Learning high-dimensional quantum entanglement through physics-guided neural networks

Abstract

High-gain spontaneous parametric down-conversion (SPDC) produces bright squeezed vacuum with rich high-dimensional entanglement, but its output is inherently multimodal and non-perturbative, making the full modal characterization a major computational bottleneck. We propose a physics-guided deep neural network that reconstructs the source's modal fingerprint: the high-dimensional correlation signature across radial and azimuthal indices. We designed a FiLM-modulated convolutional architecture that predicts the joint (m,l) distribution, and training is driven by a hybrid loss that couples data-driven metrics (JSD, KL, MSE, Wasserstein) with a soft orbital-angular-momentum (OAM) conservation term, providing an essential inductive bias toward physically consistent solutions. Across gain regimes, our method achieves high-fidelity reconstruction with average JSD of 1.96e-3, WEMD of 1.54e-3, and KL divergence of 7.85e-3, delivering an approximate 128-fold speedup over full numerical simulation and more than 30% accuracy gains over U-Net baselines. These results demonstrate that physics-guided learning, via a soft OAM-conservation regularizer and physically generated training targets, enables rapid and data-efficient modal characterization. Compared with traditional numerical simulation, our mesh-free method has demonstrated good generalization with limited or contaminated training data and has enabled fast "online" prediction of the quantum dynamics of a high-dimensional entanglement system for real-world experimental implementation.

Figures (6)

• Figure 1: Physics-guided deep learning neural network (PGNN) architecture for spatial mode distribution prediction.a. Concepts of physics-guided training for SPDC modal structure estimation. In a degenerate type-I SPDC process, a crystal with a second-order nonlinearity, $\chi^{(2)}$, is pumped by a short-wavelength laser beam with a given spatial profile, $E_p(\mathbf{r})$. A signal-idler pair is generated at a lower optical frequency and its spatial mode distribution is governed by the two-photon wavefunction $\Phi(\mathbf{q}_s,\mathbf{q}_i)$. The joint spatial mode distribution, or the spatial mode structure, can be obtained through Schmidt decomposition of the two-photon wavefunction or through direct experimental measurement. The simulation and measurement results, together with soft physical regularizers (e.g. OAM conservation), are fed to the neural network during the training stage. b. Estimation of the mode structure using a trained PGNN. The PGNN, once properly trained with physical constraints, takes in a set of physical parameters (e.g. pump intensity profile, crystal orientation, crystal length, etc.) as an input vector and infers the full modal structure of an arbitrary SPDC setup. This computationally efficient process enables real-time feedback in applications in OAM-based QKD or SU(1,1) interferometry.
• Figure 2: Physics-guided deep learning architecture for OAM mode prediction.a. Continuous parameters and embedded discrete OAM indices are concatenated to form a conditioning vector that modulates a stack of dilated convolutional FiLM residual blocks. The network outputs a normalized distribution over the $(m, l)$ grid via a final $1\times1$ convolution and softmax. b. The Conditional MLP transforms the conditioning vector through two SiLU-activated fully connected layers, generating the modulation features used throughout the network. c. The input convolutional block applies a convolution, GroupNorm, and SiLU activation to extract the initial spatial features. d. A series of FiLM residual blocks with increasing dilation factors captures spatial structure across multiple scales. e. Each FiLM module computes affine modulation parameters from the conditioning features, allowing the convolutional backbone to adapt dynamically to the underlying physical conditions.
• Figure 3: Comparison between simulated OAM mode distributions and OAMNet predictions. (a--r) Representative examples showing the agreement between simulated $(m,\ell)$ intensity distributions (left of each pair) and the corresponding OAMNet predictions (right). Color denotes normalized intensity.
• Figure 4: Comparison of U-Net Vanilla (light-weight), U-Net, and OAMNet on Schmidt number prediction and KL divergence statistics.a--c. U-Net Vanilla (light-weight):a. Predicted versus true Schmidt numbers; b. Histogram of KL divergences between predicted and simulated $(m,\ell)$ distributions; c. Scatter plot of Schmidt number error $\Delta S$ versus KL divergence. d--f. U-Net: Corresponding plots showing d. predicted versus true Schmidt numbers, e. KL divergence distribution, and f. $\Delta S$ versus KL divergence. g--i. OAMNet:g. Predicted versus true Schmidt numbers, h. KL divergence histogram, and i. $\Delta S$ versus KL divergence.
• Figure 5: Comparison of OAM spectra for varying gain values $g$.a–j. show the predicted marginal OAM spectra (red dash-dotted lines), numerical simulation (orange bars), and experimental data (black solid lines) for a range of gain values $g \in [2.258, 3.000]$. Excellent agreement is observed across all gains. (k) plots the mean absolute error (MAE), defined as $\mathrm{MAE} = \langle |S_\ell^{\rm model} - S_\ell^{\rm exp}|\rangle$, for both OAMNet predictions and numerical simulations against experimental measurements. The errors remain below $6 \times 10^{-3}$ across all gains. (l) shows the cosine similarity between predictions/simulations and experimental spectra: $\mathrm{cos\_sim} = \frac{\sum_\ell S_\ell^{\rm model} S_\ell^{\rm exp}}{\|S^{\rm model}\|\;\|S^{\rm exp}\|}$, indicating high similarity ($\ge 0.991$) throughout the range.