Shareability of correlations in multiqubit states: Optimization of nonlocal monogamy inequalities

It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.

Figure 1

(a) Plot of differences in Eq. (15): ${\mathrm{\Delta }}_3$ (red diamonds) and ${\mathrm{\Delta }}_3^{\prime }$ (green plusses) vs $max\left({\langle {\mathcal{B}}_{\mathrm{ab}}\rangle }^2+{\langle {\mathcal{B}}_{\mathrm{ac}}\rangle }^2+{\langle {\mathcal{B}}_{\mathrm{bc}}\rangle }^2\right)$ for a sample of $5\times {10}^4$ random pure states of three qubits. Both bounds behave similarly. Notice that the upper linear boundary corresponds to real states, that is, with ${\langle {\sigma }_y\rangle }_i=0$. (b) Same plot as in panel (a) but only for real states and the ${\mathrm{\Delta }}_3^{\prime }$ quantity (real states define the red line). Only ${\mathrm{\Delta }}_3^{\prime }\le 1$ is shown for clarity. (c) Plot of differences in Eq. (15): ${\mathrm{\Delta }}_2$ (red diamonds) and ${\mathrm{\Delta }}_2^{\prime }$ (green plusses) vs $max\left({\langle {\mathcal{B}}_{\mathrm{ab}}\rangle }^2+{\langle {\mathcal{B}}_{\mathrm{ac}}\rangle }^2\right)$. As opposed to panel (a), there is an accumulation of states around $max\left({\langle {\mathcal{B}}_{\mathrm{ab}}\rangle }^2+{\langle {\mathcal{B}}_{\mathrm{ac}}\rangle }^2\right)=8$, the maximum possible value, which entails poorly nonlocal states, including product ones and the very special case where one pair is in a Bell state. All quantities are dimensionless. See text for details.

Figure 2

Plot of the monogamy measure ${\mathrm{\Sigma }}_3$ (14) vs the global multipartite nonlocality measure ${\mathrm{MABK}}_{max}^{n=3}$ for a sample of $3\times {10}^4$ random pure states of three qubits. All quantities are dimensionless. See text for details.

Figure 3

Plot of the possible values for ${\mathrm{\Sigma }}_3$ (14) vs $a$ and $b$ for $W$ states (16). Extremal points such as $a=0,\pm 1$ or $b=0,\pm 1$ fall into separable states (and hence ${\mathrm{\Sigma }}_3=12$). The value ${\mathrm{\Sigma }}_3=8$ is attained in a circle on the surface. Also, in the inset, the upper blue (gray) curve represents the states given by Eq. (17) that possess maximum ${\mathrm{\Sigma }}_3$. All quantities are dimensionless. See text for details.

Figure 4

. Plot of probability density distributions $P\left(X\right)$ for ${10}^6$ pure states of three qubits randomly generated according to the Haar measure. The one to the left corresponds to $X={\mathrm{\Sigma }}_3$, and the other two to the right to the bound $X=12-4[{\langle {\sigma }_y{\sigma }_y\rangle }_{\mathrm{ab}}^2+{\langle {\sigma }_y{\sigma }_y\rangle }_{\mathrm{ac}}^2+{\langle {\sigma }_y{\sigma }_y\rangle }_{\mathrm{bc}}^2]$ (middle curve) and $X=12-4[{\langle {\sigma }_y\rangle }_a^2+{\langle {\sigma }_y\rangle }_b^2+{\langle {\sigma }_y\rangle }_c^2]$ (rightmost curve). All quantities are dimensionless. See text for details.

Figure 5

Plot of the monogamy measure ${\mathrm{\Sigma }}_4$ (14) vs the global multipartite nonlocality measure ${\mathrm{MABK}}_{max}^{n=4}$ for a sample of $3\times {10}^4$ random pure states of four qubits. All quantities are dimensionless. See text for details.

Figure 6

Plot of probability density distributions for ${10}^6$ pure states of four qubits randomly generated according to the Haar measure. The generalized multipartite monogamy measure ${\mathrm{\Sigma }}_4$ clearly does not display any structure, just a de facto effective range, ${\mathrm{\Sigma }}_4\in [5,16]$, instead of the full one, ${\mathrm{\Sigma }}_4\in [0,24]$. Notice the peak around the value of 8. All quantities are dimensionless. See text for details.

Figure 7

Plot of probability density distributions for 33 329 pure states of six qubits randomly generated according to the Haar measure. The generalized multipartite monogamy measure ${\mathrm{\Sigma }}_6$ clearly does not display any structure. Notice the peak around the value of ${\mathrm{\Sigma }}_6=4$. All quantities are dimensionless. See text for details.