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arXiv:2101.02426 · Physical Review A

Class of Bell-Clauser-Horne inequalities for testing quantum nonlocality

Chen Qian, Yang-Guang Yang, Qun Wang, Cong‐Feng Qiao

Published Jun 1, 2021• 4 citations
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Abstract

Quantum nonlocality, one of the most important features of quantum mechanics, is normally connected in experiments with the violation of Bell-Clauser-Horne (Bell-CH) inequalities. We propose effective methods for the rearrangement and linear inequality to prove a large variety of Bell-CH inequalities. We also derive a set of Bell-CH inequalities using these methods which can be violated in some quantum entangled states.

DOI
10.1103/physreva.103.062203

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Quantum nonlocality, one of the most important features of quantum mechanics, is normally connected in experiments with the violation of Bell-Clauser-Horne (Bell-CH) inequalities. We propose effective methods for the rearrangement and linear inequality to prove a large variety of Bell-CH inequalities. We also derive a set of Bell-CH inequalities using these methods which can be violated in some quantum entangled states.
Quantum nonlocality can be tested in experiments via the violation of the Bell inequality. We propose novel methods of the rearrangement and linear inequality to prove all the known Bell-CH inequalities of Imn22Imn22Imn22-type (m,n≤5m,n\le5m,n≤5). We also derive a set of new Bell-CH inequalities by using these methods which should be violated in quantum entangled states.
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 derive an efficient CH-type inequality. Quantum mechanics violates our proposed inequality independent of the detection-efficiency problem. In photonic Bell-type experiments [1], when photon pairs with parallel linear polarizations are emitted, one can consider a Clauser-Horne (CH) inequality [2], at the level of hidden variables, in the form where − 1 ≤ Srq,HV (â, ̂ b, â ′ , ̂ b ′ , λ) ≤ 0 (1) Srq,HV (â, ̂b, â ′ , ̂ b ′, λ) = p (1) r (â, λ) p (2) q (̂b, λ) − p (2) q ( ̂ b ′, λ)] +p (1) r ( â ′ , λ) [ p (2) q ( ̂b, λ) + p (2) q ( ̂ b ′, λ)] −p (1) r (â ′ , λ) − p (2) q ( ̂ b, λ) (2) In (2), we are considering four sub-ensemble of photon pairs with linear polarizations along (â, ̂ b), (â, ̂ b ′), ( â ′ , ̂ b), and ( â ′ , ̂ b ′ ) in which one registers
We present a general method for obtaining all Bell inequalities for a given experimental setup. Although the algorithm runs slowly, we apply it to two cases. First, the Greenberger-Horne-Zeilinger setup with three observers each performing one of two possible measurements. Second, the case of two observers each performing one of three possible experiments. In both cases we obtain hundreds of inequalities. Since this is the set of all inequalities, the one that is maximally violated in a given quantum state must be among them. We demonstrate this fact with a few examples. We also note the deep connection between the inequalities and classical logic, and their violation with quantum logic.
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.
Bell-type inequalities and violations thereof reveal the fundamental differences between standard probability theory and its quantum counterpart. In the course of previous investigations ultimate bounds on quantum mechanical violations have been found. For example, Tsirelson's bound constitutes a global upper limit for quantum violations of the Clauser-Horne-Shimony-Holt (CHSH) and the Clauser-Horne (CH) inequalities. Here we investigate a method for calculating the precise quantum bounds on arbitrary Bell-type inequalities by solving the eigenvalue problem for the operator associated with each Bell-type inequality. Thereby, we use the min-max principle to calculate the norm of these self-adjoint operators from the maximal eigenvalue yielding the upper bound for a particular set of measurement parameters. The eigenvectors corresponding to the maximal eigenvalues provide the quantum state for which a Bell-type inequality is maximally violated.