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arXiv:0911.0770 · New Journal of Physics

Extreme nonlocality with one photon

Libby Heaney, Adán Cabello, Marcelo França Santos, Vlatko Vedral

Published May 27, 2011• 82 citations
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Abstract

Quantum nonlocality is typically assigned to systems of two or more well-separated particles, but nonlocality can also exist in systems consisting of just a single particle when one considers the subsystems to be distant spatial field modes. Single particle nonlocality has been confirmed experimentally via a bipartite Bell inequality. In this paper, we introduce an N -party Hardy-like proof of the impossibility of local elements of reality and a Bell inequality for local realistic theories in the case of a single particle superposed symmetrically over N spatial field modes (i.e. N qubit W state). We show that, in the limit of large N , the Hardy-like proof effectively becomes an all-versus-nothing (or Greenberger–Horne–Zeilinger (GHZ)-like) proof, and the quantum-classical gap of the Bell inequality tends to be the same as that in a three-particle GHZ experiment. We describe how to test the nonlocality in realistic systems.

DOI
10.1088/1367-2630/13/5/053054

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Quantum nonlocality is typically assigned to systems of two or more well-separated particles, but nonlocality can also exist in systems consisting of just a single particle when one considers the subsystems to be distant spatial field modes. Single particle nonlocality has been confirmed experimentally via a bipartite Bell inequality. In this paper, we introduce an N -party Hardy-like proof of the impossibility of local elements of reality and a Bell inequality for local realistic theories in the case of a single particle superposed symmetrically over N spatial field modes (i.e. N qubit W state). We show that, in the limit of large N , the Hardy-like proof effectively becomes an all-versus-nothing (or Greenberger–Horne–Zeilinger (GHZ)-like) proof, and the quantum-classical gap of the Bell inequality tends to be the same as that in a three-particle GHZ experiment. We describe how to test the nonlocality in realistic systems.
A recent paper [L. Heaney, A. Cabello, M. F. Santos, and V. Vedral, New J. Phys. 13, 053054 (2011)] revealed that a single quantum symmetrically delocalized over NNN modes, namely a WWW state, effectively allows for all-versus-nothing proofs of nonlocality in the limit of large NNN. Ideally, this finding opens up the possibility of using the robustness of the WWW states while realizing the nonlocal behavior previously thought to be exclusive to the more complex class of Greenberger-Horne-Zeilinger states. We show that in practice, however, the slightest decoherence or inefficiency of the Bell measurements on WWW states will degrade any violation margin gained by scaling to higher NNN. The nonstatistical demonstration of nonlocality is thus proved to be impossible in any realistic experiment.
This thesis addresses some of the consequences of giving quantum mechanics a realist interpretation. We consider three main topics: wave-particle duality, locality, and Lorentz invariance. First we show that classical particles alone or classical waves alone cannot explain all single particle quantum phenomena. Then we consider the possibility that a quantum particle is composed of a particle and a wave, both being taken to exist objectively. We are able to demonstrate the reality of empty waves (that is, waves without a particle) if we make three realist motivated assumptions. The main part of this thesis concerns locality. In 1964 Bell demonstrated that a local realist interpretation of quantum mechanics is not possible by deriving a set of inequalities that apply to two particle systems. More recently Greenberger, Home, and Zeilinger have demonstrated this for systems with more than two particles without the need for inequalities. We present a new way to derive Bell inequalities for two particles and show how this can be extended to systems with more than two particles. A number of proposals for experiments to test local realism are put forward. In particular, we show how it is possible to demonstrate the nonlocality of a single photon. A new demonstration of Bell's theorem is presented for two particles but without inequalities. A realizable quantum optical version is proposed and inequalities are proposed which would be required in a non-ideal experiment. Finally, the question of Lorentz invariance is considered. We define a condition for the existence of elements of reality and a condition for the Lorentz invariance of these elements of reality. Then we show that, by considering a particular gedanken experiment, we obtain a contradiction demonstrating that Lorentz-invariant realistic interpretations of quantum theory are not possible.
We show, using a Bell inequality argument, that quantum correlations cannot be regarded as local elements of reality. It is well known since the work of J.S. Bell, that one cannot have locally preexistent values for all physical quantities, whether they are deterministic or stochastic. The Bell inequality argument we present here shows this is also impossible for correlations among subsystems of an individual isolated composite system. Thus there is no philosophical difference with respect to questions of locality and realism between correlations on the one hand and values of quantities on the other hand that could allow one to have a different view about their ontological status. Neither of them can be used to build up a world consisting of some local realistic structure. This argument has an important advantage over others who use so-called Greenberger-Horne-Zeilinger and Hardy type arguments because it does not need perfect correlations but only statistical correlations. It can therefore easily be tested in currently feasible experiments using four particle entanglement.
The proofs of quantum nonlocality due to Greenberger, Horne and Zeilinger and due to Hardy are qualitatively different from that of Bell insofar as they rely only on a consideration of whether events are possible or impossible, rather than relying on specific experimental probabilities. We consider the scenario of a bipartite nonlocality experiment, in which two separated experimenters each have access to some measurements they can perform on a system. In a physical theory, some outcomes of this experiment will be labelled possible, others impossible, and an assignment of the values 0 (impossible) and 1 (possible) to these different outcomes forms a table of possibilities. Here, we consider the computational task of determining whether or not a given table of possibilities constitutes a departure from possibilistic local realism. By considering the case in which one party has access to measurements with two outcomes and the other three, it is possible to see at exactly which point this task becomes computationally difficult.
Entanglement, the Einstein-Podolsky-Rosen (EPR) paradox and Bell’s failure of local-hiddenvariable (LHV) theories are three historically famous forms of “quantum nonlocality”. We give experimental criteria for these three forms of nonlocality in multi-particle systems, with the aim of better understanding the transition from microscopic to macroscopic nonlocality. We examine the nonlocality of N separated spin J systems. First, we obtain multipartite Bell inequalities that address the correlation between spin values measured at each site, and then we review spin squeezing inequalities that address the degree of reduction in the variance of collective spins. The latter have been particularly useful as a tool for investigating entanglement in Bose-Einstein condensates (BEC). We present solutions for two topical quantum states: multi-qubit Greenberger-Horne-Zeilinger (GHZ) states, and the ground state of a two-well BEC.
Bell showed 50 years ago that quantum theory is nonlocal via his celebrated inequalities, turning the issue of quantum nonlocality from a matter of taste into a matter of test. Years later, Hardy proposed a test for nonlocality without inequality, which is a kind of "something-versus-nothing" argument. Hardy's test for nnn particles induces an nnn-partite Bell's inequality with two dichotomic local measurements for each observer, which has been shown to be violated by all entangled pure states. Our first result is to show that the Bell-Hardy inequality arising form Hardy's nonlocality test is tight for an arbitrary number of parties, i.e., it defines a facet of the Bell polytope in the given scenario. On the other hand quantum theory is not that nonlocal since it forbids signaling and even not as nonlocal as allowed by non-signaling conditions, i.e., quantum mechanical predictions form a strict subset of the so called non-signaling polytope. In the scenario of each observer measuring two dichotomic observables, Fritz established a duality between the Bell polytope and the non-signaling polytope: tight Bell's inequalities, the facets of the Bell polytope, are in a one-to-one correspondence with extremal non-signaling boxes, the vertices of the non-signaling polytope. Our second result is to provide an alternative and more direct formula for this duality. As an example, the tight Bell-Hardy inequality gives rise to an extremal non-signaling box that serves as a natural multipartite generalization of Popescu-Rohrlich box.