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arXiv:2008.08426 · Physical Review A

Continuous-variable Clauser-Horne Bell-type inequality: A tool to unearth the nonlocality of continuous-variable quantum-optical systems

Chandan Kumar, Gaurav Saxena, Arvind

Published Apr 28, 2021• 9 citations
arXivHTMLPDF

Abstract

We consider a continuous-variable Clauser-Horne Bell-type inequality to study nonlocality in four-mode continuous-variable systems, which goes beyond two-photon states and can be applied to mixed as well as to states with fluctuating photon number. We apply the inequality to a wide variety of states such as pure and mixed Gaussian states (including squeezed thermal states) and non-Gaussian states. We consider beam splitters as a model for leakage and show that the inequality is able to detect nonlocality of noisy Gaussian states as well. Finally, we investigate nonlocality in pair-coherent states and entangled coherent states, which are prominent examples of nonclassical, non-Gaussian states.

DOI
10.1103/physreva.103.042224

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We consider a continuous-variable Clauser-Horne Bell-type inequality to study nonlocality in four-mode continuous-variable systems, which goes beyond two-photon states and can be applied to mixed as well as to states with fluctuating photon number. We apply the inequality to a wide variety of states such as pure and mixed Gaussian states (including squeezed thermal states) and non-Gaussian states. We consider beam splitters as a model for leakage and show that the inequality is able to detect nonlocality of noisy Gaussian states as well. Finally, we investigate nonlocality in pair-coherent states and entangled coherent states, which are prominent examples of nonclassical, non-Gaussian states.
Local unitary operations cannot affect the quantum correlations between two systems sharing an entangled state although they do influence the outcomes of local measurements. By considering local squeezing operations we introduce an extended family of observables allowing violation of the Clauser-Horne-Shimony-Holt Bell inequality for two-mode Gaussian systems. We show that local squeezing can enable or enhance the identification of nonlocal two-mode states. In particular, we show that local squeezing followed by photon--no-photon discrimination can suffice to reveal nonlocality in a broad ensemble of pure and mixed two-mode Gaussian states.
Quantum networks enable forms of nonlocality beyond the standard Bell scenario, with a multitude of potential applications. Continuous-variable (CV) platforms are particularly attractive for large-scale networks, offering deterministic entanglement generation and favorable prospects for long-distance distribution. Here, we present a formalism to study CV network nonlocality using pseudospin measurements. Considering the linear chain and star configurations, we derive the maximal violations of the corresponding network locality inequalities for arbitrary two-mode states. Using two-mode squeezed vacuum states, we show that the strength of nonlocality in the star configuration remains independent of the network size. Moreover, the nonlocal correlations persist even at arbitrarily high temperatures provided the squeezing exceeds a critical threshold. Further, we demonstrate non-Gaussianity as an enhancer of network nonlocality through illustrations of various classes of non-Gaussian resources. Remarkably, a coherent superposition of single-photon subtractions across modes achieves maximal violation for arbitrarily small squeezing. Finally, we provide schematics of an experimentally feasible implementation of CV network nonlocality based on the isomorphism between pseudospin and spatial parity observables.
We present a phase-space formalism to evaluate Bell inequality violations in continuous variable systems. By doing so we can generalize previous analyses (which have dealt only with pure states) to arbitrary mixed states. We leverage these results to analyze the effect of temperature on violations of Bell inequalities in a two-mode squeezed thermal state, which can become useful in tests of local realism in the presence of thermal noise. We also explore the nonmonotonic relationship between the violations of Bell inequalities and the amount of entanglement present in this family of mixed states. Additionally, we discuss the optimal choices of pseudospin operators for states beyond the two-mode squeezed vacuum.
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Abstract Optical hybrid entanglement can be created between two qubits, one encoded in a single photon and another one in coherent states with opposite phases. It opens the path to a variety of quantum technologies, such as heterogeneous quantum networks, merging continuous- and discrete-variable encoding, and enabling the transport and interconversion of information. However, reliable characterization of the non-local nature of this quantum state is limited so far to full quantum state tomography. Here, we perform a thorough study of Clauser–Horne–Shimony–Holt Bell inequality tests, enabling practical verification of quantum nonlocality for optical hybrid entanglement. We show that a practical violation of this inequality is possible with simple photon number on/off measurements if detection efficiencies stay above 82%. Another approach, based on photon-number parity measurements, requires 94% efficiency but works well in the limit of higher photon populations. Both tests use no postselection of the measurement outcomes and they are free of the fair-sampling hypothesis. Our proposal paves the way to performing loophole-free tests using feasible experimental tasks such as coherent state interference and photon counting.
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