  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.
