Proposed Experiment to Test the Bounds of Quantum Correlations

The combination of quantum correlations appearing in the Clauser-Horne-Shimony-Holt inequality can give values between the classical bound, 2, and Tsirelson’s bound, $2\sqrt{2}$. However, for a given set of local observables, there are values in this range which no quantum state can attain. We provide the analytical expression for the corresponding bound for a parametrization of the local observables introduced by Filipp and Svozil, and describe how to experimentally trace it using a source of singlet states. Such an experiment will be useful to identify the origin of the experimental errors in Bell’s inequality-type experiments and could be modified to detect hypothetical correlations beyond those predicted by quantum mechanics.

Figure 1

Values of the CHSH operator for the maximally entangled states $|\varphi (\xi )⟩=\cos \xi |{\varphi }^{+}⟩+\sin \xi |{\psi }^{-}⟩$, for $0\le \xi <2\pi$, for the local measurements (5, 6, 7, 8) parametrized by $\theta$. The analytical expression of the quantum bound (thick line) is given by (10). Given a particular set of local measurements (i.e., a particular value of $\theta$), the quantum bound is reached by the maximally entangled states with $\xi (\theta )$ given by (12). No other quantum state (and, presumably, no other preparation of physical systems) gives values outside these bounds. Dashed lines represent Bell’s classical bounds ($\pm 2$) and Tsirelson’s bounds ($\pm 2\sqrt{2}$).

Figure 2

Scheme of the experiment to test the bounds of quantum correlations. It consists of a source of two-qubit singlet states $|{\psi }^{-}⟩$, a unitary operation $U(\theta )$ [given by (15, 12)] on qubit $I$, and the local measurements $A(\theta )$ [given by (5)] and (alternatively) $a(\theta )$ [given by (7)] on qubit $I$, and $B(\theta )$ [given by (6)] and (alternatively) $b(\theta )$ [given by (8)] on qubit $II$. The experiment consists of a set of $N$ Bell’s inequality-type tests for $N$ different values of $\theta$ in the range $[0,\pi ]$.