Machine learning-enhanced entanglement characterization in Bi-partite Ququart systems

We present a systematic comparative analysis of machine learning and traditional approaches for quantum entanglement characterization in bi-partite ququart systems. Quantifying entanglement traditionally requires full quantum state tomography, necessitating $N_{mub}^2D^2N_{cop}$ measurements, where $N_{mub}$ is the number of mutually unbiased bases, $D$ is the subsystem dimensionality, and $N_{cop}$ is the number of identical copies. For ququart systems with $D=4$, this translates to hundreds of distinct measurement settings, each requiring multiple copies ($N_{cop}$ typically in thousands)—resulting in hundreds of thousands of total measurements in practice. Our research demonstrates that neural networks achieve comparable or superior accuracy with orders of magnitude fewer measurement settings. We evaluate three architectures—Multi-Layer Perceptron (MLP), Convolutional Neural Network (CNN), and Transformer—against Maximum Likelihood Estimation (MLE) and Bayesian methods across measurement counts ranging from 10 to 400. Neural approaches achieve up to 1650× faster computation times compared to traditional methods while maintaining competitive accuracy. At 100 measurements, the Transformer achieves Mean Squared Error (MSE) of $1.49 \times 10^{-1}$, while MLE yields $2.04 \times 10^{-1}$—over 10× higher error—despite taking 178 times longer. All neural methods in our study show error reduction scaling as approximately $1/\sqrt{N}$ with increased measurements. While this observed scaling might be influenced by our specific architectures, it provides important practical guidance for experimental design. The Transformer architecture demonstrates exceptional sample efficiency, achieving with 100 measurements what traditional methods require 250-400 measurements to accomplish—a significant advantage for resource-constrained quantum experiments. This research provides a viable pathway for real-time entanglement characterization in higher-dimensional quantum systems where traditional methods become computationally prohibitive.