Better Bell-inequality violation by collective measurements

The standard Bell-inequality experiments test for violation of local realism by repeatedly making local measurements on individual copies of an entangled quantum state. Here we investigate the possibility of increasing the violation of a Bell inequality by making collective measurements. We show that the nonlocality of bipartite pure entangled states, quantified by their maximal violation of the Bell-Clauser-Horne inequality, can always be enhanced by collective measurements, even without communication between the parties. For mixed states we also show that collective measurements can increase the violation of Bell inequalities, although numerical evidence suggests that the phenomenon is not common as it is for pure states.

Figure 1

(Color online) Best known Bell-CH inequality violation of pure two-qubit states obtained from (3), plotted as a function of $\varphi$, which gives a primitive measure of entanglement; $\varphi =0$ for bipartite pure product state and $\varphi =45°$ for bipartite maximally entangled state. The curves from right to left represent increasing numbers of copies. The dotted horizontal line at $1∕\sqrt{2}-1∕2$ is the maximal possible violation of Bell-CH inequality; correlations allowed by local realistic theories have values less than or equal to zero.

Figure 2

(Color online) Best known expectation value of the Bell-CH, Bell-3322, and Bell-2244 operators with respect to the two-dimensional Werner states; $p$ represents the overlap with a singlet state. Also included is the upper bound on the maximal $⟨{\mathcal{B}}_{CH}{⟩}_{{\rho }_W^{\otimes 2}}$ obtained from the UB algorithm [15].

Figure 3

(Color online) Distribution of two-qubit states sampled for improved Bell-CH violation by collective measurements. The maximally entangled mixed states, which demarcate the boundary of the set of density matrices on this concurrence-entropy plane [39, 40], are represented by the solid line. Note that as a result of the chosen distribution over mixed states [37] this region is not well sampled. The region bounded by the solid line and the horizontal dashed line (with concurrence equal to $1∕\sqrt{2}$) only contain two-qubit states that violate the Bell-CH inequality [42]; the region bounded by the solid line and the vertical dashed line (with normalized linear entropy equal to $2∕3$) only contain states that do not violate the Bell-CH inequality [41, 42]. Two-qubit states found to give a better three-copy Bell-CH violation are marked with red crosses.

Figure 4

(Color online) Best known expectation value of the Bell-CH and Bell-3322 operators with respect to the three-dimensional isotropic states; $p$ is the fraction of maximally entangled two-qutrit state in the mixture. Also included is the upper bound on the maximal $⟨{\mathcal{B}}_{CH}{⟩}_{{\rho }_{I_3}}$ obtained from the UB algorithm [15].