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arXiv:2110.13379 · Physical Review A

Detecting the steerability bounds of generalized Werner states via a backpropagation neural network

Jun Zhang, Kan He, Ying Zhang, Yu-yang Hao, Jinchuan Hou, Fangpeng Lan, Baoning Niu

Published Mar 3, 2022• 16 citations
arXivHTMLPDF

Abstract

We use an error backpropagation (BP) neural network to determine whether an arbitrary two-qubit quantum state is steerable and optimize the steerability bounds of the generalized Werner state. Results show that, regardless of how we select the features for the quantum states, we can use the BP neural network to construct several models to obtain high-performance quantum steering classifiers compared with the support vector machine. Moreover, we predict the steerability bounds of the generalized Werner states using the classifiers that are newly constructed by the BP neural network; that is, the predicted steerability bounds are closer to the theoretical bounds. In particular, high-performance quantum steering classifiers with partial information about the quantum states that we need to measure in only three fixed measurement directions are obtained.

DOI
10.1103/physreva.105.032408

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Zhongguo kexue. Wulixue Lixue Tianwenxue•Oct 23, 2024•1 citation
We use an error backpropagation (BP) neural network to determine whether an arbitrary two-qubit quantum state is steerable and optimize the steerability bounds of the generalized Werner state. Results show that, regardless of how we select the features for the quantum states, we can use the BP neural network to construct several models to obtain high-performance quantum steering classifiers compared with the support vector machine. Moreover, we predict the steerability bounds of the generalized Werner states using the classifiers that are newly constructed by the BP neural network; that is, the predicted steerability bounds are closer to the theoretical bounds. In particular, high-performance quantum steering classifiers with partial information about the quantum states that we need to measure in only three fixed measurement directions are obtained.
In this paper, we establish an aggregate class distribution neural network (AGGNN) structure to determine whether an arbitrary two-qubit quantum state is steerable. Compared to the classification results obtained using a support vector machine (SVM) and a backpropagation neural network (BPNN), we obtain higher-accuracy quantum-steering classification models via the AGGNN, as well as steerability bounds of generalized Werner states, which are more similar to the theoretical bounds. In particular, when we only know partial information about the quantum states, higher-performance quantum-steering classifiers are obtained compared to those via SVM and BPNN.
Abstract Quantum steering is an important nonlocal resource and has a wide range of applications in quantum information processing. Although a lot of steering criteria have been proposed, it is still very difficult to efficiently detect quantum steering in experiment. Here we employ machine learning techniques to tackle the problem of quantum steering detection in two-qubit system. The quantum steering and un-steering inequalities are combined together, so as to construct quantum steering classifiers for the generalized Werner states via artificial neural networks. More steerable and unsteerable quantum states can be identified by the classifiers proposed here than by the quantum steering inequality or un-steering inequality, which provides a new way to detect steering with only partial information of the given quantum states. We consider two types of artificial neural networks, one is the single-layer perceptron and the other is the multi-layer perceptron. The result shows that the multi-layer perceptron outperforms the single-layer perceptron in terms of accuracy. Compared with the existing quantum steering criteria, our methods do not require the whole information of the quantum state, and the steering of it is detected by using state-independent measurements, so it is easy to realize in experiment.
Quantum steering is an important nonclassical resource for quantum information processing. However, even though lots of steering criteria exist, it is still very difficult to efficiently determine whether an arbitrary two-qubit state shared by Alice and Bob is steerable or not, because the optimal measurement directions of Alice are unknown. In this work, we provide an efficient quantum steering detection scheme for an arbitrary two-qubit state with the help of machine learning, where Alice and Bob only need to measure in a few fixed measurement directions. In order to prove the validity of this method, we first realize a high-performance quantum steering classifier with the whole information. Furthermore, a high-performance quantum steering classifier with partial information is realized, where Alice and Bob only need to measure in three fixed measurement directions. Our method outperforms the existing methods in generic cases in terms of both speed and accuracy, opening up the avenues to explore quantum steering via the machine learning approach.
Not all entangled states can exhibit quantum steering, and determining whether a given entangled state is steerable is a crucial problem in quantum information theory. The main challenge lies in verifying the existence of a local hidden-state (LHS) model capable of reproducing all post-measurement assemblages generated by arbitrary measurements. To address this, we propose a machine learning-based framework that employs batch sampling of measurements and gradient-based optimization to construct an optimal LHS model. We validate our method by analyzing the steerability of two-qubit Werner and two-qutrit isotropic states. For Werner states, our approach saturates the analytical visibility bounds under three Pauli measurements, arbitrary projective measurements (PVMs), and arbitrary positive operator-valued measurements (POVMs). For isotropic states, we achieve the known analytical bounds under arbitrary PVMs. We further investigate the steerability of this class of states under arbitrary POVMs, and our results suggest that POVMs can offer an advantage over PVMs in revealing the steerability of such states.
Only a few states in high-dimensional systems can be identified as (un)steerable using existing theoretical or experimental methods. We utilize semidefinite programming (SDP) to construct a dataset for steerability detection in qutrit-qutrit systems. For the full-information feature F1{F}_{1}F1​, artificial neural networks achieve high classification accuracy and generalization, and perform better than the support vector machine. As feature engineering plays a pivotal role, we introduce a steering ellipsoidlike feature, F2{F}_{2}F2​, which significantly enhances the performance of each of our models. To address quantum steering detection in isotropic states, partially entangled states, and random states, we explore, respectively, the most tailored feature and classifier. Given that the SDP method provides only a sufficient condition for steerability detection, we establish the first rigorously constructed, accurately labeled dataset based on theoretical foundations. This dataset enables models to exhibit outstanding accuracy and generalization capabilities, independent of the choice of features. As applications, we investigate the steerability boundaries of isotropic states and partially entangled states, discover new steerable states, and determine their parameter ranges. This work not only advances the application of machine learning for probing quantum steerability in high-dimensional systems but also deepens the theoretical understanding of quantum steerability itself.
Inspired by the research on detecting the steerability of two-qubit states via machine learning, this study explores the detection of steerability in qubit-qutrit systems using various machine learning methods. It is found that, (1) for random states, Werner states, and UN states (a newly constructed class of entangled states where Alice cannot steer Bob), there always exist supervised or semi-supervised machine learning methods with a steering detectability accuracy rate of over 95%; (2) the steering bounds predicted by the supervised machine learning methods are mostly higher than the non-steerable bounds provided by theory and lower than the steering bounds determined by the semidefinite programming (SDP). This demonstrates that the machine learning methods are reliable for detecting the steerability of quantum states in qubit-qutrit systems and can detect more steering states compared with the SDP method. It lays the foundation for utilizing machine learning methods to detect quantum states' steerability in bipartite and high-dimensional systems.