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New Journal of Physics

Violation of Leggett inequalities in orbital angular momentum subspaces

Jacquiline Romero, Jonathan Leach, Barry Jack, Stephen Mark Barnett, Miles J. Padgett, Sonja Franke‐Arnold

Published Dec 7, 2010• 42 citations
Publisher

Abstract

We report an experimental test of Leggett's non-local hidden variable theory in an orbital angular momentum (OAM) state space of light. We show that the correlations we observe are in conflict with Leggett's model, thus excluding a particular class of non-local hidden variable theories for the first time in a non-polarization state space. It is known that the violation of the Leggett inequality becomes stronger as more detection settings are used. The required measurements become feasible in an OAM subspace, and we demonstrate this by testing the inequality using three and four settings. We observe excellent agreement with quantum predictions and a violation of five and six standard deviations, respectively, compared to Leggett's non-local hidden variable theory.

DOI
10.1088/1367-2630/12/12/123007

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We report an experimental test of Leggett's non-local hidden variable theory in an orbital angular momentum (OAM) state space of light. We show that the correlations we observe are in conflict with Leggett's model, thus excluding a particular class of non-local hidden variable theories for the first time in a non-polarization state space. It is known that the violation of the Leggett inequality becomes stronger as more detection settings are used. The required measurements become feasible in an OAM subspace, and we demonstrate this by testing the inequality using three and four settings. We observe excellent agreement with quantum predictions and a violation of five and six standard deviations, respectively, compared to Leggett's non-local hidden variable theory.
Leggett inequality states that nonlocal hidden-variable models might still be incompatible with the predictions of quantum physics. However, its theoretical and experimental demonstration is only in the scenario of 2-dimensional maximally entangled systems. An open question remains as to whether the Leggett inequality can be violated by nonmaximally entangled states. Here, we answer this question both in theory and experiment. Specifically, from the point of view of geometry, we theoretically map the problem of maximizing the correlation measure in the Leggett inequality to maximizing the sum of an ellipse’s diameter and semi-diameter axes, accordingly, demonstrating that the violation of the Leggett inequality requires a more robust entanglement than that of Bell’s theory. Experimentally, by leveraging the controllable photonic orbital angular momentum entanglement, we demonstrate the violation of Leggett-type inequalities by more than 8.7 and 4.5 standard deviations under concurrence C = 0.95 and 0.9, respectively. Our observations indicate that, the requirement for quantum correlation should be increased to exclude a particular class of non-local hidden variable theories that abide by Leggett’s model, providing insights into the boundaries of quantum correlation and the limitations imposed by non-local hidden variables.
We report the experimental violation of Leggett-type inequalities for a hybrid entangled state of spin and orbital angular momentum of a single photon. These inequalities give a physical criterion to verify the possible validity of a class of hidden-variable theories, originally named ``crypto nonlocal,'' that are not excluded by the violation of Bell-type inequalities. In our case, the tested theories assume the existence of hidden variables associated with independent degrees of freedom of the same particle, while admitting the possibility of an influence between the two measurements, i.e., the so-called contextuality of observables. We observe a violation of the Leggett inequalities for a range of experimental inputs, with a maximum violation of seven standard deviations, thus ruling out this class of hidden-variable models with a high level of confidence.
I impose previously neglected symmetry conditions in Leggett's non-local hidden variable theory that are required by the conditions in a recent experimental test of this theory. These conditions lead to an inequality for the polarization correlation function that has a maximal violation from the quantum predictions that is 2.5 times larger than reported in this article. Furthermore, when these symmetry conditions are applicable, Leggett's non-local theory also cannot model the observed quantum correlations in previous experiments.
Bell's theorem states that no local hidden-variable model is compatible with quantum mechanics. Surprisingly, even if we release the locality constraint, certain nonlocal hidden-variable models, such as the one proposed by Leggett, may still be at variance with the predictions of quantum physics. Here, we report an experimental test of Leggett's nonlocal model with solid-state spins in a diamond nitrogen-vacancy center. We entangle an electron spin with a surrounding weakly coupled 13C^{13}\mathrm{C}13C nuclear spin and observe that the entangled states violate Leggett-type inequalities by more than four and seven standard deviations for six and eight measurement settings, respectively. Our experimental results are in full agreement with quantum predictions and violate Leggett's nonlocal hidden-variable inequality with a high level of confidence.
Entanglement in higher dimensions is an attractive concept that is a chal- lenge to realise experimentally. To this end, the entanglement of the orbital angular momentum (OAM) of photons holds promise. The OAM state-space is discrete and theoretically unbounded. In the work that follows, we investi- gate various aspects of OAM entanglement. We show how the correlations in OAM and its conjugate variable, angular position, are determined by phase- matching and the shape of the pump beam in spontaneous parametric down- conversion. We implement tests of quantum mechanics which have been previously done for other variables. We show the Einstein-Podolsky-Rosen paradox for OAM and angle, supporting the incompatibility of quantum me- chanics with locality and realism. We demonstrate violations of Bell-type inequalities, thereby discounting local hidden variables for describing the correlations we observe. We show the Hardy paradox using OAM, again highlighting the nonlocal nature of quantum mechanics. We demonstrate violations of Leggett-type inequalities, thereby discounting nonlocal hidden variables for describing correlations. Lastly, we have looked into the entan- glement of topological vortex structures formed from a special superposition of OAM modes and show violations of Bell-type inequalities confined to a finite, isolated volume.
We analyze the class of nonlocal realistic theories that was originally considered by Leggett [Found. Phys. 33, 1469 (2003)10.1023/A:1026096313729] and tested by us in a recent experiment [Nature (London) 446, 871 (2007)10.1038/nature05677]. We derive an incompatibility theorem that works for finite numbers of polarizer settings and that does not require the previously assumed rotational symmetry of the two-particle correlation functions. The experimentally measured case involves seven different measurement settings. Using polarization-entangled photon pairs, we exclude this broader class of nonlocal realistic models by experimentally violating a new Leggett-type inequality by 80 standard deviations.