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arXiv:0909.2907 · Physical Review A

Observation of tunable Popescu-Rohrlich correlations through postselection of a Gaussian state

D. S. Tasca, Stephen Patrick Walborn, Fabricio Toscano, P. H. Souto Ribeiro

Published Sep 14, 2009• 99 citations
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Abstract

We show that nonlocal Popescu-Rohrlich correlations can be observed in the postselected results of binned position measurements on a two-party Gaussian state. Our experiment is based on the spatial correlations of entangled photons and lens systems. We obtain a maximum violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality of 3.42, which corresponds to the implementation of a nonlocal AND gate with success probability of 0.93. These results do not conflict with quantum mechanics due to the postselection required and open up the possibility of experimental investigation of fundamental aspects of Popescu-Rohrlich nonlocality with a reliable and simple experimental setup.

DOI
10.1103/physreva.80.030101

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We show that nonlocal Popescu-Rohrlich correlations can be observed in the postselected results of binned position measurements on a two-party Gaussian state. Our experiment is based on the spatial correlations of entangled photons and lens systems. We obtain a maximum violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality of 3.42, which corresponds to the implementation of a nonlocal AND gate with success probability of 0.93. These results do not conflict with quantum mechanics due to the postselection required and open up the possibility of experimental investigation of fundamental aspects of Popescu-Rohrlich nonlocality with a reliable and simple experimental setup.
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We consider quantum ensembles which are determined by pre- and post-selection. Unlike the case of only preselected ensembles, we show that in this case the probabilities for measurement outcomes at intermediate times satisfy causality only rarely; such ensembles can in general be used to signal between causally disconnected regions. We show that under restrictive conditions, there are certain nontrivial bipartite ensembles which do satisfy causality. These ensembles give rise to a violation of the Clauser-Horne-Shimony-Holt inequality, which exceeds the maximal quantum violation given by Tsirelson's bound ${B}_{\mathrm{CHSH}}\ensuremath{\le}2\sqrt{2}$ and obtains the Popescu-Rohrlich bound for the maximal violation, ${B}_{\mathrm{CHSH}}\ensuremath{\le}4$. This may be regarded as an a posteriori realization of supercorrelations, which have recently been termed Popescu-Rohrlich boxes.
Witnessing continuous-variable Bell nonlocality is a challenging endeavor, but Bell himself showed how one might demonstrate this nonlocality. Although Bell nearly showed a violation using the Clauser-Horne-Shimony-Holt (CHSH) inequality with sign-binned position-momentum statistics of entangled pairs of particles measured at different times, his demonstration is subject to approximations not realizable in a laboratory setting. Moreover, he does not give a quantitative estimation of the maximum achievable violation for the wave function he considers. In this article, we show how his strategy can be reimagined using the transverse positions and momenta of entangled photon pairs measured at different propagation distances, and we find that the maximum achievable violation for the state he considers is actually very small relative to the upper limit of 222\sqrt{2}22​. Although Bell's wave function does not produce a large violation of the CHSH inequality, other states may yet do so.
Local unitary operations cannot affect the quantum correlations between two systems sharing an entangled state although they do influence the outcomes of local measurements. By considering local squeezing operations we introduce an extended family of observables allowing violation of the Clauser-Horne-Shimony-Holt Bell inequality for two-mode Gaussian systems. We show that local squeezing can enable or enhance the identification of nonlocal two-mode states. In particular, we show that local squeezing followed by photon--no-photon discrimination can suffice to reveal nonlocality in a broad ensemble of pure and mixed two-mode Gaussian states.
We demonstrate an experimental test of the Clauser–Horne– Shimony–Holt (CHSH) Bell inequality which seemingly exhibits correlations beyond the limits imposed by quantum mechanics. Inspired by the idea of Fourier synthesis, we design analysers that measure specific superpositions of orbital angular momentum (OAM) states, such that when one analyser is rotated with respect to the other, the resulting coincidence curves are similar to a square-wave. Calculating the CHSH Bell parameter, S , from these curves result to values beyond the Tsirelson bound of . We obtain S = 3.99 ± 0.02, implying almost perfect nonlocal Popescu–Rohrlich correlations. The 'super-quantum' values of S is only possible in our experiment because our experiment, subtly, does not comply with fair-sampling. The way our Bell test fails fair-sampling is not immediately obvious and requires knowledge of the states being measured. Our experiment highlights the caution needed in Bell-type experiments based on measurements within high-dimensional state spaces such as that of OAM, especially in the advent of device-independent quantum protocols.