Violations of Bell inequalities from random pure states

We consider the expected violations of Bell inequalities from random pure states. More precisely, we focus on a slightly generalized version of the Collins-Gisin-Linden-Massar-Popescu inequality, which concerns Bell experiments of two parties, two measurement options, and $N$ outcomes, and analyze their expected quantum violations from random pure states for varying $N$, assuming the conjectured optimal measurement operators. It is seen that for small $N$ the Bell inequality is not violated on average, while for larger $N$ it is. Both ensembles of unstructured as well as structured random pure states are considered. Using techniques from random matrix theory this is obtained analytically for small and large $N$ and numerically for intermediate $N$. The results show a beautiful interplay of different aspects of random matrix theory, ranging from the Marchenko-Pastur distribution and fixed-trace ensembles to the $O\left(n\right)$ model.

Figure 1

Illustration of the Bloch sphere. Any pure state in $\mathcal{H}$ corresponds to a point on the sphere, while mixed states correspond to points within the unit ball. A random pure state of the Hilbert-Schmidt ensemble is obtained by choosing a point on the sphere uniformly at random. This can be done by applying a random rotation $U$ to the references state $|0\rangle$, here chosen to be the north pole.

Figure 2

Mean minimal value of ${\mathcal{A}}_N$ under the measure of random pure states from the HS ensemble (red circles) in comparison to the corresponding value for a maximally entangled state (blue squares), both as a function of $N$. The dotted lines correspond to their asymptotic values. In addition, the mean minimal value of ${\mathcal{A}}_N$ under the measure of random pure states from the structured ensemble (orange triangles) is plotted as a function of $N$ for $k=12,6,3,2$ (from top to bottom). Indicated in gray is the region where the Bell inequality is not violated.

Figure 3

Mean minimal value of ${\mathcal{A}}_N$ (solid line) plus or minus its standard deviation (dashed lines) under the measure of random pure states from the HS ensemble. The insets show the histograms from $N=2$ and 488.