Computation Method For The US-eigenvalues Of A Symmetric Tensor

Geometric measure of quantum entanglement,as a hotspot problem in quantum in-formation,has applications in various different topics,including theoretical physics,quan-turn computation,condensed matter systems,quantum channel capacities and so on.The rank-one approximation to higher-order tensors problem,which is related with quantum entanglement closely,also has attracted much attention.The rank-one approximation to higher-order tensors problem can be viewed as the problem of computing the largest US-eigenvalues in the complex field.This paper is aimed at presenting a fast and effective algorithm for computing the US-eigenvalues of a higher-order complex symmetric tensor.The computation of US-eigenvalue can be regard as an unconstrained nonlinear opti-mization problem in convex field.In this article,we convert the US-eigenvalue system to an unconstrained complex symmetric nonlinear optimization problem.Generally,uncon-strained nonlinear optimization methods will almost always need a first-or second-order derivative of the objective function.However,the objective function we considered is a real valued function with complex variables,it is not necessarily analytic in its argument,say,the Taylor polynomial is not necessarily exist.Hence,the classic optimization methods cannot be applied to real functions with complex variables.To overcome this problem,we notice that the objective function is not analytic about the individual variables though,it is analytic about the variable and its conjugate variable as a whole.We will obtain the gra-dient of a real-valued function with complex variables.Then we can generalize the classic optimization to the complex field.Finally,we propose a quasi-Newton method for comput-ing the US-eigenvalue.And its iterative sequence is norm descent,which guarantees the quasi-Newton direction is the descent direction at the iteration point.We also prove the algorithm is global convergent and the rate of convergence is super-linear theoretically.Meanwhile,large amount of numerical experiment confirm this method is fast and effective.