Generalized geometric measure of entanglement for multiparty mixed states

Computing entanglement of an arbitrary bipartite or multipartite mixed state is in general not an easy task as it usually involves complex optimization. Here we show that, exploiting symmetries of certain multiqudit mixed states, we can compute a genuine multiparty entanglement measure, the generalized geometric measure, for these classes of mixed states. The chosen states have different ranks and consist of an arbitrary number of parties.

Figure 1

GGM of ${\rho }_N^2(x,\text{sym})=x|{\psi }_N\rangle \langle {\psi }_N|+(1-x)|{\psi }_N^{\perp }\rangle \langle {\psi }_N^{\perp }|$ against $x$. All the quantities are dimensionless.

Figure 2

A plot of the GGM of ${\rho }_3^3(x_1,x_2)=x_1|{\text{GHZ}}_3^{+}\rangle \langle {\text{GHZ}}_3^{+}|+x_2|D^1\rangle \langle D^1|+(1-x_1-x_2)|D^2\rangle \langle D^2|$ with the state parameters $x_1$ and $x_2$. All the axes are dimensionless.

Figure 3

Plot corresponds to GGM of $|{\psi }_3^{3,g}\rangle$ vs the mixing parameters $x_1$ and $x_2$. Here, $\alpha =0.55$ for the $|g{\text{GHZ}}_3\rangle$ state. Both convex and nonconvex regions are seen. The convex part corresponds to the GGM of ${\rho }_3^{3,g}(x_1,x_2)$. All quantities are dimensionless.

Figure 4

Plot corresponds to GGM of $|{\psi }_3^{3,g}\rangle$ vs $x_1$, for two values of $r=\frac{x_2}{1-x_1}$. Here, $\alpha =0.55$ for the $|g{\text{GHZ}}_3\rangle$ state. These are given by the dotted lines. The straight lines corresponds to the convexified quantities. All quantities are dimensionless.

Figure 5

The plot of GGM for ${\rho }_5^3(x_1,x_2)=x_1|{\text{GHZ}}_5^{+}\rangle \langle {\text{GHZ}}_5^{+}|+x_2|D^1\rangle \langle D^1|+(1-x_1-x_2)|D^4\rangle \langle D^4|$ against $x_1$ and $x_2$ whenever it is convex. All axes are dimensionless.

Figure 6

GGM of ${\rho }_5^5=x_{1}P\left[{\text{GHZ}}_5^{+}\right]+\frac{x_2}{2}(P\left[D^1\right]+P\left[D^2\right])+\frac{1-x_1-x_2}{2}(P\left[D^3\right]+P\left[D^4\right])$. All axes are dimensionless.

Figure 7

Plot of GGM of ${\rho }_3^4$ with respect to the parameters, $x$ and $y$. The GGM of the corresponding unique pure state, $|{\psi }_3^4(x,y)\rangle =\sqrt{x}|{\zeta }_1\rangle -i\sqrt{y/2}\left(\right|{\zeta }_2\rangle -|{\zeta }_3\rangle )+\sqrt{1-x-y}|{\zeta }_4\rangle$, has a kink along the lines shown on the surface, in the plot. The GGM of the pure state is nonconvex around these lines, and hence convexifications are required to obtain the GGM of ${\rho }_3^4$ along those lines.

Figure 8

Plot of GGM of ${\rho }_{3,3}^3$ against $x_1$ and $x_2$. The GGM of the corresponding unique pure state, $|{\mathrm{\Psi }}_{3,3}^3\rangle =\sqrt{x_1}{|\mathrm{\Psi }\rangle }_1+e^{i{\varphi }_2}\sqrt{x_2}{|\mathrm{\Psi }\rangle }_2+e^{i{\varphi }_3}\sqrt{1-x_1-x_2}{|\mathrm{\Psi }\rangle }_3$, is plotted with ${\varphi }_1={\varphi }_2=0$. The GGM of the pure state is convex everywhere, as evident from this plot and hence $\mathcal{E}\left(\right|{\mathrm{\Psi }}_{3,3}^3\rangle )=\mathcal{G}\left({\rho }_{3,3}^3\right)$.