Riemannian Manifold Optimization Algorithm For The Best Rank One Approximation Of Complex Symmetric Tensors

In quantum information systems,quantum entanglement has became a research hot point in the field of quantum information.With the rapid development of science and technology and the information age,quantum information technology has also became an important guarantee for national information security,quantum entanglement is an important resource in the field of quantum information,therefore,judging and measuring quantum entanglement are key issues in the field of quantum information.Closely related to the quantum entanglement problem is the best rank one approximation problem for complex symmetric tensors,with the introduction of the concept of complex tensor U-eigenvalue and complex symmetric tensor US-eigenvalue,the problem of the best rank one approximation of a complex symmetric tensor is equivalent to the problem of calculating the maximum US-eigenvalue of a complex symmetric tensor,therefore,the eigenvalue problem of complex tensors has important research significance in the field of quantum entanglement.In this paper,the pure state quantum entanglement is studied by calculating the maximum US-eigenvalue in the complex number domain.Firstly,the theoretical knowledge of Wirtinger calculus is introduced,and the gradient of multiple complex variable real-valued functions and Hessian expressions are given;It also introduces the basic knowledge of Riemannian manifold optimization in the real number domain,and presents related concepts such as tangent vector,tangent space,Riemann metric,Riemann gradient,retraction,vector transport,etc.in the real Riemannian manifold.Secondly,extend the basic knowledge of Riemannian manifold optimization in the real number domain to the complex number domain,the maximum US-eigenvalue problem for complex symmetric tensors is an optimization problem with unit sphere constraints,consider the unit ball constraint on the complex number field as a complex Riemannian manifold,based on Wirtinger calculus knowledge,give specific expressions of related concepts such as tangent vector,tangent space,Riemann metric,Riemann gradient,retraction,vector transport,etc.in the complex unit spherical manifold,it lays the foundation for the proposed complex Riemannian manifold algorithm.Then,convert the maximum US-eigenvalue problem of the complex symmetric tensor into an unconstrained optimization problem in the complex Riemannian manifold space,the complex Riemann Newton method and the complex Riemann steepest descent method are proposed to calculate the maximum US-eigenvalue of the complex symmetry tensor,it also gives a theoretical analysis of the convergence of the algorithm,prove that our proposed complex Riemann Newton algorithm converges locally,and the convergence rate is super-linear or even quadratic;according to the convergence theory of complex Riemann's Newton method,first,use the solution obtained by the complex Riemann steepest descent method as the initial point of the complex Riemann Newton method,thus,the stability of complex Riemann Newton method is improved.Finally,perform numerical experiments on the algorithm to verify the effectiveness.