 2D causal masking patterns used during training. In each iteration, the model predicts the probe state [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Tensor network for constructing [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Numerical calculations of average entanglement negativity between probe qubits [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Measurement-induced mixed-state entanglement in one-dimensional cluster states. As in the main text, the probe [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

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Forward citations

Cited by 2 Pith papers

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    On the final row, perform measurements on all qubits except the probesAandB. Using this scheme, we can determine the post-measurement state of the two probe qubits while only ever storing a quantum state of 2Lqubits (rather than the fullL 2). In Fig. 10 we characterize the pur...

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.