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arXiv:2310.20570 · Physical Review Letters

Correlation-Pattern-Based Continuous Variable Entanglement Detection through Neural Networks

Xiaoting Gao, Mathieu Isoard, Feng-Xiao Sun, Carlos E. Lopetegui, Yu Xiang, Valentina Parigi, Qiongyi He, Mattia Walschaers

Published May 29, 2024• 17 citations
arXivHTMLPDF

Abstract

Entanglement in continuous-variable non-Gaussian states provides irreplaceable advantages in many quantum information tasks. However, the sheer amount of information in such states grows exponentially and makes a full characterization impossible. Here, we develop a neural network that allows us to use correlation patterns to effectively detect continuous-variable entanglement through homodyne detection. Using a recently defined stellar hierarchy to rank the states used for training, our algorithm works not only on any kind of Gaussian state but also on a whole class of experimentally achievable non-Gaussian states, including photon-subtracted states. With the same limited amount of data, our method provides higher accuracy than usual methods to detect entanglement based on maximum-likelihood tomography. Moreover, in order to visualize the effect of the neural network, we employ a dimension reduction algorithm on the patterns. This shows that a clear boundary appears between the entangled states and others after the neural network processing. In addition, these techniques allow us to compare different entanglement witnesses and understand their working. Our findings provide a new approach for experimental detection of continuous-variable quantum correlations without resorting to a full tomography of the state and confirm the exciting potential of neural networks in quantum information processing.

DOI
10.1103/physrevlett.132.220202

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Entanglement in continuous-variable non-Gaussian states provides irreplaceable advantages in many quantum information tasks. However, the sheer amount of information in such states grows exponentially and makes a full characterization impossible. Here, we develop a neural network that allows us to use correlation patterns to effectively detect continuous-variable entanglement through homodyne detection. Using a recently defined stellar hierarchy to rank the states used for training, our algorithm works not only on any kind of Gaussian state but also on a whole class of experimentally achievable non-Gaussian states, including photon-subtracted states. With the same limited amount of data, our method provides higher accuracy than usual methods to detect entanglement based on maximum-likelihood tomography. Moreover, in order to visualize the effect of the neural network, we employ a dimension reduction algorithm on the patterns. This shows that a clear boundary appears between the entangled states and others after the neural network processing. In addition, these techniques allow us to compare different entanglement witnesses and understand their working. Our findings provide a new approach for experimental detection of continuous-variable quantum correlations without resorting to a full tomography of the state and confirm the exciting potential of neural networks in quantum information processing.
A major challenge in quantum information is characterizing entanglement, for which entanglement witnesses offer effective means of detecting quantum correlations. We introduce a hybrid quantum-classical framework that learns a nonlinear entanglement witness directly from quantum data using continuous-variable quantum neural networks (CV-QNNs). Our architecture combines variational interferometers, squeezers and non-Gaussian Kerr gates with a small classical neural head to output a scalar witness value. Numerical simulations were conducted on two- and three-mode families, including Gaussian and non-Gaussian states in both pure and mixed forms. We observed over 99% classification accuracy and a robust performance gap compared to strong classical baselines, especially when scaling from two to three modes. Robustness to photon loss is further quantified under a finite number of measurement shots. On the theory side, we show that when the quantum measurement stage is informationally complete, the hybrid model can approximate any continuous witness-like functional on compact sets of states.Our findings highlight CV-QNNs as a promising framework for data-driven quantum state characterization and propose specific benchmarks where near-term photonic platforms offer tangible advantages.
Abstract Characterizing entanglement in high-dimensional continuous-variable (CV) systems remains a formidable challenge, particularly when full quantum state tomography is prohibitively resource-intensive. In this work, a hybrid quantum–classical framework is introduced that learns nonlinear entanglement witnesses directly from sparse measurement data via CV quantum neural networks (CV-QNNs). The architecture combines variational interferometers, squeezing operations, and non-Gaussian Kerr nonlinearities with a classical neural head to output a scalar witness value. To rigorously isolate the information-theoretic sample-complexity advantage, a Coherence Trap benchmark is devised in which genuinely entangled and separable states are engineered to exhibit identical photon-number statistics, thereby rendering them indistinguishable to classical classifiers operating on intensity histograms. Numerical simulations show that while classical models fail to classify these states without full tomography, the CV-QNN achieves 100 % accuracy by extracting phase-dependent quantum correlations prior to measurement. To address scalability and optimization under realistic finite-shot measurement noise, a variance-reduced simultaneous perturbation stochastic approximation algorithm is employed, demonstrating sustained convergence and consistent performance across the primary tested cutoffs up to a Fock space cutoff of d = 5 for the 3 -mode CV system, alongside a proof-of-concept indication of algorithmic trainability at d = 7 . Moreover, the generalization of this framework to the detection of bound entangled states with zero negativity is demonstrated, establishing a fundamental advantage in sample complexity over standard linear witnesses. Robustness to photon loss and statistical noise is further quantified under realistic finite-shot measurement conditions, confirming the model’s stability in noisy environments. Collectively, these findings provide a rigorous proof-of-principle for the necessity of non-Gaussian operations and demonstrate a clear information-theoretic advantage in data-driven quantum state characterization.
Quantum correlations are at the core of quantum information processing and are necessary to achieve a quantum computational advantage. In the context of continuous-variable quantum systems, another necessary resource for quantum advantages is non-Gaussianity. In this work, we propose a witness based on previously known relations between metrological power and quantum correlations, to detect a strong form of entanglement that only non-Gaussian states possess and that cannot be undone by passive optical operations, i.e., entanglement in all mode bases. The strength of our witness is two-fold: it only requires measurements in one basis to check entanglement in any arbitrary mode basis; it can be made applicable experimentally using homodyne measurements and without requiring a full tomography of the state.
We present how nonclassicality and entanglement can be characterized and detected efficiently for continuous variable systems. Of particular interest is the use of homodyne detections to measure quadrature amplitudes at minimum level to confirm nonclassicality and entanglement beyond Gaussian states. We introduce a systematic method for a functional form of uncertainty relations, which can be efficiently employed to experimentally detect non-Gaussian states comprehensively. Our approach for quantum correlations unifies a framework for quantum entanglement and quantum steering, which include the known results for Gaussian states and provides a better tool for non-Gaussian states than existing methods, e.g. entropic uncertainty relations.
Based on the theoretical basis of quantum computation, entanglement has been further explored as one of the key resources. An outstanding problem in quantum computation is the detecting of entanglement, but no closed-form algorithm exists. According to an advanced matrix rearrangement approach for detecting multipartite quantum states, we give some examples to verify the effectiveness and practicability of this new criterion. Meanwhile, this paper shows how techniques from the quantum neural network can be utilized to detect entanglement. Two models, known as the discrete-variable and the continuous-variable quantum neural network, are applied to solve the separability problem. We demonstrate the utility that such a discrete-variable quantum neural network can be trained to detect the entanglement with great exactness. And the complexity of network and computation is greatly reduced. Besides, the continuous-variable quantum neural network is applied for detecting two-qumode Gaussian state. Compared with the results of the traditional neural network, this deep quantum neural network demonstrates its capability and adaptability as well.
Continuous-variables (CV) quantum optics is a natural formalism for neural networks (NNs) due to its ability to reproduce the information processing of such trainable interconnected systems. In quantum optics, Gaussian operators induce affine mappings on the quadratures of optical modes while non-Gaussian resources (the challenging piece for physical implementation) originate the nonlinear effects, unlocking quantum analogs of an artificial neuron. This work presents a novel experimentally-feasible framework for continuous-variable quantum optical neural networks (QONNs) developed with available photonic components: coherent states as input encoding, a general Gaussian transformation followed by multi-mode photon subtractions as the processing layer, and homodyne detection as outputs readout. The closed-form expressions of such architecture are derived demonstrating the family of adaptive activations and the quantum-optical neurons that emerge from the amount of photon-subtracted modes, proving that the proposed design satisfies the Universal Approximation Theorem within a single layer. To classically simulate the QONN training, the high-performance QuaNNTO library has been developed based on Wick-Isserlis expansion and Bogoliubov transformations, allowing multi-layer exact expectation values of non-Gaussian states without truncating the infinite-dimensional Hilbert space. Experiments on supervised learning and state-preparation tasks show balanced-resource efficiency with strong expressivity and generalization capabilities, illustrating the potential of the architecture for scalable photonic quantum machine learning and for quantum applications such as complex non-Gaussian gate synthesis.