Learning entanglement from tomography data: contradictory measurement importance for neural networks and random forests

Jaros{\l}aw Paw{\l}owski; Katarzyna Roszak; Mateusz Krawczyk; Pavel Bal\'a\v{z}

REVIEW 5 major objections 6 minor 51 references

Learning entanglement from tomography data: contradictory measurement importance for neural networks and random forests

T0 review · 5 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A neural network and a random forest learn two-qubit entanglement from opposite measurements.

desk verdict A genuinely new empirical observation—NN and RF rank tomography blocks oppositely for entanglement—but the global RF claim is overstrong given the class-dependent Shapley results; deserves refereeing after harmonizing the perturbation metrics. read the letter →

arxiv 2505.03371 v2 pith:KJ5WKVB3 submitted 2025-05-06 quant-ph

Pavel Baláž , Mateusz Krawczyk , Jaros{l}aw Paw{l}owski , Katarzyna Roszak This is my paper · ORCID

classification quant-ph PACS 03.67.Mn03.67.-a

keywords entanglementquantificationtwo-qubittomographyrandomforestsneuralnetworksfeatureimportanceShapleyvaluesconcurrencemeasurement

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

The paper claims that when a neural network and a random forest are trained to quantify two-qubit entanglement from standard projective tomography data, they reach comparable predictions but assign importance to opposite kinds of measurements. The neural network treats measurements in block $M_D$, which carry information about non-local coherences, as most important; the random forest treats the occupation measurements in block $M_A$ as most important. The paper argues this is not a quirk of one dataset: the same ordering appears under mean-decrease-in-impurity, perturbation-based, and Shapley-value analyses, and it disappears when the models are trained on density-matrix elements directly. The result matters because experimental tomography data are often noisy or incomplete, so knowing which measurements a model actually relies on determines how much one can trust its prediction. The authors suggest this difference can be exploited to obtain more reliable entanglement information from noisy tomography data.

What carries the argument

The load-bearing mechanism is the linear scattering of information in two-qubit projective tomography, encoded in the four measurement blocks of Eq. (6). The measurement outcomes are linear functions of the density-matrix elements, Eqs. (7)--(10): $M_A$ gives the diagonal elements directly, $M_B$ and $M_C$ mix diagonals with local coherences, and $M_D$ mixes diagonals, local coherences, and non-local coherences. Because the diagonals must be known to convert $M_B$ and $M_C$ into actual off-diagonal elements, the $M_A$ block functions as a gatekeeper for all coherence information in the measurement representation; this is what the random forest's split-based learning exploits. The neural network, by contrast, learns an internal representation that recovers the direct role of $M_D$. The supporting analytic comparison is the Shapley-value analysis of the concurrence computed from quantum-state reconstruction, which shows the same ranking as the neural network. The disappearance of the contradiction for density-matrix inputs is the control that isolates the scattering mechanism.

What would settle it

Retrain both models from scratch after deleting each of the four measurement blocks in turn (leave-block-out). The paper's claim predicts that removing $M_A$ damages the random forest most while the neural network is damaged most by removing $M_D$; if retraining produces the same damage ranking for both models, the reported contradiction is an artifact of the attribution tools rather than a genuine difference in how the models learn.

Watch

Extended reading notes

Core claim

The central discovery is a model-class-dependent reversal in feature importance for entanglement quantification. For two-qubit states labeled by concurrence $C(\rho)=\max(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4)$, the 16 projective measurements split naturally into four blocks defined in Eq. (6): block $M_A$ contains the occupation probabilities $m_{00},m_{01},m_{10},m_{11}$, blocks $M_B$ and $M_C$ contain information about local coherences, and block $M_D$ contains information about non-local coherences, which the paper identifies as critical for entanglement. The neural network, like the analytic quantum-state reconstruction, ranks $M_D$ first, then $M_B,M_C$, then $M_A$; the random forest ranks $M_A$ first and $M_D$ last. The paper traces the inversion to the tomography equations (7)--(10): the diagonal outcomes in $M_A$ enter the reconstruction of every off-diagonal density-matrix element, so the random forest, which builds splits on single features, latches onto $M_A$ as the gatekeeper for all coherence information, while the neural network effectively inverts the linear relations and reads the entanglement-relevant coherences directly. When both models are fed density-matrix elements instead of raw measurements, their importance rankings coincide.

Load-bearing premise

The conclusion rests on the assumption that the feature-importance measures, especially the random forest's mean-decrease-in-impurity and the perturbation-based accuracy drops, correctly reveal which measurements the models actually use even though the tomography features are strongly correlated with one another.

Editorial extensions

If this is right

On full, clean tomography data, the neural network classifies two-qubit entanglement more accurately than the random forest (accuracy 0.922 versus 0.879 on the test set), so the better learner and the more interpretable learner are not the same model.
When a single measurement is corrupted by noise, the random forest degrades more gracefully at high noise levels than the neural network, because many trees vote over different feature subsets; the neural network is more accurate at low noise.
The block-importance rankings provide a practical criterion for choosing a model: if occupation measurements are the reliable part of a tomography run, the random forest's output should be preferred, while the neural network is the safer predictor when coherence measurements carry the signal.
Feature importance in machine-learning entanglement quantification is not a property of the data alone; it depends on the model family, so interpretability claims should be stated relative to the learner.
The same models trained on density-matrix elements agree on importance, confirming that the discrepancy is caused by the dispersed, interdependent structure of measurement data rather than by different notions of entanglement.

Reading between the lines

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

A testable extension is to train the same model families on three-qubit or higher-dimensional partial tomography and compare leave-one-block-out retraining; the prediction would be that tree ensembles again shift importance onto directly measured diagonal blocks while networks track coherence-relevant blocks, with the gap widening as Hilbert-space dimension grows.
The same informational-scattering mechanism should appear in any sensor-fusion task where raw features are linear mixtures of latent variables and one class of features is needed to calibrate the others; random forests would be expected to key on the calibrating features, neural networks on the latent-relevant ones.
A practical diagnostic for noisy experiments is to compare model predictions under targeted measurement noise and use disagreement between the two models as a flag that the tomography data are unreliable in a particular block.
Because Shapley values for the random forest differ between separable and strongly entangled samples (block $M_D$ dominating for separable, block $M_A$ for entangled), model confidence could be recalibrated separately for the two regimes, potentially improving low-concurrence classification.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

Here's the bottom line: this paper reports that a neural network and a random forest trained on the same two-qubit tomography data attribute opposite importance to measurement blocks—the NN leans on non-local coherences (block MD), the RF on occupation measurements (block MA)—and that this difference vanishes when both models are trained on density matrix elements instead. That is a new empirical result in the ML-entanglement literature, and it is the sort of finding that should make experimentalists think twice before trusting a single model's feature importance ranking as a window into the physics.

The paper earns credit for not relying on one interpretability tool. The RF's MA dominance shows up in MDI, perturbation, and Shapley analyses (with the class-dependent caveat below), and the density-matrix control is a sensible check. The qualitative story—that occupations are needed to invert the tomography equations, so the RF builds its decision rules around them—is plausible and worth testing further.

The soft spots are manageable but real. The perturbation measure for the RF (Eq. 17) is a drop in classification accuracy, while for the NN (Eq. 18) it is the mean absolute error of the regression output, baseline included. That makes cross-model magnitudes meaningless, though within-model rankings over blocks remain informative because the baseline is fixed. The authors should harmonize the measure or explicitly compare relative increases over baseline. More substantively, the Shapley values reveal that the RF's preferred block depends on the class: for separable states, MD is the most important block for the RF; MA dominates only for strongly entangled states. The paper's global claim that "RF signifies dominance of occupations" is therefore overstated—it is an average over a test set in which entangled states are slightly overrepresented. The class-resolved Shapley results deserve a more central place in the narrative.

Two smaller issues: the NN classification threshold τ_NN is tuned on the test set, which inflates the reported accuracy edge, and no code, data, or error bars are provided. The threshold tuning does not affect the regression-based feature importance, but it should still be fixed.

I left this paper feeling the central qualitative finding—model-class-dependent measurement importance rooted in the structure of tomography—survives a careful read. It deserves a serious referee. My recommendation is conditional acceptance: harmonize the perturbation metrics, re-frame the RF claim to reflect the class dependence, and clean up the test-set tuning.

Referee Report

5 major / 6 minor

Summary. The paper trains a two-hidden-layer MLP regressor and several random-forest classifiers to predict entanglement (concurrence) of two-qubit states from the 16 measurement outcomes of the James-Kwiat-Munro-White tomography protocol. On a synthetic data set of 460,000 training and 46,000 test states, the MLP achieves higher classification accuracy (0.922 vs. 0.879 for the best RF). Using three interpretability tools (MDI, noise-perturbation importance, and Shapley values), the authors report that the RF attributes most importance to the occupation-measurement block MA, whereas the NN attributes most importance to the non-local-coherence block MD. They interpret this as a model-class-dependent effect caused by information scattering in the linear tomography equations, and they claim the discrepancy disappears when the models are trained directly on density-matrix elements.

Significance. If the reported discrepancy is real, the paper provides a useful cautionary example that feature-importance rankings in machine-learned entanglement witnesses can be model-class-dependent rather than reflecting the intrinsic physical importance of the measurements. The work combines three interpretability routes and includes hyperparameter details and class-conditional Shapley analyses, which is a strength. However, several methodological mismatches (between the RF and NN perturbation metrics, between the SHAP targets, and in the threshold-selection procedure) currently prevent the headline claim from being established at the level of rigor required for a journal publication. The results are nonetheless amenable to correction within the manuscript's scope.

major comments (5)

[Sec. VI B 2, Eqs. (17) and (18)] The perturbation importance measures for the RF and NN are not commensurable. E_P^RF in Eq. (17) is the drop in classification accuracy caused by perturbing one measurement, whereas E_P^NN in Eq. (18) is the mean absolute error between the perturbed NN output and the ground-truth concurrence without subtracting the unperturbed baseline error. Consequently, the block orderings in Fig. 5(a) and Fig. 5(c) may differ simply because E_P^NN includes the model's baseline error, not because the models use the measurements differently. The authors should redefine E_P^NN as a drop in accuracy (using the same classification threshold as for the RF) or as the increase in error relative to the unperturbed case, and use identical metrics for both model classes.
[Sec. VI B 3, Fig. 6] The Shapley values for the RF are computed for the class-probability output (with TreeExplainer, as stated in Appendix C), while those for the NN are computed for the continuous concurrence output, and the SHAP explainer used for the MLP is not specified. These are different target functions, so the averaged SV patterns in Figs. 6(a,b) and Figs. 6(e,f) are not directly comparable. Additionally, Fig. 6(a) shows that for separable states the RF's most important block is MD, contradicting the unqualified claim that the RF always favors MA; the MA dominance appears only for strongly entangled states (Fig. 6(b)). The claims should be made class-conditional and the SHAP settings for the NN should be reported.
[Sec. VI A, Eq. (14)] The NN classification threshold tau_NN is chosen by maximizing the area under the precision-recall curve, but the manuscript does not state whether this is done on the test set Stest or on a validation set. If the threshold is optimized on the test set, the accuracy, precision, and recall values for the MLP in Table I are optimistically biased relative to the RF classifiers, whose decision threshold is not tuned. The threshold-selection procedure should be described precisely and performed on a validation split.
[Sec. VI B 1, Fig. 4] Mean Decrease in Impurity is known to be biased for correlated predictors, and the tomography measurement features are strongly interdependent through the linear relations in Eqs. (7)-(10). The paper does not discuss this limitation or provide a complementary permutation-based importance for the RF, so the MDI ranking in Fig. 4 should be treated with caution as evidence for the claim that occupations are the most important features for the RF.
[Sec. VI B 2, final paragraph] The statement that the discrepancy disappears when the models are trained on density-matrix elements is a central piece of the proposed information-scattering explanation, but no quantitative result, figure, or table is presented for this control experiment. The authors should provide the corresponding feature-importance results for both models trained on density-matrix inputs, or explicitly mark this as a claim requiring further verification.

minor comments (6)

[Abstract] In the abstract, 'measurements which arthe most important' contains a typo; it should read 'are the most important'.
[Eq. (18)] The subscript notation E_jk in Eq. (18) is not defined; specify that j indexes test samples and k indexes independent noise realizations.
[Figs. 5 and 10] The vertical axis labels in Figs. 5 and 10 are not fully defined; Fig. 10 in particular should state whether the ordinate is classification accuracy or the perturbation measure E_P, and the color scale in Fig. 5 should be labeled.
[Sec. V B] The sentence 'utilizing a ReLU activation function' should be plural because each hidden layer has its own ReLU activation.
[Sec. VI B 2] The phrase 'it's results' should be 'its results'.
[Appendix B, Table II] The statement that 'The explanations of these hyperparameters can be found within the main text' is inaccurate; the hyperparameters are defined in the appendix itself, so the cross-reference should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the feature-importance claims are post-hoc empirical properties of models trained on independently computed labels and measurements.

full rationale

The paper's derivation chain is empirical rather than deductive, and no claimed prediction is defined in terms of its own output. Concurrence labels are computed by the Wootters formula, Eq. (1), directly from sampled density matrices, while the features are the projective-measurement outcomes of Eq. (4); the two are generated independently, and neither is a fitted parameter. The trained NN and RF models are then probed by MDI, perturbation measures, and Shapley values, all of which are post-hoc properties of the fitted models. None of those importance scores enters the definition of the target C(rho) or the feature construction, so there is no fitted input renamed as a prediction. The nearest concern is that Eq. (18) is an absolute error rather than a perturbation-induced change, making the RF/NN importance comparison imperfect; however, that is a comparability or correctness issue, not a circular reduction, because neither Eq. (17) nor Eq. (18) is derived from the other and the block ordering remains a property of the trained models. The only self-citation that could be scrutinized is Ref. [11] for the quantum-circuit sampling method; that citation supplies a data-sampling procedure, not an assumption equivalent to the entanglement-importance result, and the dataset also includes Haar-random and explicitly added Bell states. The inversion-based explanation that occupation measurements are needed to infer other density-matrix elements is a direct consequence of the linear relations in Eqs. (7)-(10), and the density-matrix-input control in Sec. VI B tests rather than assumes that explanation.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central claims rest on standard concurrence theory, an ideal projective-measurement model, a representative random-state dataset, and several unvalidated tooling assumptions about feature importance measures. No new physical entities are introduced. The main fitted parameter is the NN classification threshold, tuned on the test set.

free parameters (6)

tau_NN (NN classification threshold) = 0.03
Chosen to maximize NN accuracy via the precision-recall curve on the test set (Sec. VI A, Eq. 14); tuning on test data can inflate the reported NN-RF accuracy gap.
tau (separability threshold) = 1e-6
Defines separable versus entangled labels and is used to balance the dataset (Sec. IV); arbitrary numerical cutoff.
Perturbation noise amplitude sigma = 0.05 for importance maps; 0.01 for reconstruction propagation
Chosen by hand for the perturbation analysis (Sec. VI B 2, Fig. 5); Fig. 10 varies sigma, so the qualitative RF-vs-NN noise trend is not tied to a single value, but exact importance magnitudes depend on it.
Dataset balance and Bell-state oversampling = 53% entangled / 47% separable; 20,000 maximally entangled states
Sampling design chosen to balance classes (Sec. IV); affects accuracy and Shapley value magnitudes.
RF1 hyperparameters = n_estimators=1000, max_features=4, max_depth=None, min_samples_leaf=None
Selected by grid search with cross-validation (Table II); all three RF variants show the same feature-importance ordering, so the core discrepancy is robust to these choices.
MLP architecture = 2 hidden layers, 128 units each, ReLU, linear output, MSE loss
Chosen by hand (Sec. V B); a 10-layer variant gave comparable results, so the NN importance result is not tightly coupled to the exact architecture. Optimizer, learning rate, epochs and seeds are not reported.

assumptions (7)

standard math Wootters concurrence (Eqs. 1 and 2) is a valid entanglement measure and the labels are ground truth.
Used to label all training and test states; standard result cited to Wootters, not derived in the paper.
domain assumption Tomography measurements are ideal projective expectation values m_ij = Tr{rho mu_i ⊗ mu_j} with no shot noise or basis misalignment in the training data.
Eqs. (7)-(10) assume perfect measurements; realistic Poissonian shot noise is not part of the training distribution, so the noisy-data conclusions use synthetic additive noise.
domain assumption The random sampling methods (quantum circuits and Haar measure) generate a representative distribution of two-qubit states.
The paper relies on the generated dataset to stand for generic two-qubit states (Sec. IV).
ad hoc to paper Mean Decrease in Impurity is a reliable feature-importance indicator for the RF despite the strong correlations among measurement features.
MDI is known to be biased for correlated predictors; the paper uses it as primary RF evidence (Sec. VI B 1) without discussing this bias.
ad hoc to paper The SHAP explainer used for the neural network produces faithful Shapley values for the MLP.
The paper states SHAP was used (Sec. VI B 3) but does not name the explainer or its assumptions for MLPs.
ad hoc to paper The perturbation measures in Eqs. (17) and (18) are comparable across model types.
RF importance is measured as a drop in classification accuracy, NN importance as mean absolute error in predicted concurrence; these are different scales and can yield different rankings.
ad hoc to paper The RF's preference for occupations is caused by the interdependence of measurement data in the linear tomography equations.
The paper asserts this causal mechanism in the abstract and Sec. VI B 2, but feature-importance patterns alone do not establish the mechanism; it is a plausible post-hoc interpretation.

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Cite this review

Pith. "Pith review of Learning entanglement from tomography data: contradictory measurement importance for neural networks and random forests." pith.science (2026). https://pith.science/paper/KJ5WKVB3

@misc{pith2026250503371,
  author       = {Pith},
  title        = {Pith review of: Learning entanglement from tomography data: contradictory measurement importance for neural networks and random forests},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KJ5WKVB3}},
  note         = {Machine review of arXiv:2505.03371}
}

read the original abstract

We study the effectiveness of two distinct machine learning techniques, neural networks and random forests, in the quantification of entanglement from two-qubit tomography data. Although we predictably find that neural networks yield better accuracy, we also find that the way that the two methods reach their prediction is starkly different. This is seen by the measurements which arthe most important for the classification. Neural networks follow the intuitive prediction that measurements containing information about non-local coherences are most important for entanglement, but random forests signify the dominance of information contained in occupation measurements. This is because occupation measurements are necessary for the extraction of data about all other density matrix elements from the remaining measurements. The same discrepancy does not occur when the models are used to learn entanglement directly from the elements of the density matrix, so it is the result of the scattering of information and interdependence of measurement data. As a result, the models behave differently when noise is introduced to various measurements, which can be harnessed to obtain more reliable information about entanglement from noisy tomography data.

Figures

Figures reproduced from arXiv: 2505.03371 by the authors.

**Figure 2.** FIG. 2. Measurement perturbations: (a) methodology, (b) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** FIG. 4. Mean Decrease in Impurity (MDI) of RF1 model [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: summarizes the prediction fidelity when a single measurement characterized by indices i, j = 0, 1, 2, 3 is FIG. 5. Perturbation measure E p for random noise applied to a single measurement outcome with σ = 0.05 for (a) RF1 model trained using Strain, (b) RF1 model trai…

**Figure 6.** Figure 6: FIG. 6. Shapley values Φ [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. Cumulative explained variance ratio of PCA per [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8. Measurement vectors projected into 2D space using [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 10.** Figure 10: FIG. 10. Perturbation measure for (a) random forest classi [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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In an RF, feature importance is typically determined by measuring the impact of each feature on the model’s predictive accuracy, e.g

Mean Decrease in Impurity One of the key advantages of RF models is their ability to assess feature importance, providing insights into the contributions of different variables in predicting the tar- get outcome. In an RF, feature importance is typically determined by measuring the impact of each feature on the model’s predictive accuracy, e.g. using Mean...

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To this end, we evaluate the importance of features of both models in the same way

Measurement importance via perturbations Since RF and NN models are fundamentally different approaches, our main goal is to determine how the NN predictor works compared to ensemble RF models. To this end, we evaluate the importance of features of both models in the same way. The employed methodology for testing the importance of features (measurements) i...

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Thus, we com- pute the so-called Shapley values [41] to explain how the models learn and predict

Shapley additive explanations In the previous section we have shown that RFs and NNs make their predictions in a way which is sensitive to different measurements at the input. Thus, we com- pute the so-called Shapley values [41] to explain how the models learn and predict. SVs, originally developed in cooperative game theory, provide a fair way to distrib...

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