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A CH-type Inequality For Real Experiments

A. Shafiee

Published Jan 6, 2004• 3 citations
OpenAlex

Abstract

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

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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
It is demonstrated that the conjunction of the premises of Einstein-Podolsky-Rosen paper with a supplementary assumption lead to validity of an inequality that is sometimes grossly violated by the quantum-mechanical probabilities in the case of real experiments. The supplementary assumption is as follows: For every photon in the state \ensuremath{\lambda}, the sum of the detection probabilities in the ordinary and in the extraordinary beams emerging from a two-way polarizer is less than or equal to the detection probability with the polarizer removed. This assumption, which always holds for an ensemble of photons, is weaker and more general than the assumption of Garuccio and Rapisarda [Nuovo Cimento A 65, 269 (1981)] and the assumption of Clauser and Horne [Phys. Rev. D 10, 526 (1974)]. Hence an experiment based on this assumption refutes a larger family of hidden-variable series than an experiment based on the assumption of Garuccio and Rapisardo or on the assumption of Clauser and Horne.
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.
Einstein's locality is invoked to derive a correlation inequality. In the case of ideal experiments, this inequality is equivalent to Bell's original inequality of 1965 which, as is well known, is violated by a maximum factor of 1.5. The crucial point is that even in the case of real experiments where polarizers and detectors are non-ideal, the present inequality is violated by a factor of 1.5, whereas previous inequalities such as Clauser-Horne-Shimony-Holt inequality of 1969 and Clauser-Horne inequality of 1974 are violated by a factor of 2\sqrt 22​. The larger magnitude of violation can be of importance for the experimental test of locality. Moreover, the supplementary assumption used to derive this inequality is weaker than Garuccio-Rapisarda assumption. Thus an experiment based on this inequality refutes a larger family of hidden variable theories than an experiment based on Garuccio-Rapisarda inequality.
In the analysis of experiments designed to reveal violation of Bell-type inequalities, it is usually assumed that any hidden variables associated with the nthn\mathrm{th}nth particle pair would be independent of measurement choices and outcomes for the first $(n\ensuremath{-}1)$ pairs. Models which violate this assumption exploit what we call the memory loophole. We focus on the strongest type of violation, which uses the two-sided memory loophole, in which the hidden variables for pair n can depend on the previous measurement choices and outcomes in both wings of the experiment. We show that the two-sided memory loophole allows a systematic violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality when the data are analyzed in the standard way, but cannot produce a violation if a CHSH expression depending linearly on the data is used. In the first case, the maximal CHSH violation becomes small as the number of particle pairs tested becomes large. Hence, although in principle the memory loophole implies a slight flaw in the existing analyses of Bell experiments, the data still strongly confirm quantum mechanics against local hidden variables. We consider also a related loophole, the simultaneous measurement loophole, which applies if all measurements on each side are carried out simultaneously. We show that this can increase the probability of violating the linearized CHSH inequality as well as other Bell-type inequalities.