Learning Entanglement Quasiprobability from Noisy and Incomplete Data

Abstract

Negativities in quasiprobability distributions, a foundational concept originating in quantum optics, serve as a fundamental signature of quantum nonclassicality, with entanglement quasiprobabilities offering a necessary and sufficient criterion for entanglement. However, practical reconstruction of entanglement quasiprobabilities conventionally requires full quantum state tomography, severely limiting scalability. Here, we propose a deep-learning framework that reconstructs entanglement quasiprobabilities directly from incomplete local projective measurements, bypassing full state reconstruction. Using a residual neural network, partial measurement outcomes are mapped to high-fidelity entanglement quasiprobabilities. Numerical benchmarks up to three qubits show more than a $30\times$ reduction in reconstruction error compared with state-of-the-art tomographic methods. Experimental validation on photonic entangled states demonstrates reconstruction and entanglement detection with substantially reduced measurement resources. Our results establish machine-learning-assisted reconstruction of entanglement quasiprobabilities as a scalable and practical tool for entanglement characterization in quantum optical systems.

Figures (4)

• Figure 1: Architecture of deep-learning-assisted EQP reconstruction from incomplete measurements. Incomplete local measurement outcomes (1) are mapped by a trained network to entanglement quasiprobabilities (2). The framework exhibits strong generalization to out-of-distribution states (3), robustness to experimental noise (4), and enables accurate estimation of fidelity, purity, and other state properties (5).
• Figure 2: EQP reconstruction error versus measurement cost. Root-mean-square error as a function of the number of projectors for (a) generic two-qubit states, (b) noisy Werner states, and (c) three-qubit systems. The deep-learning-based approach is shown by purple squares, standard maximum-likelihood estimation (MaxLik) by red circles, and maximum-entropy maximum-likelihood estimation (MLME) by blue triangles. Error bars indicate statistical uncertainty.
• Figure 3: Evolution of EQPs reconstruction under varying projector number in experimental data. (a) A schematic diagram of our experimental setup for generation of entangled-photons. BD, beam displacer; HWP, half-wave plate; dHWP, dual-path half-wave plate; BBO, $\beta-$Barium Borate crystal. (b) The reconstructed EQP using our deep-learning method. (c) The reconstructed EQP using the maximum likelihood method at 2, 10, 22, and 36 projectors. The open columns and solid columns represent theoretical and experimental values, respectively. Light coral red denotes negative components (indicating entanglement), and light olive green denotes positive components.
• Figure 4: Performance comparison for predicting Fidelity (a) and Purity (b) from EQPs. The RMSE of the deep learning approach trained on specific incomplete measurements is depicted by purple squares, the standard MaxLik method by red circles, and the MaxLik method with maximum entropy regularization by blue triangles.