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arXiv:0812.0758 · New Journal of Physics

Couplers for non-locality swapping

Paul Skrzypczyk, Nicolas Brunner

Published Jul 3, 2009• 16 citations
arXivHTMLPDF

Abstract

Studying generalized non-signaling theories brings insight into the foundations of quantum mechanics. Here we focus on a dynamical process in such general theories, namely non-locality swapping, the analogue of quantum entanglement swapping. In order to implement such a protocol, one needs to define a coupler , which performs the equivalent of quantum joint measurements on generalized 'box-like' states. Establishing a connection to Bell inequalities, we define consistent couplers for theories containing an arbitrary amount of non-locality, which leads us to introduce the concepts of perfect and minimal couplers. Remarkably, Tsirelson's bound for quantum non-locality naturally appears in our study.

DOI
10.1088/1367-2630/11/7/073014

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Studying generalized non-signaling theories brings insight into the foundations of quantum mechanics. Here we focus on a dynamical process in such general theories, namely non-locality swapping, the analogue of quantum entanglement swapping. In order to implement such a protocol, one needs to define a coupler , which performs the equivalent of quantum joint measurements on generalized 'box-like' states. Establishing a connection to Bell inequalities, we define consistent couplers for theories containing an arbitrary amount of non-locality, which leads us to introduce the concepts of perfect and minimal couplers. Remarkably, Tsirelson's bound for quantum non-locality naturally appears in our study.
By studying generalized non-signalling theories, the hope is to find out what makes quantum mechanics so special. In the present paper, we revisit the paradigmatic model of non-signalling boxes and introduce the concept of a genuine box. This will allow us to present the first generalized non-signalling model featuring quantum-like dynamics. In particular, we present the coupler, a device enabling non-locality swapping, the analogue of quantum entanglement swapping, as well as teleportation. Remarkably, part of the boundary between quantum and post-quantum correlations emerges in our study.
By studying generalized nonsignaling theories, the hope is to find out what makes quantum mechanics so special. In the present Letter, we revisit the paradigmatic model of nonsignaling boxes and introduce the concept of a genuine box. This will allow us to present the first generalized nonsignaling model featuring quantumlike dynamics. In particular, we present the coupler, a device enabling nonlocality swapping, the analogue of quantum entanglement swapping, as well as teleportation. Remarkably, part of the boundary between quantum and post-quantum correlations emerges in our study.
We consider an analogue of entanglement-swapping for a set of black boxes with the most general non-local correlations consistent with relativity (including correlations which are stronger than any attainable in quantum theory). In an attempt to incorporate this phenomenon, we consider expanding the space of objects to include not only correlated boxes, but `couplers', which are an analogue for boxes of measurements with entangled eigenstates in quantum theory. Surprisingly, we find that no couplers exist for two binary-input/binary-output boxes, and hence that there is no analogue of entanglement-swapping for such boxes.
Nonlocality swapping of bipartite binary correlated boxes can be realized by a \emph{coupler} (χχχ) in nonsignaling models. By studying the swapping process we find that the previous bipartite coupler can be applied to the swapping of two multipartite boxes, and then generate a multipartite box with more users than that of any of the boxes before swapping. Here quantum bound still appears in the scheme. The bipartite coupler also can be applied to a hybrid scheme of generating a multipartite extremal box from many PR boxes. As the analogue of multipartite entanglement swapping, we generalize the nonlocality swapping of bipartite binary boxes to multipartite binary boxes by using a multipartite coupler χNχ_{N}χN​, and get the probability of success by connecting the coupler to the generalized Svetlichny inequality. The multipartite coupler acting on many multipartite boxes makes multipartite nonlocality swapping be a more efficient device to manipulate nonlocality between many users. The results show that Tsirelson's bound for quantum nonlocality emerges only when two of the nnn boxes involved in the coupler process are noisy ones.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
We consider an analog of entanglement-swapping for a set of black boxes with the most general nonlocal correlations consistent with relativity (including correlations which are stronger than any attainable in quantum theory). In an attempt to incorporate this phenomenon, we consider expanding the space of objects to include not only correlated boxes, but ``couplers,'' which are an analog for boxes of measurements with entangled eigenstates in quantum theory. Surprisingly, we find that no couplers exist for two binary-input--binary-output boxes, and hence that there is no analog of entanglement swapping for such boxes.