Calculating Entanglement Eigenvalues for Nonsymmetric Quantum Pure States Based on the Jacobian Semidefinite Programming Relaxation Method

Mengshi Zhang (), Xinzhen Zhang () and Guyan Ni ()
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Mengshi Zhang: National University of Defense Technology
Xinzhen Zhang: Tianjin University
Guyan Ni: National University of Defense Technology

Journal of Optimization Theory and Applications, 2019, vol. 180, issue 3, No 6, 787-802

Abstract: Abstract The geometric measure of entanglement is a widely used entanglement measure for quantum pure states. The key problem of computation of the geometric measure is to calculate the entanglement eigenvalue, which is equivalent to computing the largest unitary eigenvalue of a corresponding complex tensor. In this paper, we propose a Jacobian semidefinite programming relaxation method to calculate the largest unitary eigenvalue of a complex tensor. For this, we first introduce the Jacobian semidefinite programming relaxation method for a polynomial optimization with equality constraint and then convert the problem of computing the largest unitary eigenvalue to a real equality constrained polynomial optimization problem, which can be solved by the Jacobian semidefinite programming relaxation method. Numerical examples are presented to show the availability of this approach.

Keywords: Jacobian semidefinite programming relaxation; Entanglement eigenvalue; Unitary eigenvalue; Polynomial optimization; Complex tensor; 15A18; 15A69; 81P40 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-018-1357-7

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