University of Cambridge > Talks.cam > CQIF Seminar >  CHSH Bell Inequalities made simple(r): Linear functions, loopholes, and how to post-select data without causing one.

CHSH Bell Inequalities made simple(r): Linear functions, loopholes, and how to post-select data without causing one.

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This work began as an investigation into whether the cosmetic similarities between Bell inequality experiments and measurement-based quantum computing might reflect a deeper connection. What we find is that the ``computational viewpoint’’ is remarkably well-suited to the study of CHSH -type Bell inequalities, and provides a very simple way of characterising the correlations permitted by local hidden variable (LHV) theories. While mathematically equivalent to previous work (see Werner and Wolf’s paper cited below for the original derivation of this set of correlations) this provides a cleaner and simpler operational definition, which, in particular, make ’’loopholes’’ easy to characterise.

Loopholes in Bell inequality experiments arise when imperfections in the experimental setup mean that it does not concord precisely with the assumptions under which Bell inequalities are derived. Often those flaws mean that local hidden variable theories would be able to access correlations which in a strict Bell setup would be forbidden to them, and hence a quantum mechanical reproduction of that correlation cannot refute local hidden variable theories – a “loophole” in the argument.

The simplicity of the characterisation of the LHV correlations in our model allow us to pin down explicit mechanisms by which loopholes arise. For example, a well-known loophole is the detector loophole, where, due to inefficient detectors, data must be post-selected – only when both detectors fire can the data be used. The origin of the detector loophole can be cleanly understood within our framework, and constructing LHV models which fake an imperfect quantum detector while maximally violating Bell inequalities (to the bounds previously identified by Garg and Mermin) is straightforward. We see that there are many ways in which post-selection of data can cause loopholes.

In addition to providing a simpler (and quantum information friendly) way to understand previous results (hence “for beginners”), we can use our model for new investigations. For example, we can characterise post-selection strategies where no loopholes arise – and consider the effect of this post-selection upon quantum correlations. We find that there is a broad family of non-loophole inducing post-selection strategies which one can adopt. Surprisingly, we see that while not expanding the region of correlations accessible by LHV theories, such post-selection can expand the region of correlations accessible by quantum theories. In other words, performing this post-selection allows quantum measurements to achieve correlations which were previously impossible, without creating a loophole. This effect becomes apparent in the multi-partite setting (the smallest example we have is for 6 qubits), and does not enhance the bi-partite CHSH inequality, therefore, it is currently unclear whether this post-selection will aid current Bell inequality experiments. However, the larger multi-partite region now includes new types of quantum correlation previously overlooked in the Bell inequality setting, most notably (post-selected simulations of) the adaptive measurements which arise in measurement-based quantum computation. We expect that these results will give new insights into measurement-based quantum computation and related areas and will be valuable in the search for information theoretic characterisations of the set of quantum correlations which go beyond the bi-partite setting.

These results were developed in collaboration with Matty Hoban.

Previous work in this direction: J. Anders and D.E. Browne, arXiv:0805.1002; M. Hoban et al, arXiv:1009.5213

Main reference for these results: M. Hoban and D. E. Browne, in preparation (hopefully on the arxiv very soon)

Other reading: Multi-partite Bell inequalities, Werner and Wolf, Phys. Rev. A 64 , 032112 (2001); Detector Loophole, Garg and Mermin, Phys. Rev. D 35 , 3831–3835 (1987)

This talk is part of the CQIF Seminar series.

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