Violation of CHSH inequality and marginal laws in mixed sequential measurements with order effects


We model a typical Bell-test experimental situation by considering that Alice and Bob perform incompatible measurements in a sequential way, with mixed orders of execution. After emphasizing that order effects will generally produce a violation of the marginal laws, we derive an upper limit for the observed correlations. More precisely, when Alice’s and Bob’s measurements are compatible, the marginal laws are obeyed and Tsirelson’s bound limits the quantum correlations in the Bell-CHSH inequality to \(2\sqrt{2}\). On the other hand, when Alice and Bob perform incompatible mixed sequential measurements, the marginal laws are typically violated and the upper limit for the correlations is pushed up to \(2\sqrt{3}\). Considering that significant violations of the marginal laws (also called no-signaling conditions) have been observed in the data of numerous Bell-test experiments, the present analysis provides a possible mechanism for their appearance, when the protocols are such that Alice’s and Bob’s measurements can be assumed to be performed in a mixed sequential way. We, however, emphasize that this does not imply that a communication with superluminal effective speed would be possible.

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    Another assumption would, however, be possible: that Alice’s and Bob’s measurements would be part of bigger joint measurements that cannot be decomposed into sequential measurements (Aerts and Sozzo 2014a, b; Aerts et al. 2019).


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I thank with pleasure Sandro Sozzo for his useful comments in relation to the content of this article.

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Correspondence to Massimiliano Sassoli de Bianchi.

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Sassoli de Bianchi, M. Violation of CHSH inequality and marginal laws in mixed sequential measurements with order effects. Soft Comput 24, 10231–10238 (2020).

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  • Bell’s inequalities
  • Tsirelson’s bound
  • Marginal laws
  • No-signaling conditions
  • Order effects