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arXiv:1409.7290

Entropic Version of the Greenberger-Horne-Zeilinger Paradox

Sadegh Raeisi, Paweł Kurzyński, Dagomir Kaszlikowski

Published Sep 25, 2014• 1 citation
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Abstract

Consider four binary +-1 variables A, B, C and D for which classical reasoning implies ABCD = 1. In this case the knowledge of A, B, C automatically provides knowledge of D because D = ABC. However, the Greenberger-Horne-Zeilinger paradox shows that despite classical prediction one can find quantum states and observables with well defined outcomes for which D = -ABC. In this work we formulate an information-theoretic version of this paradox. We show that for a tripartite quantum system one can find a set of four properties for which classical reasoning implies that D = ABC, yet quantum theory predicts that one can know everything about A, B, C and nothing about D.

DOI
10.48550/arxiv.1409.7290

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Consider four binary +-1 variables A, B, C and D for which classical reasoning implies ABCD = 1. In this case the knowledge of A, B, C automatically provides knowledge of D because D = ABC. However, the Greenberger-Horne-Zeilinger paradox shows that despite classical prediction one can find quantum states and observables with well defined outcomes for which D = -ABC. In this work we formulate an information-theoretic version of this paradox. We show that for a tripartite quantum system one can find a set of four properties for which classical reasoning implies that D = ABC, yet quantum theory predicts that one can know everything about A, B, C and nothing about D.
The Greenberger-Horne-Zeilinger (GHZ) version of the Einstein-Podolsky-Rosen (EPR) paradox is widely regarded as a conclusive logical argument that rules out the possibility of describing quantum phenomena within the framework of any model of local hidden variables in which the observers are free to choose their own experimental settings. In this paper, we show, however, that the GHZ argument implicitly relies on a subtle, though crucial, assumption that is not required by fundamental physical principles, and which had gone unnoticed. Namely, we note that the argument implicitly assumes the existence of an absolute angular frame of reference with respect to which the polarization properties of the hypothetical hidden configurations of the entangled particles, as well as the orientations of the measurement devices that test them, can be defined. We further note that such an absolute frame of reference may not exist if the hidden configurations of the entangled particles spontaneously break the gauge rotational symmetry. Indeed, by skipping the disputed requirement we are able to build a model of local hidden variables for the GHZ state, which complies with the 'free-will' hypothesis and reproduces the quantum mechanical predictions. Our model, thus, completes-in the EPR sense-the quantum description of the GHZ state.
The Greenberger-Horne-Zeilinger~(GHZ) version of the Einstein-Podolsky-Rosen~(EPR) paradox is widely regarded as a conclusive logical argument that rules out the possibility of describing quantum phenomena within the framework of a local and realistic model of hidden variables in which the observers are free to choose their own experimental settings. In this paper we show, however, that the GHZ argument implicitly relies on an additional crucial assumption, which is not required by fundamental physical principles and had gone unnoticed. Namely, we note that the argument implicitly assumes the existence of an absolute angular frame of reference with respect to which the polarization properties of the hypothetical hidden configurations of the entangled particles as well as the orientation of the measurement apparatus that test the system can be defined. We further note that such an absolute frame of reference would not exist if the hidden configurations of the entangled particles spontaneously break the gauge rotational symmetry. Indeed, by skipping this unnecessary additional assumption we are able to build an explicitly local and realistic model of hidden variables for the GHZ state, which complies with the 'free-will' hypothesis and reproduces the quantum mechanical predictions, and thus completes the description of the system in the EPR sense.
The Greenberger-Horne-Zeilinger (GHZ) paradox provides an all-versus-nothing test for the quantum nonlocality. In most of the GHZ paradoxes known so far each observer is allowed to measure only two alternative observables. Here we present a general construction for GHZ paradoxes in which each observer measures more than two observables given that the system is prepared in the nnn-qudit GHZ state. By doing so we are able to construct a multisetting GHZ paradox for the nnn-qubit GHZ state, with nnn being arbitrary, which is genuine nnn-partite; i.e., no GHZ paradox exists when restricted to a subset of a number of observers for a given set of Mermin observables. Our result fills up the gap of the absence of a genuine GHZ paradox for the GHZ state of an even number of qubits, especially the four-qubit GHZ state as used in GHZ's original proposal.
The paradox of Greenberger-Horne-Zeilinger (GHZ) disproves directly the concept of EPR elements of reality, based on the EPR correlations, in an all-versus-nothing way. A three-qubit experimental demonstration of the GHZ paradox was achieved nearly 20 years ago, followed by demonstrations for more qubits. Still, the GHZ contradictions underlying the tests can be reduced to a three-qubit one. We show an irreducible four-qubit GHZ paradox, and report its experimental demonstration. The bound of a three-setting-per-party Bell-GHZ inequality is violated by $7\ensuremath{\sigma}$. The fidelity of the GHZ state was around 8181%81, and an entanglement witness reveals a violation of the separability threshold by $19\ensuremath{\sigma}$.