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Photonics Research

Entanglement quantification via weak measurements assisted by deep learning

Mu Yang, Xiao Ya, Ze-Yan Hao, Yuwei Liao, Jia-He Cao, Kai Sun, Enhui Wang, Zhenghao Liu, Yutaka Shikano, Jin‐Shi Xu, Chuan‐Feng Li, Guang‐Can Guo

Published Dec 11, 2023• 4 citations
Publisher

Abstract

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.

DOI
10.1364/prj.498498

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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.
Detecting and quantifying quantum entanglement of a given unknown state poses problems that are fundamentally important for quantum information processing. Surprisingly, no direct (i.e., without quantum tomography) universal experimental implementation of a necessary and sufficient test of entanglement has been designed even for a general two-qubit state. Here we propose an experimental method for detecting a collective universal witness, which is a necessary and sufficient test of two-photon polarization entanglement. It allows us to detect entanglement for any two-qubit mixed state and to establish tight upper and lower bounds on its amount. A different element of this method is the sequential character of its main components, which allows us to obtain relatively complicated information about quantum correlations with the help of simple linear-optical elements. As such, this proposal realizes a universal two-qubit entanglement test within the present state of the art of quantum optics. We show the optimality of our setup with respect to the minimal number of measured quantities.
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.
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.
Entanglement not only plays a crucial role in quantum technologies, but is key to our understanding of quantum correlations in many-body systems. However, in an experiment, the only way of measuring entanglement in a generic mixed state is through reconstructive quantum tomography, requiring an exponential number of measurements in the system size. Here, we propose a machine-learning-assisted scheme to measure the entanglement between arbitrary subsystems of size N_{A} and N_{B}, with O(N_{A}+N_{B}) measurements, and without any prior knowledge of the state. The method exploits a neural network to learn the unknown, nonlinear function relating certain measurable moments and the logarithmic negativity. Our procedure will allow entanglement measurements in a wide variety of systems, including strongly interacting many-body systems in both equilibrium and nonequilibrium regimes.
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 30×30\times30× 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.
Detecting and quantifying quantum entanglement of a given unknown state poses problems that are fundamentally important for quantum information processing. Surprisingly, no direct (i.e., without quantum tomography) universal experimental implementation of a necessary and sufficient test of entanglement has been designed even for a general two-qubit state. Here we propose an experimental method for detecting a collective universal witness, which is a necessary and sufficient test of two-photon polarization entanglement. It allows us to detect entanglement for any two-qubit mixed state, and to establish tight upper and lower bounds on its amount. A novel element of this method is the sequential character of its main components, which allows us to obtain relatively complicated information about quantum correlations with the help of simple linear-optical elements. As such, this proposal is the first to realize a universal two-qubit entanglement test within the present state of the art of quantum optics. We show the optimality of our setup with respect to the minimal number of measured quantities.