Multipartite entanglement accumulation in quantum states: Localizable generalized geometric measure

Multiparty quantum states are useful for a variety of quantum information and computation protocols. We define a multiparty entanglement measure based on local measurements on a multiparty quantum state and an entanglement measure averaged on the postmeasurement ensemble. Using the generalized geometric measure as the measure of multipartite entanglement for the ensemble, we demonstrate, in the case of several well-known classes of multipartite pure states, that the localized multipartite entanglement can exceed the entanglement present in the original state. We also show that measurement over multiple parties may be beneficial in enhancing localizable multipartite entanglement. We point out that localizable generalized geometric measure faithfully signals quantum critical phenomena in well-known quantum spin models even when considerable finite-size effect is present in the system.

Figure 1

(a) Plot of $E_L^1$ vs $\mathcal{G}$ for three-qubit $\mathrm{g}W$ states. To obtain the scatter diagram, ${10}^5$ three-qubit $\mathrm{g}W$ states are generated Haar uniformly. The solid line represents the line $E_L^1=\mathcal{G}$, while the dashed line corresponds to $2E_L^1+\mathcal{G}=1$. (b) Plot of $E_L^1$ vs $\mathcal{G}$ in the case of Haar-uniformly generated generalized superposition of Dicke states, as given in Eq. (10). To obtain the scatter diagram, ${10}^5$ states of the form $|D_g^N\rangle$ are generated Haar-uniformly for each of the cases $N=4$ and $N=5$. All quantities plotted in both figures are dimensionless.

Figure 2

(a) Plot of $\mathcal{G},E_L^1$, and $E_L^{\{1,2\}}$ as functions of $a$ in case of the four-qubit $\mathrm{g}W$ state $|W^4{\rangle }_g$ with $a_1=a,a_2=\sqrt{(1-a^2)/5},a_3=\sqrt{3(1-a^2)/10}$, and $a_4=\sqrt{(1-a^2)/2}$. (b) Plot of $E_L^1$ and $E_L^{\{1,2\}}$ with $k$ for $N=7$ and $N=8$ in the case of Dicke states. All quantities plotted are dimensionless.

Figure 3

Scatter plots. (a) $E_L^1$ vs $\mathcal{G}$ for three-qubit pure states belonging to the GHZ class and the $W$ class. (b) $E_L^1$ vs $\mathcal{G}$ for the four-qubit classes represented by $|{\mathrm{\Psi }}_i^4\rangle ,i=1,\cdots ,6$. (c) $E_L^2$ vs $\mathcal{G}$ for the four-qubit classes represented by $|{\mathrm{\Psi }}_i^4\rangle ,i=3,5,6$. (d) $E_L^1$ and $E_L^{\{1,2\}}$ vs $\mathcal{G}$ in the case of arbitrary four-qubit pure states. (e) $E_L^1$ and $E_L^{\{1,2\}}$ vs $\mathcal{G}$ for four-qubit states of the form $|{\mathrm{\Psi }}_1^4\rangle$. (f) $E_L^1$ and $E_L^{\{1,2\}}$ vs $\mathcal{G}$ for four-qubit symmetric states of the form $|D_g^N\rangle$. Each plot constitutes of ${10}^5$ Haar-uniformly generated pure states. All quantities plotted are dimensionless.

Figure 4

(a) Variations of $E_L^1$ (left panel) and $d{E}_L^1/d\lambda$ (right panel) as functions of $\lambda =J/h$ for the transverse-field Ising model with $N=16$. (b) Variation of $\mathcal{G},E_L^1$, and $E_L^{\{1,2\}}$ as functions of $\lambda$ with $\mathrm{\Delta }=0.5$ (left panel) and $\mathrm{\Delta }$ with $\lambda =0$ (right panel) for $N=20$. All quantities plotted are dimensionless.