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arXiv:2205.01462 · Science Advances

Deep learning of quantum entanglement from incomplete measurements

Dominik Koutný, Laia Ginés, Magdalena Moczała-Dusanowska, Sven Höfling, Christian Schneider, Ana Predojević, Miroslav Ježek

Published Jul 19, 2023• 49 citations
arXivHTMLPDF

Abstract

The quantification of the entanglement present in a physical system is of paramount importance for fundamental research and many cutting-edge applications. Now, achieving this goal requires either a priori knowledge on the system or very demanding experimental procedures such as full state tomography or collective measurements. Here, we demonstrate that, by using neural networks, we can quantify the degree of entanglement without the need to know the full description of the quantum state. Our method allows for direct quantification of the quantum correlations using an incomplete set of local measurements. Despite using undersampled measurements, we achieve a quantification error of up to an order of magnitude lower than the state-of-the-art quantum tomography. Furthermore, we achieve this result using networks trained using exclusively simulated data. Last, we derive a method based on a convolutional network input that can accept data from various measurement scenarios and perform, to some extent, independently of the measurement device.

DOI
10.1126/sciadv.add7131

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Entanglement has been recognized as being crucial when implementing various quantum information tasks. Nevertheless, quantifying entanglement for an unknown quantum state requires nonphysical operations or post-processing measurement data. For example, evaluation methods via quantum state tomography require vast amounts of measurement data and likely estimation. Although a direct entanglement determination has been reported for the unknown pure state, it is still tricky for the mixed state. In this work, assisted by weak measurement and deep learning technology, we directly detect the entanglement (namely, the concurrence) of a class of two-photon polarization-entangled mixed states both theoretically and experimentally according to the local photon spatial distributions after weak measurement. In this way, the number of projective bases is much smaller than that required in quantum state tomography.
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Abstract 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 Nmub2D2NcopN_{mub}^2D^2N_{cop}Nmub2​D2Ncop​ measurements, where NmubN_{mub}Nmub​ is the number of mutually unbiased bases, DDD is the subsystem dimensionality, and NcopN_{cop}Ncop​ is the number of identical copies. For ququart systems with D=4D=4D=4, this translates to hundreds of distinct measurement settings, each requiring multiple copies (NcopN_{cop}Ncop​ 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 1.49×10−11.49 \times 10^{-1}1.49×10−1, while MLE yields 2.04×10−12.04 \times 10^{-1}2.04×10−1—over 10× higher error—despite taking 178 times longer. All neural methods in our study show error reduction scaling as approximately 1/N1/\sqrt{N}1/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.
We revisit quantum tomography in an informationally incomplete scenario and propose improved state reconstruction methods using deep neural networks. In the first approach, the trained network predicts an optimal linear or quadratic reconstructor with coefficients depending only on the collection of (already taken) measurement operators. This effectively refines the undercomplete tomographic reconstructor based on pseudoinverse operation. The second, based on an LSTM recurrent network performs state reconstruction sequentially. It can also optimize the measurement sequence, which suggests a no-free-lunch theorem for tomography: by narrowing the state space, we gain the possibility of more efficient tomography by learning the optimal sequence of measurements. Numerical experiments for a 2-qubit system show that both methods outperform standard maximum likelihood estimation and also scale to larger 3- and 4-qubit systems. Our results demonstrate that neural networks can effectively learn the underlying geometry of multi-qubit states and use it for their reconstruction.