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Physical Review A

Microscopic analogs of the Greenberger-Horne-Zeilinger experiment

Seth Lloyd

Published Mar 1, 1998• 62 citations
Publisher

Abstract

This paper shows how analogs of the Greenberger-Horne-Zeilinger experiment, which demonstrates the impossibility of certain types of local hidden-variable theories in quantum mechanics, can be performed using nuclear magnetic resonance on spins in molecules at finite temperature. Because of their microscopic nature, these experiments cannot rule out the presence of nonstandard interactions that could cause classical variables to imitate quantum variables. Instead, they are designed to reveal the peculiarly quantum-mechanical correlations on which the Greenberger-Horne-Zeilinger effect is based.

DOI
10.1103/physreva.57.r1473

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This paper shows how analogs of the Greenberger-Horne-Zeilinger experiment, which demonstrates the impossibility of certain types of local hidden-variable theories in quantum mechanics, can be performed using nuclear magnetic resonance on spins in molecules at finite temperature. Because of their microscopic nature, these experiments cannot rule out the presence of nonstandard interactions that could cause classical variables to imitate quantum variables. Instead, they are designed to reveal the peculiarly quantum-mechanical correlations on which the Greenberger-Horne-Zeilinger effect is based.
The Greenberger-Horne-Zeilinger (GHZ) effect provides an example of quantum correlations that cannot be explained by classical local hidden variables. This paper reports on the experimental realization of GHZ correlations using nuclear magnetic resonance (NMR). The NMR experiment differs from the originally proposed GHZ experiment in several ways: it is performed on mixed states rather than pure states, and instead of being widely separated, the spins on which it is performed are all located in the same molecule. As a result, the NMR version of the GHZ experiment cannot entirely rule out classical local hidden variables. It nonetheless provides an unambiguous demonstration of the ``paradoxical'' GHZ correlations, and confirms the predictions of quantum mechanics.
A scheme is proposed to implement microscopic analogs of the Greenberger-Horne-Zeilinger experiment, as described by Lloyd, on a nuclear magnetic resonance (NMR) quantum computer. This scheme includes how to prepare the effective pure spin state |0000〉 (a state with all the spins pointing along the direction of the magnetic field) from a state at thermal equilibrium, how to transform it into states of the form, e.g., $(|000〉\ensuremath{-}|111〉)|0〉$ (which comprises the first three spins in an entangled state and the fourth register-spin), and how to measure the eigenvalue of the operator product like ${\ensuremath{\sigma}}_{x}^{1}{\ensuremath{\sigma}}_{y}^{2}{\ensuremath{\sigma}}_{y}^{3}$ for such three-spin entangled states, where ${\ensuremath{\sigma}}^{i}$ is the Pauli matrix for the ithi\mathrm{th}ith spin, and store the result on the fourth spin. Also proposed is a general method to implement the controlled-not gate, an essential gate for any quantum algorithm, on n-qubit NMR quantum computers with n> 4.n>~4.n> 4.
In all local realistic theories worked out till now, locality is considered as a basic assumption. Most people in the field consider the inconsistency between local realistic theories and quantum mechanics to be a result of non-local nature of quantum mechanics. In this Paper, we derive the Greenberger-Horne-Zeilinger type theorem for particles with instantaneous (non-local) interactions at the hidden-variable level. Then, we show that the previous contradiction still exists between quantum mechanics and non-local hidden variable models.