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arxiv: 2508.07652 · v1 · submitted 2025-08-11 · 🪐 quant-ph · cond-mat.dis-nn· cond-mat.str-el

Estimating classical mutual information between quantum subsystems with neural networks

D. A. Konyshev , V. V. Mazurenko This is my paper

Pith reviewed 2026-05-19 00:08 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nncond-mat.str-el
keywords mutual informationneural networksquantum Ising modelprojective measurementsphase diagramentropyquantum correlations
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0 comments X

The pith

Neural networks reconstruct classical mutual information from limited projective measurements in quantum systems.

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

The paper shows that neural networks can estimate classical mutual information and specific entropy in a quantum system using only a limited number of projective measurements rather than complete state statistics. This is demonstrated on the antiferromagnetic quantum Ising model in transverse and longitudinal fields, a system of interest in condensed matter and quantum computing. The approach remains reliable even for paramagnetic wave functions that are delocalized across the state space. The same estimates are then used to reconstruct the phase diagram while distinguishing different types of disordered states.

Core claim

A neural network trained on a limited number of projective measurements can accurately reconstruct the classical mutual information and specific entropy for the antiferromagnetic quantum Ising model, including for delocalized paramagnetic wave functions, and this enables reconstruction of the phase diagram with emphasis on disordered states.

What carries the argument

Neural network that maps partial projective measurement data to estimates of classical mutual information and entropy.

If this is right

  • Reliable mutual information estimates become possible without enumerating all system states.
  • The phase diagram of the quantum Ising model can be recovered while separating distinct disordered phases.
  • The method applies to systems studied in both condensed matter physics and quantum computing.
  • Delocalized states do not prevent the neural network from producing accurate information estimates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Fewer measurements could suffice in future quantum experiments to extract correlation information.
  • The technique may extend to other many-body models where full tomography is costly.
  • Training on synthetic data from one model could inform analysis of real-device measurements in quantum simulators.

Load-bearing premise

A neural network trained on limited projective measurements can accurately reconstruct the full classical mutual information without needing complete state statistics or failing on delocalized wave functions.

What would settle it

Compare the neural network output directly to the exact classical mutual information computed from full state probabilities for the Ising model in the paramagnetic regime.

Figures

Figures reproduced from arXiv: 2508.07652 by D. A. Konyshev, V. V. Mazurenko.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of the procedure for estimating classical mutual information between quantum subsystems [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic representation of the system decomposi [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Probability distributions of the different phases: [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Entropy-based quantities calculated on the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Phase diagram of the Ising model in the longitudinal [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Example that demonstrates training a neural network [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Convergence of [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Size scaling of the exact classical mutual information [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
read the original abstract

Characterizing correlations in a quantum system on the basis of the results of the projective measurements can be performed with different means including the calculation of the classical mutual information. Generally, estimating such information-entropy-based quantities requires having complete statistics of the system's states. Here we explore the possibility to reconstruct the classical mutual information and specific entropy of a quantum system with neural network approach on the basis of limited number of projective measurements. As a prominent example we consider the antiferromagnetic quantum Ising model in transverse and longitudinal magnetic fields which is in demand in both condensed matter physics and quantum computing. We show that the neural network approach gives reliable estimates of the classical mutual information even in the case of paramagnetic wave functions delocalized in the state space. In addition, the phase diagram of the considered quantum system is reconstructed with a special focus on discriminating various types of disordered states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that neural networks can be used to reconstruct classical mutual information and specific entropy from a limited number of projective measurements in quantum systems. Using the antiferromagnetic quantum Ising model as an example, the authors show reliable estimates even for delocalized paramagnetic wave functions and reconstruct the phase diagram with focus on disordered states.

Significance. This method could be significant for practical characterization of quantum correlations in systems where complete measurement statistics are unavailable, offering a scalable alternative for information entropy calculations in quantum many-body physics and quantum computing applications. The success on delocalized states highlights its potential robustness.

minor comments (3)
  1. Include explicit details on the neural network architecture, training dataset size, loss function, and validation strategy in the methods section to support reproducibility of the reported estimates.
  2. The figures showing phase diagram reconstruction should include error bars or uncertainty estimates derived from the neural network predictions.
  3. Consider discussing potential overfitting issues or generalization challenges when applying the trained network to states with different degrees of delocalization.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. We are pleased that the potential significance for practical characterization of quantum correlations via neural networks is recognized, particularly the robustness for delocalized paramagnetic states in the antiferromagnetic quantum Ising model.

Circularity Check

0 steps flagged

No significant circularity detected in neural network estimation approach

full rationale

The paper applies a neural network as an external approximator to reconstruct classical mutual information and entropy from a limited set of projective measurements on the antiferromagnetic quantum Ising model. This is a standard supervised learning setup on measurement data rather than a closed mathematical derivation. No load-bearing steps reduce by the paper's own equations to fitted inputs by construction, nor do self-citations or ansatzes form the central claim. The numerical demonstrations for delocalized paramagnetic states rely on the network's generalization capability, which is independent of the target quantities and does not create self-referential loops. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of quantum mechanics for the Ising model and on the capacity of supervised neural networks to generalize from partial measurement data; no new entities or free parameters are introduced in the abstract.

axioms (2)
  • domain assumption The antiferromagnetic quantum Ising model in transverse and longitudinal fields accurately represents the target quantum system for testing the method.
    The abstract selects this model as the prominent example without deriving its Hamiltonian from more fundamental principles.
  • standard math Projective measurements on the system produce statistics from which classical mutual information can be defined.
    This is the standard information-theoretic definition invoked implicitly when the abstract discusses estimating the quantity.

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