TY - JOUR

T1 - Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors

AU - Hu, Shenglong

AU - Qi, Liqun

AU - Zhang, Guofeng

PY - 2016/1/6

Y1 - 2016/1/6

N2 - The geometric measure of entanglement for pure states has attracted much attention. On the other hand, the spectral theory of non-negative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of non-negative tensors can be applied to the study of the geometric measure of entanglement for pure states. For symmetric pure multipartite qubit or qutrit states an elimination method is given. For symmetric pure multipartite qudit states, a numerical algorithm with randomization is presented. We also show that a nonsymmetric pure state can be augmented to a symmetric one whose amplitudes can be encoded in a non-negative symmetric tensor, so the geometric measure of entanglement can be calculated. Several examples, such as mGHZ states, W states, inverted W states, qudits, and nonsymmetric states, are used to demonstrate the power of the proposed methods. Given a pure state, one can always find a change of basis (a unitary transformation) so that all the probability amplitudes of the pure state are non-negative under the new basis. Therefore, the methods proposed here can be applied to a very wide class of multipartite pure states.

AB - The geometric measure of entanglement for pure states has attracted much attention. On the other hand, the spectral theory of non-negative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of non-negative tensors can be applied to the study of the geometric measure of entanglement for pure states. For symmetric pure multipartite qubit or qutrit states an elimination method is given. For symmetric pure multipartite qudit states, a numerical algorithm with randomization is presented. We also show that a nonsymmetric pure state can be augmented to a symmetric one whose amplitudes can be encoded in a non-negative symmetric tensor, so the geometric measure of entanglement can be calculated. Several examples, such as mGHZ states, W states, inverted W states, qudits, and nonsymmetric states, are used to demonstrate the power of the proposed methods. Given a pure state, one can always find a change of basis (a unitary transformation) so that all the probability amplitudes of the pure state are non-negative under the new basis. Therefore, the methods proposed here can be applied to a very wide class of multipartite pure states.

UR - http://www.scopus.com/inward/record.url?scp=84954547010&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.93.012304

DO - 10.1103/PhysRevA.93.012304

M3 - Journal article

VL - 93

JO - Physical Review A - Atomic, Molecular, and Optical Physics

JF - Physical Review A - Atomic, Molecular, and Optical Physics

SN - 1050-2947

IS - 1

M1 - 012304

ER -