# Non-locality: sheaf cohomology

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Here are a few wonderful papers which describe non-locality and contextuality in detail using a combination of sheaf theory, graph theory and algebraic topology.

Abramsky et al. 2011, The Sheaf-Theoretic Structure Of Non-Locality and Contextuality
Abstract said:
We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, in a setting which generalizes the familiar probability tables used in non-locality theory to arbitrary measurement covers; this includes Kochen-Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections. We describe a linear algebraic approach to computing these obstructions, which allows a systematic treatment of arguments for non-locality and contextuality. We distinguish a proper hierarchy of strengths of no-go theorems, and show that three leading examples --- due to Bell, Hardy, and Greenberger, Horne and Zeilinger, respectively --- occupy successively higher levels of this hierarchy. A general correspondence is shown between the existence of local hidden-variable realizations using negative probabilities, and no-signalling; this is based on a result showing that the linear subspaces generated by the non-contextual and no-signalling models, over an arbitrary measurement cover, coincide. Maximal non-locality is generalized to maximal contextuality, and characterized in purely qualitative terms, as the non-existence of global sections in the support. A general setting is developed for Kochen-Specker type results, as generic, model-independent proofs of maximal contextuality, and a new combinatorial condition is given, which generalizes the `parity proofs' commonly found in the literature. We also show how our abstract setting can be represented in quantum mechanics. This leads to a strengthening of the usual no-signalling theorem, which shows that quantum mechanics obeys no-signalling for arbitrary families of commuting observables, not just those represented on different factors of a tensor product.
Abramsky et al. 2015, Contextuality, Cohomology and Paradox
Abstract said:
Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in quantum theory [New Journal of Physics 13 (2011) 113036]. However, contextual phenomena are found in other fields as well, for example database theory. In this paper, we shall develop this unified view of contextuality. We provide two main contributions: firstly, we expose a remarkable connection between contexuality and logical paradoxes; secondly, we show that an important class of contextuality arguments has a topological origin. More specifically, we show that "All-vs-Nothing" proofs of contextuality are witnessed by cohomological obstructions.
Carù 2018, Towards a complete cohomology invariant for non-locality and contextuality
Abstract said:
The sheaf theoretic description of non-locality and contextuality by Abramsky and Brandenburger sets the ground for a topological study of these peculiar features of quantum mechanics. This viewpoint has been recently developed thanks to sheaf cohomology, which provides a sufficient condition for contextuality of empirical models in quantum mechanics and beyond. Subsequently, a number of studies proposed methods to detect contextuality based on different cohomology theories. However, none of these cohomological descriptions succeeds in giving a full invariant for contextuality applicable to concrete examples. In the present work, we introduce a cohomology invariant for possibilistic and strong contextuality which is applicable to the vast majority of empirical models.
First, it must be understood that non-locality is a form of (measurement) contextuality. The key point is then that contextuality is equivalent to the non-existence of global sections for a family of probability distributions.

Secondly, using this sheaf theoretic framework, Feynman's interpretation that negative probability characterizes QM is incorrect; negative probability characterizes all models with no-signalling.

Lastly, and perhaps most surprisingly, is that the incompatibility of measurements - which in QM is usually a postulate taken to be specific to the non-commuting observables formalism - can be shown to be derived from a theory-independent structural impossibility result of certain families of empirical distributions.

DarMM and fresh_42

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Seeing the thread hasn't gotten a lot of replies yet, I want to make some clarifying comments about what is meant by nonlocality and why cohomology (a topological concept) is relevant w.r.t. the topic of nonlocality.

There seems to be a notorious semantic issue - frequently leading to huge yet completely avoidable misconceptions and misunderstandings - when discussing Bell's theorem and the related nonlocality which is also characteristic of some aspects of QM.

First, we should make a clear distinction between I) mathematical locality - i.e. locality as nearness or proximity in a geometric or topological sense, let's call it topologic locality - and II) physical locality - i.e. locality referring to SR's speed of light ##c## as the maximum speed limit at which influences can travel, let's call it relativistic locality.

One might argue that it is somewhat peculiar to refer to there being a finite ##c## as 'locality' and they wouldn't be mistaken; it is a somewhat strange and unfortunate accident of history that this custom became standard terminology in SR and physics following Einstein, Lorentz et al.

It is however absolutely paramount to recognize and realize that definition I is far more general than definition II, i.e. topologic locality is independent of relativistic locality but not vice versa; this is because relativistic locality necessarily refers to temporally occurring processes in space(time) while topologic locality can be completely atemporal, i.e. a necessary consequence of the topology of the space.

I think not carefully acknowledging this distinction directly leads to much of the confusion surrounding the issue of nonlocality in QM; I think it is quite clear that the nonlocality in QM is a form of geometric or topologic nonlocality, perhaps one even (fully) consistent with relativistic locality.

The reason for this claim is that nonlocal influences in QM (e.g. entanglement) seem to be occurring instantaneously between coupled distant objects while no information can be transmitted using such instantaneous effects; this implies that this nonlocality is a consequence of properties i.e. the topology of spacetime itself.

This is where cohomology comes in in the present story. The problem is that topology - even without focussing on cohomology - is a difficult and abstruse topic, which many physicists simply may not be as familiar with as with other forms of mathematics, unfortunately halting further enquiry.

Luckily, the authors in the second paper in the OP (Abramsky et al. 2015) not only carefully explained why cohomology is so important in this context, but they even referred to a much more accessible paper which must be the best exposition of cohomology I have ever seen: Penrose 1992, On the cohomology of impossible figures.

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Connections between sheaf theory and non commutative geometry, anyone?

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Connections between sheaf theory and non commutative geometry, anyone?
The only connection I know of is in an application, namely twistor theory: twistor wavefunctions ##f(Z^{\alpha})## are elements of holomorphic sheaf cohomology, while twistors ##Z^{\alpha}## and their duals ##\bar Z_{\alpha}## do not commute i.e. ##[Z^{\alpha},\bar Z_{\beta}]=\hbar \delta^{\alpha}_{\beta}##.

Geometrically, twistors ##Z^{\alpha}## are points in projective twistor space ##\mathbb {PT}## (a ##\mathbb {CP}^3##), while dual twistors ##\bar Z_{\alpha}## are planes in ##\mathbb {PT}##. Analytically, dual twistors ##\bar Z_{\alpha}## are complex conjugated twistors, while algebraically they are also basically first quantized operators i.e. ##\bar Z_{\alpha} \mapsto -h\frac {\partial} {\partial Z^{\alpha}}##, making the entire scheme holomorphic.