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Physical Review A

Violation of Bell’s inequality with continuous spatial variables

Ayman F. Abouraddy, Timothy M. Yarnall, Bahaa E. A. Saleh, Malvin Carl Teich

Published May 30, 2007• 58 citations
Publisher

Abstract

The Einstein-Podolsky-Rosen (EPR) argument revealed the paradoxical properties of a two-particle system entangled continuously in the spatial parameter. Yet a direct test of quantum nonlocality exhibited by this state, via a violation of Bell's inequality, has not been forthcoming. In this paper, we identify and construct experimental arrangements comprising simple optical components, without nonlinearities or moving parts, that implement operators in the spatial-parity space of single-photon fields that correspond to the familiar Pauli spin operators. We achieve this by first establishing an isomorphism between the single-mode multiphoton electromagnetic-field space spanned by a Fock-state basis and the single-photon multimode electromagnetic-field space spanned by a spatial-eigenmode basis. We then proceed to construct a Hilbert space with a two-dimensional basis of spatial even-odd parity modes. In particular, we describe an arrangement that implements a rotation in the parity space of each photon of an entangled-photon pair, allowing for a straightforward experimental test of Bell's inequality using the EPR state. Finally, the violation of a Bell inequality is quantified in terms of the physical parameters of the two-photon source.

DOI
10.1103/physreva.75.052114

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The Einstein-Podolsky-Rosen (EPR) argument revealed the paradoxical properties of a two-particle system entangled continuously in the spatial parameter. Yet a direct test of quantum nonlocality exhibited by this state, via a violation of Bell's inequality, has not been forthcoming. In this paper, we identify and construct experimental arrangements comprising simple optical components, without nonlinearities or moving parts, that implement operators in the spatial-parity space of single-photon fields that correspond to the familiar Pauli spin operators. We achieve this by first establishing an isomorphism between the single-mode multiphoton electromagnetic-field space spanned by a Fock-state basis and the single-photon multimode electromagnetic-field space spanned by a spatial-eigenmode basis. We then proceed to construct a Hilbert space with a two-dimensional basis of spatial even-odd parity modes. In particular, we describe an arrangement that implements a rotation in the parity space of each photon of an entangled-photon pair, allowing for a straightforward experimental test of Bell's inequality using the EPR state. Finally, the violation of a Bell inequality is quantified in terms of the physical parameters of the two-photon source.
We report the first experimental violation of Bell's inequality in the spatial domain using the Einstein-Podolsky-Rosen state. Two-photon states generated via optical spontaneous parametric down-conversion are shown to be entangled in the parity of their one-dimensional transverse spatial profile. Superpositions of Bell states are prepared by manipulation of the optical pump's transverse spatial parity-a classical parameter. The Bell-operator measurements are made possible by devising simple optical arrangements that perform rotations in the one-dimensional spatial-parity space of each photon of an entangled pair and projective measurements onto a basis of even-odd functions. A Bell-operator value of 2.389+/-0.016 is recorded, a violation of the inequality by more than 24 standard deviations.
We discuss an approach to encoding qubits in the spatial distribution of single photons and entangled-photon pairs based on utilizing the spatial parity of the photon wave fronts. Using simple optical components, that do not require nonlinearities or moving parts, we discuss implementing rotations in the Hilbert space of spatial parity and measurement of the state of parity. Using entangled photon pairs, we use spatial parity to demonstrate quantum nonlocality by violating a Bell inequality using an EPR state. We also discuss generalizations of this scheme that may allow for a larger number of qubits to be encoded per photon.
A generalization of the 1935 Einstein-Podolsky-Rosen (EPR) argument for measurements with continuous variable outcomes is presented to establish criteria for the demonstration of the EPR paradox, for situations where the correlation between spatially separated subsystems is not perfect. Two types of criteria for EPR correlations are determined. The first type are based on measurements of the variances of conditional probability distributions and are necessary to reflect directly the situation of the original EPR paradox. The second weaker set of EPR criteria are based on the proven failure of (Bell-type) local realistic theories which could be consistent with a local quantum description for each subsystem. The relationship with criteria sufficient to prove entanglement is established, to show that any demonstration of EPR correlations will also signify entanglement. It is also shown how a demonstration of entanglement between two spatially separated subsystems, if not able to interpreted as a violation of a Bell-type inequality, may be interpreted as a demonstration of the EPR correlations. In particular it is explained how the experimental observation of two-mode squeezing using spatially separated detectors will signify not only entanglement but EPR correlations defined in a general sense.
We discuss violation of Bell inequalities by the regularized Einstein-Podolsky-Rosen (EPR) state, which can be produced in a quantum optical parametric down-conversion process. We propose an experimental photodetection scheme to probe nonlocal quantum correlations exhibited by this state. Furthermore, we show that the correlation functions measured in two versions of the experiment are given directly by the Wigner function and the Q function of the EPR state. Thus, the measurement of these two quasidistribution functions yields a novel scheme for testing quantum nonlocality.
We propose an EPR inequality based on an entropic uncertainty relation for complementary continuous variable observables. This inequality is more sensitive than the previously established EPR inequality based on inferred variances, and opens up the possibility of EPR tests of quantum nonlocality in a wider variety of quantum states. We experimentally test the inequality using spatially entangled photons. For a particular quantum state, our experimental results show a violation of the entropic EPR inequality, while the variance EPR inequality is not violated.