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Microwave and Optical Technology Letters

Photon‐pair states and violation of CHSH inequality

Luís P. Martins, Álvaro J. Almeida, Paulo Sérgio de Brito André, Armando N. Pinto

Published Aug 24, 2012• 2 citations

Abstract

Abstract In addition to Bell's original inequality, John Clauser, Michael Horne, Abner Shimony, and Richard Holt (CHSH) developed a special form of Bell's inequality which gives classical limits to the expected correlation between two particles. We present a theoretical demonstration of the maximum value for violation of CHSH inequality, S = 2 $\sqrt{2}$ , for the pure entangled quantum state. Using the spontaneous four‐wave mixing process, we generate polarization‐entangled photon pairs in a highly nonlinear fiber (HNLF) loop, for experimental verification. Using a nonentangled quantum state, we show theoretically that the violation of CHSH inequality is only a property of entangled states. We also show the theoretical solution of violation of the CHSH inequality for a general Greenberger–Horne–Zeilinger state and for an arbitrary state of two particles. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:2454–2461, 2012; View this article online at wileyonlinelibrary. com. DOI 10.1002/mop.27127

DOI: 10.1002/mop.27127

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Abstract In addition to Bell's original inequality, John Clauser, Michael Horne, Abner Shimony, and Richard Holt (CHSH) developed a special form of Bell's inequality which gives classical limits to the expected correlation between two particles. We present a theoretical demonstration of the maximum value for violation of CHSH inequality, S = 2

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, for the pure entangled quantum state. Using the spontaneous four‐wave mixing process, we generate polarization‐entangled photon pairs in a highly nonlinear fiber (HNLF) loop, for experimental verification. Using a nonentangled quantum state, we show theoretically that the violation of CHSH inequality is only a property of entangled states. We also show the theoretical solution of violation of the CHSH inequality for a general Greenberger–Horne–Zeilinger state and for an arbitrary state of two particles. © 2012 Wiley Periodicals, Inc. Microwave Opt Technol Lett 54:2454–2461, 2012; View this article online at wileyonlinelibrary. com. DOI 10.1002/mop.27127

A violation of Bell's inequalities using quantum entanglement indicates the nonlocality of two particles such as photons and electrons. We observe a violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, a particular Bell's inequality, in polarization-entangled photon pairs in the 1550 nm band. For photon pair generation based on spontaneous parametric down conversion, a fiber-optic two-photon interferometer with two periodically poled lithium niobate waveguides is adopted. The obtained S value of the CHSH inequality is 2.44 ±0.02 without subtraction of accidental coincidences, yielding a violation by ∼22 standard deviations in an hour. Two-photon interference with a visibility of 0.885 is also observed. If those coincidences are subtracted, the visibility becomes almost unity.

Linearly polarized projections are the tacit means for performing Clauser–Horne–Shimony–Holt (CHSH) Bell-inequality tests using polarization-entangled photon pairs. The inequality is valid for all states on the Poincaré–Bloch sphere, but few laboratory studies have investigated violations with the full sphere. In this article, we explore the experimental verifications of the predicted violations of the CHSH inequality with Bell and non-Bell states with same and different linear and elliptically polarized basis states for each photon. We find that Bell states violate CHSH when using the same basis for both photons, regardless of their ellipticity, whereas they show no violations for photon projections in different bases. We found non-Bell maximally entangled states for which the converse is true.

Linearly polarized projections are the tacit means for performing Clauser-Horne-Shimony-Holt (CHSH) Bell-inequality tests using polarization-entangled photon pairs. The inequality is valid for all states on the Poincaré-Bloch sphere, but few laboratory studies have investigated violations with the full sphere. In this article, we explore the experimental verifications of the predicted violations of the CHSH inequality with Bell and non-Bell states with same and different linear and elliptically polarized basis states for each photon. We find that Bell states violate CHSH when using the same basis for both photons, regardless of their ellipticity, whereas they show no violations for photon projections in different bases. We found non-Bell maximally-entangled states for which the converse is true.

Witnessing continuous-variable Bell nonlocality is a challenging endeavor, but Bell himself showed how one might demonstrate this nonlocality. Although Bell nearly showed a violation using the Clauser-Horne-Shimony-Holt (CHSH) inequality with sign-binned position-momentum statistics of entangled pairs of particles measured at different times, his demonstration is subject to approximations not realizable in a laboratory setting. Moreover, he does not give a quantitative estimation of the maximum achievable violation for the wave function he considers. In this article, we show how his strategy can be reimagined using the transverse positions and momenta of entangled photon pairs measured at different propagation distances, and we find that the maximum achievable violation for the state he considers is actually very small relative to the upper limit of

2\sqrt{2}

. Although Bell's wave function does not produce a large violation of the CHSH inequality, other states may yet do so.

We construct a Bell inequality from the Clauser-Horne-Shimony-Holt inequality for two qubits that provides a stronger bound on the correlations of entangled states than allowed by the CHSH inequality. The argument involved here can be generalized to

n

qubits. The inequalities obtained are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits.

Cirel'son inequality states that the absolute value of the combination of quantum correlations appearing in the Clauser-Horne-Shimony-Holt (CHSH) inequality is bound by 2 square root of (2). It is shown that the correlations of two qubits belonging to a three-qubit system can violate the CHSH inequality beyond 2 square root of (2). Such a violation is not in conflict with Cirel'son's inequality because it is based on postselected systems. The maximum allowed violation of the CHSH inequality, 4, can be achieved using a Greenberger-Horne-Zeilinger state.