Arbitrary violation of the Tsirelson bound

We present a simple hidden-variable model for the singlet state of a pair of qubits, characterized by two kinds of hierarchically ordered hidden variables. The model is such that, averaging over both types of variables, one reproduces all the quantum mechanical correlations of the singlet state. In contrast, averaging only over the hidden variables of the lower level, one obtains a general formal theoretical scheme exhibiting correlations stronger than the quantum ones, but with faster-than-light communication forbidden. By varying the parameters of the model, it is possible to obtain an arbitrary violation of the Clauser-Horne-Shimony-Holt and Tsirelson bounds. This result is interesting by itself since it shows that a violation of the quantum bound for nonlocal correlations can be implemented in a precise physical manner and not only mathematically and it suggests that resorting to two levels of nonlocal hidden variables might lead to a deeper understanding of the physical principles at the basis of quantum nonlocality. It is useful to mention that a model having features similar to ours has already been considered in the literature. However, from the point of view of its nonlocal aspects, it is physically much less appealing since averaging over the lower-level hidden variables yields correlations that can take only a discrete set of values for any choice of the upper-level variables and not the continuous set of values characterizing our approach. We discuss this problem in the concluding section of our work.

Figure 1

Contour plot of the function $|F_{\psi ,\tau }|$ in the space of the parameters $(\alpha ,\tau )$. This space is divided in three regions, corresponding to locality $|F_{\psi ,\tau }|⩽2$, quantum nonlocality $2<|F_{\psi ,\tau }|⩽2\sqrt{2}$, and superquantum nonlocality $2\sqrt{2}<\left|F_{\psi ,\tau }\right|⩽4$, identified by the dashed lines.

Figure 2

The eight spherical regions described in the text, labeled 1–8. The four maximal circles, which define such regions, are associated with the four vectors ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_{+}$, and ${\lambda }_{-}$. As discussed in the text, if $\mathbf{a}$ and $\mathbf{b}$ are changed by being kept in the same region, the function $|F_{\psi ,\tau }|$ remains constant and takes one of the three values $\{0,2,4\}$.

Figure 3

In our model, integration over $\mu$ for a fixed $\tau$ means integration over a meridian of the sphere. Two possible choices of $\tau$ are shown, labeled ${\tau }_1$ and ${\tau }_2$. The function $E_{\psi ,\tau }(A,B)$ varies continuously with $\tau$, $\mathbf{a}$, and $\mathbf{b}$.