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Chained Clauser–Horne–Shimony–Holt inequality for Hardy’s ladder test of nonlocality

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Abstract

Relativistic causality forbids superluminal signaling between distant observers. By exploiting the non-signaling principle, we derive the exact relationship between the chained Clauser–Horne–Shimony–Holt sum of correlations \(\text {CHSH}_K\) and the success probability \(P_K\) associated with Hardy’s ladder test of nonlocality for two qubits and \(K+1\) observables per qubit. Then, by invoking the Tsirelson bound for \(\text {CHSH}_K\), the derived relationship allows us to establish an upper limit on \(P_K\). In addition, we draw the connection between \(\text {CHSH}_K\) and the chained version of the Clauser–Horne (CH) inequality.

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Notes

  1. A few technical lemmas concerning a version of the \(\text {CHSH}_K\) expression (7) have been derived in a series of papers by Colbeck and Renner (see, for instance, Refs. [12–14]). Interestingly, as shown by these authors, the use of a chained Bell inequality allows one to construct a program to rule out a broad class of hidden variable theories.

  2. In Sect. 3, it will be verified that, in fact, the quantum predictions satisfy relation (8).

  3. Incidentally, it is worth pointing out that, in this limit, a direct (“all or nothing”) contradiction between quantum mechanics and local realism emerges in Hardy’s ladder scenario [23].

  4. Note that, from relation (8), the limit \(P_K = 0.5\) can never be surpassed since this would entail that \(\text {CHSH}_K > 2K +2\), which is impossible by the very definition of \(\text {CHSH}_K\).

  5. The IC principle states that communication of m classical bits causes information gain of at most m bits. The NS principle is just IC for \(m=0\).

  6. As pointed out by one referee, robust self-testing conditions for any partially entangled two-qubit states have been derived by Bamps and Pironio [34] (see also Ref. [35] for closely related work and references therein).

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Acknowledgments

The author would like to thank the Associate Editor Michael Frey and two anonymous referees for their comments and suggestions regarding an earlier version of this paper.

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  1. Telefónica de España, Distrito Telefónica, Edificio Este 1, 28050, Madrid, Spain

    José Luis Cereceda

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Cereceda, J.L. Chained Clauser–Horne–Shimony–Holt inequality for Hardy’s ladder test of nonlocality. Quantum Inf Process 15, 1779–1792 (2016). https://doi.org/10.1007/s11128-015-1217-4

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