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arXiv:1401.2323 · Physical Review A

Testing the Clauser-Horne-Shimony-Holt inequality using observables with arbitrary spectrum

Andreas Ketterer, Arne Keller, T. Coudreau, P. Milman

Published Jan 14, 2015• 14 citations
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Abstract

The Clauser-Horne-Shimony-Holt inequality applies when measurements with binary outcomes are performed on physical systems under the assumption of local realism. Testing such inequalities in the quantum realm usually involves either measurements of two-valued quantum observables or predefining a context-dependent binning procedure. Here we establish the conditions to test the Clauser-Horne-Shimony-Holt inequality using any quantum observable. Our result applies to observables with an arbitrary spectrum, and no prior knowledge of their underlying Hilbert space's dimension is required. Finally, we demonstrate the proposed general measurement strategy, that can be seen as positive operator-valued measurements performed on the system, using the formalism of modular variables applied to the transverse degrees of freedom of single photons.

DOI
10.1103/physreva.91.012106

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The Clauser-Horne-Shimony-Holt inequality applies when measurements with binary outcomes are performed on physical systems under the assumption of local realism. Testing such inequalities in the quantum realm usually involves either measurements of two-valued quantum observables or predefining a context-dependent binning procedure. Here we establish the conditions to test the Clauser-Horne-Shimony-Holt inequality using any quantum observable. Our result applies to observables with an arbitrary spectrum, and no prior knowledge of their underlying Hilbert space's dimension is required. Finally, we demonstrate the proposed general measurement strategy, that can be seen as positive operator-valued measurements performed on the system, using the formalism of modular variables applied to the transverse degrees of freedom of single photons.
We establish a necessary and sufficient condition for the existence of a quantum state that reproduces given correlation values in the Clauser--Horne--Shimony--Holt setup for any fixed normalized observables. This result addresses a fundamental question shared by both local realism and quantum mechanics: under what conditions a given set of observed data can be reproduced by a physical model. While previous studies have mainly addressed conditions for correlations achievable without specifying the measurement settings, our result gives a finer characterization by treating the observables as fixed in advance. The resulting quantum condition strengthens previously known constraints, such as Tsirel'son's inequalities and the Tsirel'son--Landau inequality, by characterizing statistical constraints explicitly for each specified set of observables. In particular, we show that our condition applies to Bell's original scenario and reveals that whether Bell's original inequality is violated depends sensitively on the chosen observables. More broadly, this perspective offers new insights into how quantum violations of local realism depend on the measurement settings.
We establish a necessary and sufficient condition for the existence of a quantum state that reproduces given correlation values in the Clauser--Horne--Shimony--Holt (CHSH) setup for any fixed normalized observables. This result addresses a fundamental question shared by both local realism and quantum mechanics: under what conditions a given set of observed data can be reproduced by a physical model. While previous studies have mainly addressed conditions for correlations achievable without specifying the measurement settings, our result gives a finer characterization by treating the observables as fixed in advance. The resulting quantum condition strengthens previously known constraints, such as Tsirel'son's inequalities and the Tsirel'son--Landau inequality, by characterizing statistical constraints explicitly for each specified set of observables. In particular, we show that our condition applies to Bell's original scenario and reveals that whether Bell's original inequality is violated depends sensitively on the chosen observables. More broadly, this perspective offers new insights into how quantum violations of local realism depend on the measurement settings.
The Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, which is proven in the context of the local hidden-variable theory, has been used as a test to reveal the failure of the hidden-variable theory and to prove the validity of the quantum theory. We note that the violation of the BCHSH inequality is caused by the noncommutativity of quantum observables and find a systematic method of constructing generalizations of the BCHSH inequality. This method is used to invent a new quantity, which is defined in a two-qubit system and satisfies a new type of inequality. This provides a fair test of the quantum theory. Remaining problems are also discussed.
We investigate quantum nonlocality of a single-photon entangled state under feasible measurement techniques consisting of on-off and homodyne detections along with unitary operations of displacement and squeezing. We test for a potential violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, in which each of the bipartite party has a freedom to choose between two measurement settings, with each measurement yielding a binary outcome. We find that single-photon quantum nonlocality can be detected when two or less of the four total measurements are carried out by homodyne detection. The largest violation of the CHSH inequality is obtained when all four measurements are squeezed-and-displaced on-off detections. We test the robustness of violations against imperfections in on-off detectors and single-photon sources, finding that the squeezed-and-displaced measurement schemes perform better than the displacement-only measurement schemes.
A correlation inequality is derived from local realism and a supplementary assumption. Unlike Clauser-Horne (CH) inequality [or Clauser-Horne-Shimony-Holt (CHSH) inequality] which is violated by quantum mechanics by a factor of 2\sqrt 22​, this inequality is violated by a factor of 1.5. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, unlike CH (or CHSH) inequality which requires the measurement of five detection probabilities, the present inequality requires the measurement of only two detection probabilities. This inequality can therefore be used to test locality more simply than CH or CHSH inequality.
We investigate the violation of Leggett's inequality for nonlocal realism using entangled coherent states and various types of local measurements. We prove mathematically the relation between the violation of the Clauser-Horne-Shimony-Holt form of Bell's inequality and Leggett's one when tested by the same resources. For Leggett inequalities, we generalize the nonlocal realistic bound to systems in Hilbert spaces larger than bidimensional ones and introduce an optimization technique that allows one to achieve larger degrees of violation by adjusting the local measurement settings. Our work describes the steps that should be performed to produce a self-consistent generalization of Leggett's original arguments to continuous-variable states.