Using the bitstring probabilities of ground states of bipartitioned ladders of Rydberg atoms, we calculate the mutual information, which is a lower bound on the corresponding bipartite von Neumann quantum entanglement entropy
SAvN. We show that in many cases these lower bounds can be improved by removing the bitstrings with a probability lower than some value
pmin and renormalizing the remaining probabilities (filtering). We propose a heuristic based on the change of the conditional entropy under filtering that very effectively improves the estimate of
SAvN. We consider various sizes, lattice spacings and bipartitions. Our numerical investigation suggest that the filtered mutual information obtained with samples having just a few thousand bitstrings can provide reasonably close estimates of
SAvN. We briefly discuss practical implementations with QuEra's Aquila device.
Calculating the von Neumann entanglement entropy from experimental data is challenging due to its dependence on the complete wavefunction, forcing reliance on approximations like classical mutual information (MI). We propose a machine learning approach using a graph neural network for predicting the von Neumann entropy from experimentally accessible bitstrings. We tested this approach on a Rydberg ladder system and achieved a mean absolute error of
3.7×10−3 when testing it inside of its training range on a dataset with entropy ranging from 0 to 1.9, we also evaluated the mean absolute percentage error on the data that have its entropy larger than 0.01 of the maximum entropy(so we can avoid the areas of our phase space with near zero entropy) and got a relative error of 1.72%, outperforming MI-based bounds. When tested beyond this range, the model still delivers reasonable results. Furthermore, we demonstrate that fine tuning the model with a small dataset significantly improves its performance on data outside its original training range.
Matrix elements of the density matrix of small quantum systems have recently become experimentally accessible within the cold atoms simulations. Numerical estimation of these quantities is known to be computationally challenging. Tracing out a part of the studied system and hence accessing the reduced density matrices offers similar difficulties, but it is more interesting, as the reduced density matrix is the basis of entanglement entropy measures. In this work, we present a method of estimating the individual elements of the reduced density matrices that uses autoregressive neural networks and the path integral quantization approach. We use a hierarchy of neural networks capable of estimating conditional probabilities of consecutive spins to evaluate elements of reduced density matrices directly. Using the Ising chain as an example, we calculate the continuum limit of the ground state’s von Neumann and Rényi bipartite entanglement entropies of an interval built of up to 5 spins. We demonstrate that our architecture is able to estimate all the needed matrix elements with just a single training for a fixed temperature discretization and lattice volume. Our method can be applied to other types of spin chains, possibly with defects, as well as to estimating entanglement entropies of thermal states at non-zero temperature. Hence, it may offer an attractive computational approach as a counterpart to the experimental effort. Abstract Published by the Jagiellonian University 2025 authors
The empirical bitstring distribution is the most accessible observable on Rydberg-atom arrays, but the bipartite von~Neumann entropy it constrains is far costlier to obtain. We present a six-term linear closed form for the entropy, built on bitstring-derivable physics scalars, and characterize its accuracy, portability, scaling behaviour, and calibration cost. The feature set is selected with guidance from a trained graph neural network: probing the network localizes its entropy prediction to the two-point correlators on the bipartition boundary, and an exhaustive ground-truth search restricted to those boundary correlators isolates the form. It reaches
0.024~nats mean absolute error in distribution:
6.4 times the network's error, but in a form a human can read and apply without retraining. Fit once and applied unchanged, it has lower error than the base network on five of six out-of-distribution pools and ties the sixth. An independent density-matrix renormalization-group study to one hundred atoms -- five times the reach of exact diagonalization -- settles the size-extrapolation question: coefficients frozen at small size fail at scale, but the failure is structured. Refit per size the form holds to
25--
50~mnat (cross-validated); two of its six slopes follow clean inverse-size laws, one a downward curving growth, and the others are trendless; the fitted laws deploy the form label-free at roughly
40--
80~mnat. The result fixes a label-budget rule: at large sizes, a few dozen labels recalibrate the closed form to match a fine-tuned in-distribution ensemble on the same features, while nonlinear ML models pull ahead only given large labelled datasets.
Abstract Calculating the von Neumann entanglement entropy from experimental data is challenging due to its dependence on the complete wavefunction, forcing reliance on approximations such as classical mutual information (MI). We propose a machine learning approach using a graph neural network to predict the von Neumann entropy directly from experimentally accessible bitstrings. We test this approach on a Rydberg ladder system and achieve a mean absolute error of 3.6 × 10 − 3 when evaluating within the training range on a dataset with entropy values ranging from 0 to 1.9. The model achieves a mean absolute percentage error of 1.44% and outperforms MI-based bounds. When tested beyond the training range, the model maintains reasonable accuracy. Furthermore, we demonstrate that fine-tuning the model with small datasets significantly improves performance on data outside the original training range.
The entropy of a quantum system is a measure of its randomness and is useful in quantifying entanglement. We study the problem of measuring the von Neumann and Rényi entropies of an unknown mixed quantum state given access to independent copies of the state. For Rényi entropy of integral order exceeding one, we determine the order-optimal copy complexity and show that it is strictly lower than the number of copies required to learn the underlying state. The main technical innovation is a concentration result for certain polynomials that arise in the Kerov algebra of Young diagrams, which is proven using the cycle structure of compositions of certain types of permutations. For von Neumann entropy and Rényi entropy of non-integral orders, we provide upper and lower bounds on the sample complexity of the Empirical Young Diagram (EYD) algorithm, which is the analogue of the empirical plug-in estimator in classical estimation.