11 research outputs found
On the Cohomology of Contextuality
Recent work by Abramsky and Brandenburger used sheaf theory to give a
mathematical formulation of non-locality and contextuality. By adopting this
viewpoint, it has been possible to define cohomological obstructions to the
existence of global sections. In the present work, we illustrate new insights
into different aspects of this theory. We shed light on the power of detection
of the cohomological obstruction by showing that it is not a complete invariant
for strong contextuality even under symmetry and connectedness restrictions on
the measurement cover, disproving a previous conjecture. We generalise
obstructions to higher cohomology groups and show that they give rise to a
refinement of the notion of cohomological contextuality: different "levels" of
contextuality are organised in a hierarchy of logical implications. Finally, we
present an alternative description of the first cohomology group in terms of
torsors, resulting in a new interpretation of the cohomological obstructions.Comment: In Proceedings QPL 2016, arXiv:1701.0024
On Measures and Measurements: a Fibre Bundle approach to Contextuality
Contextuality is the failure of "local" probabilistic models to become global
ones. In this paper we introduce the notions of \emph{measurable fibre
bundles}, \emph{probability fibre bundles}, and \emph{sample fibre bundle}
which capture and make precise the former statement. The central notions of
contextuality are discussed under this formalism, examples worked out, and some
new aspects pointed out.Comment: 14 pages; no figures; Purdue-Winer Memorial Lectures 2018; submitted
to Phil. Trans. Roy. Soc. A. Comments are very welcom
Combining contextuality and causality: a game semantics approach
We develop an approach to combining contextuality with causality, which is
general enough to cover causal background structure, adaptive measurement-based
quantum computation, and causal networks. The key idea is to view contextuality
as arising from a game played between Experimenter and Nature, allowing for
causal dependencies in the actions of both the Experimenter (choice of
measurements) and Nature (choice of outcomes).Comment: 13 page
A Sheaf Model of Contradictions and Disagreements. Preliminary Report and Discussion
We introduce a new formal model -- based on the mathematical construct of
sheaves -- for representing contradictory information in textual sources. This
model has the advantage of letting us (a) identify the causes of the
inconsistency; (b) measure how strong it is; (c) and do something about it,
e.g. suggest ways to reconcile inconsistent advice. This model naturally
represents the distinction between contradictions and disagreements. It is
based on the idea of representing natural language sentences as formulas with
parameters sitting on lattices, creating partial orders based on predicates
shared by theories, and building sheaves on these partial orders with products
of lattices as stalks. Degrees of disagreement are measured by the existence of
global and local sections.
Limitations of the sheaf approach and connections to recent work in natural
language processing, as well as the topics of contextuality in physics, data
fusion, topological data analysis and epistemology are also discussed.Comment: This paper was presented at ISAIM 2018, International Symposium on
Artificial Intelligence and Mathematics. Fort Lauderdale, FL. January 3 5,
2018. Minor typographical errors have been correcte
Topos quantum theory with short posets
Topos quantum mechanics, developed by Isham et. al., creates a topos of
presheaves over the poset V(N) of abelian von Neumann subalgebras of the von
Neumann algebra N of bounded operators associated to a physical system, and
established several results, including: (a) a connection between the
Kochen-Specker theorem and the non-existence of a global section of the
spectral presheaf; (b) a version of the spectral theorem for self-adjoint
operators; (c) a connection between states of N and measures on the spectral
presheaf; and (d) a model of dynamics in terms of V(N). We consider a
modification to this approach using not the whole of the poset V(N), but only
its elements of height at most two. This produces a different topos with
different internal logic. However, the core results (a)--(d) established using
the full poset V(N) are also established for the topos over the smaller poset,
and some aspects simplify considerably. Additionally, this smaller poset has
appealing aspects reminiscent of projective geometry.Comment: 14 page
Continuous-variable nonlocality and contextuality
Contextuality is a non-classical behaviour that can be exhibited by quantum
systems. It is increasingly studied for its relationship to
quantum-over-classical advantages in informatic tasks. To date, it has largely
been studied in discrete variable scenarios, where observables take values in
discrete and usually finite sets. Practically, on the other hand,
continuous-variable scenarios offer some of the most promising candidates for
implementing quantum computations and informatic protocols. Here we set out a
framework for treating contextuality in continuous-variable scenarios. It is
shown that the Fine--Abramsky--Brandenburger theorem extends to this setting,
an important consequence of which is that nonlocality can be viewed as a
special case of contextuality, as in the discrete case. The contextual
fraction, a quantifiable measure of contextuality that bears a precise
relationship to Bell inequality violations and quantum advantages, can also be
defined in this setting. It is shown to be a non-increasing monotone with
respect to classical operations that include binning to discretise data.
Finally, we consider how the contextual fraction can be formulated as an
infinite linear program, and calculated with increasing accuracy using
semi-definite programming approximations.Comment: 27 pages including 6 pages supplemental material, 2 figure
Contextuality and the fundamental theorems of quantum mechanics
Contextuality is a key feature of quantum mechanics, as was first brought to
light by Bohr and later realised more technically by Kochen and Specker. Isham
and Butterfield put contextuality at the heart of their topos-based formalism
and gave a reformulation of the Kochen-Specker theorem in the language of
presheaves. Here, we broaden this perspective considerably (partly drawing on
existing, but scattered results) and show that apart from the Kochen-Specker
theorem, also Wigner's theorem, Gleason's theorem, and Bell's theorem relate
fundamentally to contextuality. We provide reformulations of the theorems using
the language of presheaves over contexts and give general versions valid for
von Neumann algebras. This shows that a very substantial part of the structure
of quantum theory is encoded by contextuality.Comment: v2: minor revisions, added definition of Bell presheaf, adjustment of
Bell's theorem in contextual for
Closing Bell: Boxing black box simulations in the resource theory of contextuality
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 construct a new 'hom' scenario built from S and T, whose
empirical models induce such functions. Our characterisation then boils down to
being induced by a non-contextual model. We also show that this construction on
scenarios provides a closed structure on the category of measurement scenarios.Comment: Corrected a mistake in Theorem 44 and other fixes stemming from it.
This supersedes the published version and should be considered the version of
referenc