Geometric entanglement from matrix product state representations

An efficient scheme to compute the geometric entanglement per lattice site for quantum many-body systems on a periodic finite-size chain is proposed in the context of a tensor network algorithm based on the matrix product state representations. It is systematically tested for three prototypical critical quantum spin chains, which belong to the same Ising universality class. The simulation results lend strong support to the previous claim [Q.-Q. Shi, R. Or\'{u}s, J. O. Fj{\ae}restad, and H.-Q. Zhou, New J. Phys \textbf{12}, 025008 (2010); J.-M. St\'{e}phan, G. Misguich, and F. Alet, Phys. Rev. B \textbf{82}, 180406R (2010)] that the leading finite-size correction to the geometric entanglement per lattice site is universal, with its remarkable connection to the celebrated Affleck-Ludwig boundary entropy corresponding to a conformally invariant boundary condition.Comment: 4+ pages, 3 figure

Additivity and non-additivity of multipartite entanglement measures

We study the additivity property of three multipartite entanglement measures, i.e. the geometric measure of entanglement (GM), the relative entropy of entanglement and the logarithmic global robustness. First, we show the additivity of GM of multipartite states with real and non-negative entries in the computational basis. Many states of experimental and theoretical interests have this property, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, the Smolin state, and the generalization of D\"{u}r's multipartite bound entangled states. We also prove the additivity of other two measures for some of these examples. Second, we show the non-additivity of GM of all antisymmetric states of three or more parties, and provide a unified explanation of the non-additivity of the three measures of the antisymmetric projector states. In particular, we derive analytical formulae of the three measures of one copy and two copies of the antisymmetric projector states respectively. Third, we show, with a statistical approach, that almost all multipartite pure states with sufficiently large number of parties are nearly maximally entangled with respect to GM and relative entropy of entanglement. However, their GM is not strong additive; what's more surprising, for generic pure states with real entries in the computational basis, GM of one copy and two copies, respectively, are almost equal. Hence, more states may be suitable for universal quantum computation, if measurements can be performed on two copies of the resource states. We also show that almost all multipartite pure states cannot be produced reversibly with the combination multipartite GHZ states under asymptotic LOCC, unless relative entropy of entanglement is non-additive for generic multipartite pure states.Comment: 45 pages, 4 figures. Proposition 23 and Theorem 24 are revised by correcting a minor error from Eq. (A.2), (A.3) and (A.4) in the published version. The abstract, introduction, and summary are also revised. All other conclusions are unchange

Finite-size geometric entanglement from tensor network algorithms

The global geometric entanglement (GE) is studied in the context of newly developed tensor network algorithms for finite systems. For onedimensional quantum spin systems it is found that, at criticality, the leading finite-size correction to the global GE per site behaves as b/n, where n is the size of the system and b a given coefficient. Our conclusion is based on the computation of the GE per spin for the quantum Ising model in a transverse magnetic field and for the spin-1/2 XXZ model. We also discuss the possibility of coefficient b being universal