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Scientific Reports

Optical simulation of a Popescu-Rohrlich Box

Wen‐Jing Chu, Xiao-Lan Zong, Yang Ming, Guo‐Zhu Pan, Zhuo‐Liang Cao

Published Jun 22, 2016• 7 citations
Publisher

Abstract

It is well known that the fair-sampling loophole in Bell test opened by the selection of the state to be measured can lead to post-quantum correlations. In this paper, we make the selection of the results after measurement, which opens the fair- sampling loophole too, and thus can lead to post-quantum correlations. This kind of result-selection loophole can be realized by pre- and post-selection processes within the "two-state vector formalism", and a physical simulation of Popescu-Rohrlich (PR) box is designed in linear optical system. The probability distribution of the PR has a maximal CHSH value 4, i.e. it can maximally violate CHSH inequality. Because the "two-state vector formalism" violates the information causality, it opens the locality loophole too, which means that this kind of results selection within "two-state vector formalism" leads to both fair- sampling loophole and locality loophole, so we call it a comprehensive loophole in Bell test. The comprehensive loophole opened by the results selection within "two-state vector formalism" may be another possible explanation of why post-quantum correlations are incompatible with quantum mechanics and seem not to exist in nature.

DOI
10.1038/srep28351

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It is well known that the fair-sampling loophole in Bell test opened by the selection of the state to be measured can lead to post-quantum correlations. In this paper, we make the selection of the results after measurement, which opens the fair- sampling loophole too, and thus can lead to post-quantum correlations. This kind of result-selection loophole can be realized by pre- and post-selection processes within the "two-state vector formalism", and a physical simulation of Popescu-Rohrlich (PR) box is designed in linear optical system. The probability distribution of the PR has a maximal CHSH value 4, i.e. it can maximally violate CHSH inequality. Because the "two-state vector formalism" violates the information causality, it opens the locality loophole too, which means that this kind of results selection within "two-state vector formalism" leads to both fair- sampling loophole and locality loophole, so we call it a comprehensive loophole in Bell test. The comprehensive loophole opened by the results selection within "two-state vector formalism" may be another possible explanation of why post-quantum correlations are incompatible with quantum mechanics and seem not to exist in nature.
Abstract The pre- and post-selection processes of the “two-state vector formalism” lead to a fair sampling loophole in Bell test, so it can be used to simulate post-quantum correlations. In this paper, we propose a physical implementation of such a correlation with the help of quantum non-demolition measurement, which is realized via the cross-Kerr nonlinear interaction between the signal photon and a probe coherent beam. The indirect measurement on the polarization state of photon is realized by the direct measurement on the phase shift of the probe coherent beam, which enhances the detection efficiency greatly and leaves the signal photon unabsorbed. The maximal violation of the CHSH inequality 4 can be achieved by pre- and post-selecting maximally entangled states. The reason why we can get the post-quantum correlation is that the selection of the results after measurement opens fair-sampling loophole. The fair-sampling loophole opened here is different from the one usually used in the currently existing simulation schemes for post-quantum correlations, which are simulated by selecting the states to be measured or enlarging the Hilbert space. So, our results present an alternative way to mimic post-quantum correlations .
An optical scheme for simulating nonlocality distillation is proposed in post-quantum regime. The nonlocal boxes are simulated by measurements on appropriately pre- and post-selected polarization entangled photon pairs, i.e. post-quantum nonlocality is simulated by exploiting fair-sampling loophole in a Bell test. Mod 2 addition on the outputs of two nonlocal boxes combined with pre- and post-selection operations constitutes the key operation of simulating nonlocality distillation. This scheme provides a possible tool for the experimental study on the nonlocality in post-quantum regime and the exact physical principle precisely distinguishing physically realizable correlations from nonphysical ones.
We consider quantum ensembles which are determined by pre and post selection. Unlike the case of only pre selected ensembles, it is shown that in this case the probabilities for measurements outcomes at intermediate times respect causality only rarely: disconnected regions can generally use such ensembles to signal each other. We show that under restrictive conditions, there are certain non-trivial bi-partite ensembles which do respect causality. Some of these ensembles give rise to a violation of the CHSH inequality which exceeds the quantum maximal violation given by Cirel'son's bound (BrmCHSHle2sqrt2B_{\\rm CHSH}\\le 2\\sqrt2BrmCHSH​le2sqrt2). We find that one of these causality respecting cases saturates the Popescu-Rohrlich bound for the maximal violation BrmCHSHle4B_{\\rm CHSH}\\le 4BrmCHSH​le4. This case can be regarded as an a posteriori realization of ``super-correlations'' which has been recently termed as ``Popescu-Rohrlich box''.
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.
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.
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.