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arXiv:quant-ph/0603017 · AIP conference proceedings

Feats, Features and Failures of the PR-box

Valerio Scarani

Published Jan 1, 2006• 13 citations
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Abstract

One of the most intriguing features of quantum physics is the non‐locality of correlations that can be obtained by measuring entangled particles. Recently, it has been noticed that non‐locality can be studied without reference to the Hilbert space formalism. I review here the properties of the basic mathematical tool used for such studies, the so called Popescu‐Rohrlich‐box, in short PR‐box. Among its feats, are the simulation of the correlations of the singlet and of other non‐local probability distributions. Among its features, the “anomaly of non‐locality” and a great power for information‐theoretical tasks. Among its failures, the impossibility of reproducing all multi‐partite distributions and the triviality of the allowed dynamics.

DOI
10.1063/1.2219371

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One of the most intriguing features of quantum physics is the non‐locality of correlations that can be obtained by measuring entangled particles. Recently, it has been noticed that non‐locality can be studied without reference to the Hilbert space formalism. I review here the properties of the basic mathematical tool used for such studies, the so called Popescu‐Rohrlich‐box, in short PR‐box. Among its feats, are the simulation of the correlations of the singlet and of other non‐local probability distributions. Among its features, the “anomaly of non‐locality” and a great power for information‐theoretical tasks. Among its failures, the impossibility of reproducing all multi‐partite distributions and the triviality of the allowed dynamics.
One of the most intriguing features of quantum physics is the non-locality of correlations that can be obtained by measuring entangled particles. Recently, it has been noticed that non-locality can be studied without reference to the Hilbert space formalism. I review here the properties of the basic mathematical tool used for such studies, the so called Popescu-Rohrlich-box, in short PR-box. Among its feats, are the simulation of the correlations of the singlet and of other non-local probability distributions. Among its features, the "anomaly of non-locality" and a great power for information-theoretical tasks. Among its failures, the impossibility of reproducing all multi-partite distributions and the triviality of the allowed dynamics.
A set of nonlocal correlations that have come to be known as a Popescu-Rohrlich (PR) box suggest themselves as a natural unit of nonlocality, much as a singlet is a natural unit of entanglement. We present two results relevant to this idea. One is that a wide class of multipartite correlations can be simulated using local operations on PR boxes only. We show this with an explicit scheme, which has the interesting feature that the number of PR boxes required is related to the computational resources necessary to represent a function defining the multipartite box. The second result is that there are quantum multipartite correlations, arising from measurements on a cluster state, that cannot be simulated with n PR boxes, for any n.
The propensity for certain quantum system to consistently violate Bell inequalities—dubbed quantum nonlocality—has recently been found to be an information theoretic resource in quantum communication that is distinct from entanglement. Accordingly, a useful measurement of “how much” of this resource is present in a given system is needed. One candidate, the PR‐box (sometimes called a nonlocal machine), has been suggested as a possible benchmark, but one of the difficulties in using it has been to determine exactly what kinds of quantum systems are simulable by it. The present work is a first step towards a graph‐theoretic understanding of PR‐boxes that may prove useful in this endeavor.
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''.
One of the most fascinating consequences of quantum theory is non‐locality, i.e., the fact that the measurement outcomes of (spatially separated) parts of a system can have a correlation unexplainable by shared classical information. Note that at the same time, these correlations are non‐signaling and do not allow for message transmission. Popescu and Rohrlich have defined a non‐local box as a “basic building block of non‐locality” and initiated a systematic study of non‐local correlations and their applications. However, they left open the question of whether any non‐signaling correlation can be simulated by such non‐local boxes. We show that the answer is yes with respect to arbitrarily accurate approximations.
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.