Quantifying entanglement for quantum states whose density matrices are unknown is a difficult task, but also becomes more and more necessary because of the fast development of quantum engineering. Machine learning provides practical solutions to this fundamental problem, where one has to train a proper machine learning model to predict entanglement measures of unknown quantum states based on experimentally measurable data, say, moments of density matrices or correlation data produced by local measurements. In this paper, we compare the performance of these two different machine learning approaches systematically. Particularly, we first show that the approach based on moments enjoys a remarkable advantage over that based on correlation data, though the cost of measuring moments is much higher. Next, since correlation data is much easier to obtain experimentally, we try to better its performance by proposing a hybrid quantum-classical machine learning framework for this problem, where the key is to train optimal local measurements, or even optimal tensor products of single-qubit observables, to generate more informative correlation data. Our numerical simulations show that the new hybrid framework brings us comparable performance with the approach based on moments to quantify entanglement, and furthermore, it turns out that the new hybrid framework has a nice noise-resistant capability when handling realistic correlation data.
Quantifying entanglement for quantum states whose density matrices are unknown is a difficult task, but also becomes more and more necessary because of the fast development of quantum engineering. Machine learning provides practical solutions to this fundamental problem, where one has to train a proper machine learning model to predict entanglement measures of unknown quantum states based on experimentally measurable data, say, moments of density matrices or correlation data produced by local measurements. In this paper, we compare the performance of these two different machine learning approaches systematically. Particularly, we first show that the approach based on moments enjoys a remarkable advantage over that based on correlation data, though the cost of measuring moments is much higher. Next, since correlation data is much easier to obtain experimentally, we try to better its performance by proposing a hybrid quantum-classical machine learning framework for this problem, where the key is to train optimal local measurements, or even optimal tensor products of single-qubit observables, to generate more informative correlation data. Our numerical simulations show that the new hybrid framework brings us comparable performance with the approach based on moments to quantify entanglement, and furthermore, it turns out that the new hybrid framework has a nice noise-resistant capability when handling realistic correlation data.
Abstract Quantifying entanglement measures for quantum states with unknown density matrices is a challenging task. Machine learning offers a new perspective to address this problem. By training machine learning models using experimentally measurable data, we can predict the target entanglement measures. In this study, we compare various machine learning models and find that the linear regression and stack models perform better than others. We investigate the model’s impact on quantum states across different dimensions and find that higher-dimensional quantum states yield better results. Additionally, we investigate which measurable data has better predictive power for target entanglement measures. Using correlation analysis and principal component analysis, we demonstrate that quantum moments exhibit a stronger correlation with coherent information among these data features.
Entanglement plays a pivotal role in numerous quantum applications, and as technology progresses, entanglement systems continue to expand. However, quantifying entanglement is a complex problem, particularly for multipartite quantum states. The currently available entanglement measures suffer from high computational complexity, and for unknown multipartite entangled states, complete information about the quantum state is often necessary, further complicating calculations. In this paper, we train neural networks to quantify unknown multipartite entanglement using input features based on squared entanglement (SE) and outcome statistics data produced by locally measuring target quantum states. By leveraging machine learning techniques to handle non-linear relations between outcome statistics and entanglement measurement SE, we achieve high-precision quantification of unknown multipartite entanglement states with a linear number of measurements, avoiding the need for global measurements and quantum state tomography. The proposed method exhibits robustness against noise and extends its applicability to pure and mixed states, effectively scaling to large-scale multipartite entanglement systems. The results of the experiment show that the predicted entanglement measures are very close to the actual values, which confirms the effectiveness of the proposed method.
k-entanglement, including entanglement relative to full separability and genuinely multipartite entanglement, is a fundamental quantum resource in multipartite quantum systems. Its identification and quantification play essential roles in quantum information processing, quantum simulation, and quantum metrology. However, the practical computation of rigorous k-entanglement measures remains highly challenging due to the need for high-dimensional optimization. In this work, we propose a machine-learning-based surrogate framework for approximating the witness-based k-entanglement measure Ew(k,n). The numerical evaluation of the computationally realized quantity E˜w(k,n)(ρ) is reformulated as a supervised regression problem, where the input is the density matrix ρ and the labels are obtained from finite witness databases. The framework combines multilayer perceptrons (MLPs), convolutional neural networks (CNNs), and light gradient boosting machine (LightGBM) through a stacking ensemble. Numerical experiments are performed for 3- and 4-qubit systems as representative demonstrations of the proposed workflow. The results show that the learned models achieve high predictive accuracy in terms of MAE, MSE, and R2, while providing millisecond-level inference for single-state evaluation. Werner state tests serve as symmetric benchmark checks, and an additional four-qubit noisy circuit-generated state family, obtained from finite-depth circuit preparation followed by local amplitude-damping noise, is used as a structured physical test beyond random density matrices. Compared with the optimization-based evaluation, the trained surrogate model significantly reduces the computational time while maintaining accuracy within the tested system sizes and data distributions. These results show that the proposed framework provides an efficient numerical surrogate for rapid approximation of witness-based k-entanglement measures, while extensions to larger systems and experimental data require further validation.
Quantum entanglement is a fundamental resource in quantum information, but its characterization becomes increasingly challenging as the size of multipartite systems grows. This thesis investigates machine learning methods to detect and quantify entanglement in pure multiqubit states using only single qubit Pauli expectation values. Two complementary tasks are considered: the distinction between fully separable and entangled states, with particular attention to weakly entangled states close to the separability boundary, and the regression of normalized multipartite classical squashed entanglement. Several machine learning models are evaluated for detection, while multilayer perceptrons are employed for quantification. The low entanglement regime is investigated through dedicated datasets and targeted data augmentation strategies. The proposed approaches are studied for systems of 5, 10, 15, and 20 qubits and are further evaluated under realistic measurement conditions, including finite shot noise, asymmetric readout errors, calibration drift, and data acquired from real quantum processors. The study assesses the scalability, robustness, and practical limitations of data-driven entanglement characterization based on a linearly scaling set of local observables.
Quantum coherence and entanglement are fundamental resources in quantum technologies, yet their efficient estimation for unknown states by employing minimal resources in experimental settings remains challenging, particularly in high-dimensional systems. We present a machine learning approach based on support vector regression (SVR) that directly estimates the coherence measures and the geometric measure of quantum entanglement using minimal experimental resources. Our method requires only the diagonal entries of the density matrix, along with the traces of the squared and cubed density matrices for quantum coherence, and additionally along with the traces of the squared and cubed reduced density matrix for estimating quantum entanglement. These quantities can be obtained through random measurements or a hybrid quantum-classical framework. This approach significantly reduces the resource overhead compared to quantum state tomography while maintaining high accuracy. {Furthermore, the support vector quantile regression (SVQR) with pinball loss is employed to prevent SVR overestimation. This model not only ensures that over 95\% of predictions are conservative lower bounds in most cases, but also maintains this lower-bound reliability for over 93\% of predictions, despite 2\% perturbations in the input features.} The proposed technique provides a practical and scalable tool for characterizing quantum resources across computation, communication, and metrology applications.
Quantum entanglement acts as a crucial part in quantum computation and quantum information, hence quantifying unknown entanglement is an important task. Due to the fact that the amount of entanglement cannot be achieved directly by measuring any physical observables, it remains an open problem to quantify entanglement experimentally. In this work, we provide an effective way to quantify entanglement for the unknown quantum states via artificial neural networks. By choosing the expectation values of measurements as input features and the values of entanglement measures as labels, we train artificial neural network models to predict the entanglement for new quantum states accurately. Our method does not require the full information about unknown quantum states, which highlights the effectiveness and versatility of machine learning in exploring quantum entanglement.