Machine Learning Achieves Coherence and Entanglement Estimation with Minimal Resources for Unknown States

Quantifying coherence and entanglement remains a significant hurdle in the development of quantum technologies, especially when dealing with complex, high-dimensional systems. Ting Lin, Zhihua Chen, and Kai Wu, from Jimei University, alongside Zhihua Guo of Shaanxi Normal University, Shao-Ming Fei from Capital Normal University, and Zhihao Ma, have now demonstrated a machine learning technique to directly estimate these crucial quantum properties. Their research introduces a support vector regression (SVR) model that accurately determines coherence and entanglement using only a minimal set of readily measurable quantities , specifically, diagonal entries and traces of density matrices. This innovative approach bypasses the need for full state tomography, dramatically reducing experimental demands while maintaining precision. By employing support vector quantile regression, the team further ensures reliable and conservative estimations, paving the way for practical applications in quantum computation, sensing, and metrology.

Minimal Data Estimates Quantum Coherence and Entanglement

Scientists have achieved a significant breakthrough in the efficient estimation of quantum coherence and entanglement, fundamental resources for emerging technologies. Their work presents a machine learning approach, utilising support vector regression (SVR), to directly estimate these quantum properties from minimal experimental data. The team demonstrated that only the diagonal entries of the density matrix, alongside traces of the squared and cubed density matrices, are required for coherence estimation, while entanglement estimation additionally requires traces of the squared and cubed reduced density matrices. Experiments revealed that this method substantially reduces the resource overhead compared to full state tomography while maintaining high accuracy in quantifying these quantum properties.

The researchers employed support vector quantile regression (SVQR) with pinball loss to prevent overestimation by the SVR model, ensuring that over 95% of predictions are conservative lower bounds. Remarkably, this lower-bound reliability is maintained for over 93% of predictions even with 2% perturbations in the input features, highlighting the robustness of the technique. Data shows the approach leverages a semi-definite programming (SDP) method to compute the geometric measure of coherence, maximizing fidelity between quantum states. The optimisation process involves reformulating the problem over non-negative real variables, allowing for the computation of the geometric measure of coherence through the SDP problem, solved using the cvxopt solver within the picos library.

Furthermore, the geometric measure of entanglement is calculated using a lower bound derived from the maximum fidelity between the density matrix and positive partial transpose (PPT) states. To train and validate their SVR model, scientists generated a dataset of 10,000 quantum states, comprising mixed, pure, and convex combinations of states for various quantum systems. This innovative technique delivers a practical and scalable tool with potential applications spanning computation, sensing, and metrology, offering a pathway to characterise quantum resources with unprecedented efficiency.

Efficient Quantum Resource Estimation via Machine Learning

This work demonstrates a machine learning approach, utilising support vector regression (SVR) and support vector quantile regression (SVQR), to efficiently estimate quantum coherence and entanglement measures in unknown states. The researchers successfully estimated these resources, including the l1 norm of coherence, relative entropy of coherence, and geometric measure of entanglement, by employing minimal experimental data, specifically diagonal entries and traces of the density matrix. This represents a significant reduction in resource overhead compared to traditional state tomography techniques while maintaining a high degree of accuracy.

The developed model consistently provides reliable lower bounds for predicted values, with over 93% of estimations remaining secure even when input data contained perturbations of up to 2%. This robustness is achieved through the implementation of SVQR with pinball loss, effectively addressing the potential for overestimation inherent in standard SVR models. While the study acknowledges variance in predictive performance between two-qutrit and 4×4 systems, the R2 values remained consistently high, indicating maintained reliability. Future work could extend this methodology to estimate other quantum correlations currently computed via semi-definite programming, without requiring prior knowledge of the quantum states themselves.