On The Computation Of US-eigenvalues Of Symmetric Complex Tensors And Geometric Measure Of Quantum Entanglement

A tensor is a multidimensional array.A vector is an order one tensor as a matrix is simply an order two tensor.A tensor whose order is greater than two is called a highorder tensor.High-order tensor has been widely applied in many fields including quantum entanglement,signal processing,stoichiometry.Theory and application of tensors have been developed quickly since tensor eigenvalues were introduced in 2005.Quantum entanglement is a important resource in quantum information.A fundamental task is how to characterize and quantify entanglement.One of most important entanglement measure is geometric measure of quantum entanglement.Geometric measure of quantum entanglement can be related to the problem of tensor eigenvalues since the U-eigenvalue and US-eigenvalue of complex tensors were introduced.In this paper,we study geometric measure of quantum entanglement by US-eigenvalues of complex symmetric tensors.The main work is as follows.1.In order to deal with real-valued functions with complex variables,we introduce Wirtinger calculus theory and propose the first-order optimization necessary condition of real-valued functions with complex variables optimization problems with equality constraints.We prove US-eigenvectors are KKT points of real-valued functions with complex variables optimization problems with sphere constraints.We find that the largest US-eigenvalue is also the entanglement eigenvalue.We propose a mathematic model to compute geometric measure of quantum entanglement.2.we study and comb the relationship between US-eigenvalues and entanglement eigenvalues.The main conclusion contains:(1).If a vector z is a US-eigenvector of S,then zmis a separable state corresponding to entanglement eigenvalue,but the inverse may be not true;(2).If z is a KKT point of the entanglement eigenvalue problem and|S*zm| = ?,then ±? is a US-eigenvalue of S;(3).If z is a US-eigenvalue of S,then z is a KKT point of the entanglement eigenvalue problem,but the inverse may be not true.3.By the relationship between US-eigenvalues and entanglement eigenvalues,we can compute geometric measure of entanglement by solving the largest US-eigenvalue problem.We covert the largest US-eigenvalue problem to a real polynomial optimization problem,then construct it to a Jacobian SDP relaxations form and compute it.The sufficient and necessary condition of its convergence is guaranteed if US-eigenvectors are finite.Comparing the numerical result with geometric measure of entanglement shows its feasibility.