Genuine multipartite entanglement in quantum phase transitions

We demonstrate that the global-entanglement (GE) measure defined by Meyer and Wallach [J. Math. Phys. 43, 4273 (2002)] is maximal at the critical point for the Ising chain in a transverse magnetic field. Our analysis is based on the equivalence of GE to the averaged linear entropy, allowing an understanding of multipartite entanglement (ME) features through a generalization of GE for bipartite blocks of qubits. Moreover, in contrast to GE, the proposed ME measure can distinguish three paradigmatic entangled states: ${\mathrm{GHZ}}_N$ (Greenberger-Horne-Zeilinger), $W_N$, and ${\mathrm{EPR}}^{\otimes N∕2}$. As such the generalized measure can detect a genuine ME and is maximal at the critical point.

Figure 1

(Color online) von Neumann entropy (dashed curve) and $\mathrm{GE}∕\mathrm{LE}$ (solid curve) as a function of $\lambda$.

Figure 2

(Color online) $E_G^{\left(1\right)}$ (solid curve) and $G(2,1)$ (dashed curve) as a function of $\lambda$. Both quantities are maximal at the critical point $\lambda =1$.

Figure 3

(Color online) $G(2,1)$ (dashed/black), $G(2,15)$ (solid/red), and $E_G^{\left(2\right)}$ (dotted-dashed/blue) as a function of $\lambda$. We see that $E_G^{\left(2\right)}$ is slightly different from $G(2,15)$ showing that $G(2,l)$ saturates as $l\to \infty$.