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Closing Bell Boxing Black Box Simulations in the Resource Theory of Contextuality

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Samson Abramsky on Logic and Structure in Computer Science and Beyond

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 25))

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Abstract

This chapter contains an exposition of the sheaf-theoretic framework for contextuality emphasising resource-theoretic aspects, as well as some original results on this topic. In particular, we consider functions that transform empirical models on a scenario S to empirical models on another scenario T, and characterise those that are induced by classical procedures between S and T corresponding to ‘free’ operations in the (non-adaptive) resource theory of contextuality. We proceed by expressing such functions as empirical models themselves, on a new scenario built from S and T. Our characterisation then boils down to the non-contextuality of these models. We also show that this construction on scenarios provides a closed structure in the category of measurement scenarios.

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Notes

  1. 1.

    At the risk of provoking a relapse into an erstwhile indulgence of Samson’s, who has been known to self-identify as a recovering philosopher, we remark that this chapter adopts an approach that is somewhat in the empiricist tradition of philosophy. As someone whose research interests and contributions are ever-evolving and indeed ever-relevant, one former student of his has also pointed out that Samson has clearly distinguished himself from another famous Samson, the dinosaur.

  2. 2.

    From now on, we shall primarily adopt the terminology measurements and outcomes.

  3. 3.

    This is similarly illustrated by the ‘impossible biscuit’ in the poster for the 2018 Lorentz Centre workshop Logical Aspects of Quantum Information, which was co-organised by Samson: https://www.lorentzcenter.nl/logical-aspects-of-quantum-information.html.

  4. 4.

    This also brings to mind the words of Álvaro de Campos, as if quantum systems were contriving to realise the motto from his futurist phase, ‘to be sincere contradicting oneself’. In the original: ‘Ser sincero contradizendo-se’. From the poem Passagem das horas (22–05–1916), in Campos (1944).

  5. 5.

    Within these frameworks, the phenomenon of non-locality as discussed by Bell may be seen as a special case of contextuality that arises in distributed or multi-party scenarios. Note that locality in Bell’s sense differs from our use of the term earlier in relation to local compatibility.

  6. 6.

    Subsequent developments are to be found in many papers including in particular the local-consistency-versus-global-inconsistency picture in Abramsky et al. (2015).

  7. 7.

    One cannot help but be reminded of the reversal of Player and Opponent rôles in games of function type in game semantics. Player in such a game plays simultaneously, and inter-dependently, two simpler games, corresponding to the output and the input types, and adopts a different rôle in each of them Abramsky et al. (1999).

  8. 8.

    Note that in Abramsky et al. (2019a, b), where we first made an observation to this effect, the source of these simulations was the trivial scenario with one measurement and a single outcome. The difference arises due to the kind of simulations we allow in each case. It is related to the fact that ‘the’ singleton set is the terminal object in the category of sets and functions (i.e. there is exactly one function from any given set to a singleton set) whereas in the category of sets and relations the terminal object is the empty set.

  9. 9.

    From Meditation XVII, in Donne (1624).

  10. 10.

    At the risk of overstretching the use of poetic metaphor, one is reminded of Blake’s, ‘[to] hold infinity in the palm of your hand, and eternity in an hour.’ From the poem ‘Auguries of Innocence’ (c. 1803), in The Ballads (or Pickering) Manuscript, published in Gilchrist and Gilchrist (1863).

  11. 11.

    In fact, the notion of morphism considered here differs slightly from those of Karvonen (2019) and Abramsky et al. (2019a). It is the appropriate notion to capture non-adaptive simulations. The minor discrepancies are discussed and explained at the end of Sect. 13.3.2.

  12. 12.

    Otherwise it would be difficult to justify calling it a measurement in the first place.

  13. 13.

    Note that for Boolean distributions the restriction to finite support is unnecessary. But here we only deal with finite sets of events, anyway.

  14. 14.

    While the word “predicate” does not quite fit with the experimental imagery evoked by much of our terminology, there is a reason for introducing an alternative word for two-valued experiments. Later, it will be useful to consider whether a given model always returns the outcome 1 in such an experiment, and to restrict attention to those models on a scenario which do so. Hence calling it a predicate serves the purpose of indicating a change of viewpoint, where we will be restricting attention to models that always satisfy a given property.

  15. 15.

    See Abramsky et al. (2017b, Sect. 4) for a more general account of state-independent contextuality phrased in similar language.

  16. 16.

    Most of this should follow from the fact that changing the basis of enrichment is a 2-functor and in our case preserves the duality involutions, so one should automatically get a functor \([-,-]\) and the required natural transformations, leaving only dinaturality and some of the axioms to be checked by hand. However, we are not aware of general results guaranteeing that change-of-basis preserves closed structure, so we sketch the hands-on proof.

  17. 17.

    Littlewood wrote of his collaboration with Mary Cartwright: Two rats fell into a can of milk. After swimming for a time one of them realised his hopeless fate and drowned. The other persisted, and at last the milk was turned to butter and he could get out.

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Acknowledgements

This work was in part carried out while RSB was based at the School of Informatics, University of Edinburgh and while SM was based at the Paris Centre for Quantum Computing, Institut de Recherche en Informatique Fondamentale, University of Paris with financial support from the Bpifrance project RISQ. RSB acknowledges financial support from EPSRC—Engineering and Physical Sciences Research Council, EP/R044759/1, Combining Viewpoints in Quantum Theory (Ext.), and from FCT—Fundação para a Ciência e a Tecnologia, CEECINST/00062/2018.

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Authors and Affiliations

  1. INL–International Iberian Nanotechnology Laboratory, Braga, Portugal

    Rui Soares Barbosa

  2. Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada

    Martti Karvonen

  3. Quandela, Palaiseau, France

    Shane Mansfield

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  1. Rui Soares Barbosa
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  2. Martti Karvonen
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  3. Shane Mansfield
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Correspondence to Rui Soares Barbosa .

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  1. Department of Ethics, Governance and Society, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

    Alessandra Palmigiano

  2. Department of Computer Science, University College London, London, UK

    Mehrnoosh Sadrzadeh

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Barbosa, R.S., Karvonen, M., Mansfield, S. (2023). Closing Bell Boxing Black Box Simulations in the Resource Theory of Contextuality. In: Palmigiano, A., Sadrzadeh, M. (eds) Samson Abramsky on Logic and Structure in Computer Science and Beyond. Outstanding Contributions to Logic, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-031-24117-8_13

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