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arXiv:2504.18163

Optimal entanglement witness of multipartite systems using support vector machine approach

Mohmoud Mahdian, Zahra Beygom Mousavi

Published Jul 30, 2024• 0 citations
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Abstract

Abstract An entanglement witness (EW) is a Hermitian operator that can distinguish an entangled state from all separable states. We developed and implemented a numerical method based on machine learning to create multipartite EW. Using a Support Vector Machine (SVM) algorithm, we construct EW's based on local orthogonal observables in the form of a hyperplane that separates the separable region from the entangled state for two, three, and four qubits Bell-diagonal mixed states which can be generalized to a multipartite mixed state. One of the important features of this method is that the EW s are optimal and are completely tangent to the separable region. Also, we generate non-decomposable EW's that can detect positive partial transpose entangled states (PPTES).

DOI
10.21203/rs.3.rs-4693906/v1

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Abstract An entanglement witness (EW) is a Hermitian operator that can distinguish an entangled state from all separable states. We developed and implemented a numerical method based on machine learning to create multipartite EW. Using a Support Vector Machine (SVM) algorithm, we construct EW's based on local orthogonal observables in the form of a hyperplane that separates the separable region from the entangled state for two, three, and four qubits Bell-diagonal mixed states which can be generalized to a multipartite mixed state. One of the important features of this method is that the EW s are optimal and are completely tangent to the separable region. Also, we generate non-decomposable EW's that can detect positive partial transpose entangled states (PPTES).
In this work, we show a correspondence between linear support vector machines (SVMs) and entanglement witnesses, and use this correspondence to generate entanglement witnesses for bipartite and tripartite qubit (and qudit) target entangled states. An SVM allows for the construction of a hyperplane that clearly delineates between separable states and the target entangled state; this hyperplane is a weighted sum of observables (``features'') whose coefficients are optimized during the training of the SVM. We demonstrate with this method the ability to obtain witnesses that require only local measurements even when the target state is a nonstabilizer state. Furthermore, we show that SVMs are flexible enough to allow us to rank features, and to reduce the number of features systematically while bounding the inference error. This allows us to derive WWW-state witnesses capable of detecting entanglement with fewer measurement terms than the fidelity method dominant in today's literature. The utility of this approach is demonstrated on quantum hardware furnished through the IBM Quantum Experience.
Abstract We propose a support vector machine (SVM) based approach for generating an entanglement witness that requires exponentially less training data than previously proposed methods. SVMs generate hyperplanes represented by a weighted sum of expectation values of local observables whose coefficients are optimized to sum to a positive number for all separable states and a negative number for as many entangled states as possible near a specific target state. Previous SVM-based approaches for entanglement witness generation used large amounts of randomly generated separable states to perform training, a task with considerable computational overhead. Here, we propose a method for orienting the witness hyperplane using only the significantly smaller set of states consisting of the eigenstates of the generalized Pauli matrices and a set of entangled states near the target entangled states. With the orientation of the witness hyperplane set by the SVM, we tune the plane’s placement using a differential program that ensures perfect classification accuracy on a limited test set as well as maximal noise tolerance. For N qubits, the SVM portion of this approach requires only O ( 6 N ) training states, whereas an existing method needs O ( 2 4 N ) . We use this method to construct witnesses of 4 and 5 qubit GHZ states with coefficients agreeing with stabilizer formalism witnesses to within 3.7 percent and 1 percent, respectively. We also use the same training states to generate novel 4 and 5 qubit W state witnesses. Finally, we computationally verify these witnesses on small test sets and propose methods for further verification.
Abstract Although entanglement is a basic resource for reaching quantum advantage in many computation and information protocols, we lack a universal recipe for detecting it, with analytical results obtained for low-dimensional systems and few special cases of higher-dimensional systems. In this work, we use a machine learning algorithm, the support vector machine with polynomial kernel, to classify separable and entangled states. We apply it to two-qubit and three-qubit systems, and we show that, after training, the support vector machine is able to recognize if a random state is entangled with an accuracy up to 92%92\%92% 92 % for the two-qubit system and up to 98%98\%98% 98 % for the three-qubit system. We also describe why and in what regime the support vector machine algorithm is able to implement the evaluation of an entanglement witness operator applied to many copies of the state, and we describe how we can translate this procedure into a quantum circuit.
We apply the support vector machine (SVM) algorithm to derive a set of entanglement witnesses (EW) to identify entanglement patterns in families of four-qubit states. The effectiveness of SVM for practical EW implementations stems from the coarse-grained description of families of equivalent entangled quantum states. The equivalence criteria in our work is based on the stochastic local operations and classical communication classification and the description of the four-qubit entangled Werner states. We numerically verify that the SVM approach provides an effective tool to address the entanglement witness problem when the coarse-grained description of a given family state is available. We also discuss and demonstrate the efficiency of nonlinear kernel SVM methods as applied to four-qubit entangled state classification.
We are entering a time in which quantum channel capacity is becoming increasingly dependent on having high-dimensional entangled systems. However, existing methods of detecting high-dimensional entanglement are hindered by their need for an unwieldy amount of measurements and computational power. Indeed, the use of entanglement witnesses circumvents this problem by only considering a small subset of the entangled states in Hilbert Space. Although efficient witnesses for 2-qubit Bell States and 3-Qubit GHZ states are well-known, the predominant “fidelity method" witnesses used for detecting multipartite W states remain relatively complex in the amount of measurements that they require. The work outlined in this thesis aims to improve on existing entanglement witnesses by drawing their correspondence with supervised learning techniques, specifically Support Vector Machines. In doing so, one can derive entanglement witnesses whose features (observables obtained via local operations and classical communications) can be ranked according to their impact on the classification of entangled and separable states. This method allows us to derive W state witnesses that require fewer measurements than those derived via the fidelity method. The process of Recursive Feature Elimination is also presented, by which we can define witnesses that can be decomposed into fewer Pauli observables at the cost of inferior tolerance to noise. Finally, we present the practical utility of our derived witnesses through physical experiments performed on the superconducting qubit devices provided by IBM.
We introduce a systematic framework for three-qubit entanglement classification using a cascaded architecture of Support Vector Machine (SVM) classifiers. Leveraging the well defined three-qubit structure with the four nested entanglement classes (S, B, W, and GHZ), we construct three distinct witness models (MB\mathcal{M}_{B}MB​, MW\mathcal{M}_{W}MW​, and MGHZ\mathcal{M}_{GHZ}MGHZ​) that sequentially discriminate between these classes. The proposed Cascaded model achieves an overall classification accuracy of 95%95\%95% on a comprehensive dataset of mixed states. The framework's robustness and generalization capabilities are confirmed through rigorous testing against out-of-distribution (OOD) entangled states and various quantum noise channels, where the model maintains high performance. A key contribution of this research is an optimization protocol based on systematic feature importance analysis. This approach yields a tunable framework that significantly reduces the number of required features, while maintaining reliable model accuracy.