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arXiv:quant-ph/0601122 · Physical Review A

Quantum-mechanical realization of a Popescu-Rohrlich box

S. Marcovitch, Benni Reznik, Lev Vaidman A

Published Feb 8, 2007• 73 citations
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Abstract

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.

DOI
10.1103/physreva.75.022102

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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 BCHSH\ensuremath≤22{B}_{\mathrm{CHSH}}\ensuremath{\le}2\sqrt{2}BCHSH​\ensuremath≤22​ and obtains the Popescu-Rohrlich bound for the maximal violation, BCHSH\ensuremath≤4{B}_{\mathrm{CHSH}}\ensuremath{\le}4BCHSH​\ensuremath≤4. This may be regarded as an a posteriori realization of supercorrelations, which have recently been termed Popescu-Rohrlich boxes.
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 consider a temporal version of the Clauser–Horne–Shimony–Holt (CHSH) scenario using projective measurements on a single quantum system. It is known that quantum correlations in this scenario are fundamentally more general than correlations obtainable with the assumptions of macroscopic realism and non-invasive measurements. In this paper, we also educe some fundamental limitations of these quantum correlations. One result is that a set of correlators can appear in the temporal CHSH scenario if and only if it can appear in the usual spatial CHSH scenario. In particular, we derive the validity of the Tsirelson bound and the impossibility of Popescu–Rohrlich (PR)-box behavior. The strength of possible signaling also turns out to be surprisingly limited, giving a maximal communication capacity of approximately 0.32 bit. We also find a temporal version of Hardy's non-locality paradox with a maximal quantum value of 1/4.
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.
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.
The key feature in correlations established by multi-party quantum entangled states is nonlocality. A quantity to measure the average nonlocality, distinguishing it from shared randomness and in a direct relation with no-signaling stochastic processes (which provide an operational interpretation of quantum correlations, without involving information transmission between the parties as to sustain causality), is proposed and resolved exhaustively for the quantum correlations established by a Clauser-Horne-Shimony-Holt setup (or CHSH box). The amount of nonlocality that is available in a CHSH box is measured by its proximity to the nearest Popescu-Rohrlich set of causal stochastic processes (aka a PR box) in the no-signaling polytope, related by polyhedral duality to Bell's correlation function. The proposed amount of average nonlocality is an entanglement monotone with a simple relation to concurrence. We provide the optimal setup vectors of a maximally nonlocal CHSH box for any entangled pair. The strongest nonlocality is the fraction [Formula: see text] of a PR box, attained by maximally entangled qubit pairs. The most economical causal stochastic process reproducing any maximally nonlocal CHSH box is developed. Data produced by a computer implementation of the simulator agrees with the quantum mechanical formulas.
Quantum probabilities differ from classical ones in many ways, e.g. by violating the well-known Bell and Clauser–Horne–Shimony–Holt inequalities or another simple inequality due to R Wright. The latter has recently regained attention because of its equivalence to a novel noncontextual inequality by Klyachko et al . On the other hand, quantum probabilities still obey many limitations which need not hold in more general probabilistic theories (super quantum probabilities). Wright, Popescu and Rohrlich identified states which are included in such theories, but impossible in quantum mechanics, and they showed this using the Hilbert space formalism. Recently, Fritz et al and Cabello detected that the impossibility of these states can be derived from very general principles (local orthogonality and global exclusive disjunction, respectively) without using Hilbert space techniques. In the paper, an alternative derivation from rather different physical principles will be presented. These are a reasonable calculus of conditional probability (i.e. a model for the quantum measurement process) and the absence of third-order interference. The concept of third-order interference was introduced by Sorkin, who also recognized its impossibility in quantum mechanics.