5 research outputs found
A complete characterisation of All-versus-Nothing arguments for stabiliser states
An important class of contextuality arguments in quantum foundations are the
All-versus-Nothing (AvN) proofs, generalising a construction originally due to
Mermin. We present a general formulation of All-versus-Nothing arguments, and a
complete characterisation of all such arguments which arise from stabiliser
states. We show that every AvN argument for an n-qubit stabiliser state can be
reduced to an AvN proof for a three-qubit state which is local
Clifford-equivalent to the tripartite GHZ state. This is achieved through a
combinatorial characterisation of AvN arguments, the AvN triple Theorem, whose
proof makes use of the theory of graph states. This result enables the
development of a computational method to generate all the AvN arguments in
on n-qubit stabiliser states. We also present new insights into
the stabiliser formalism and its connections with logic.Comment: 18 pages, 6 figure
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
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
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
Logic of local inference for contextuality in quantum physics and beyond
Contextuality in quantum physics provides a key resource for quantum information and computation. The topological approach in [3, 2] characterizes contextuality as “global inconsistency” coupled with “local consistency”, revealing it to be a phenomenon also found in many other fields. This has yielded a logical method of detecting and proving the “global inconsistency” part of contextuality. Our goal is to capture the other, “local consistency” part, which requires a novel approach to logic that is sensitive to the topology of contexts. To achieve this, we formulate a logic of local inference by using context-sensitive theories and models in regular categories. This provides a uniform framework for local consistency, and lays a foundation for high-level methods of detecting, proving, and moreover using contextuality as computational resource