Contextuality of general probabilistic theories

Farid Shahandeh

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Contextuality of general probabilistic theories

Farid Shahandeh

Quantum Physics · Submitted 25 Nov 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim It follows that in any GPT construction the three insistence of (i) the no-restriction hypothesis, (ii) the ontological noncontextuality, and (iii) multiple nonrefinable measurements for any fixed number of outcomes are incompatible.

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Generalized contextuality refers to our inability of explaining measurement statistics using a context-independent probabilistic and ontological model. On the other hand, measurement statistics can also be modeled using the framework of general probabilistic theories (GPTs). Here, starting from a construction of GPTs based on a Gleason-type theorem, we fully characterize these structures with respect to their permission and rejection of generalized (non)contextual ontological models. It follows that in any GPT construction the three insistence of (i) the no-restriction hypothesis, (ii) the ontological noncontextuality, and (iii) multiple nonrefinable measurements for any fixed number of outcomes are incompatible. Hence, any GPT satisfying the no-restriction hypothesis is ontologically noncontextual if and only if it is simplicity. We give a detailed discussion of GPTs for which the no-restriction hypothesis is violated, and show that they can always be considered as subtheories (subGPTs) of GPTs satisfying the hypothesis. It is shown that subGPTs are ontologically noncontextual if and only if they are subtheories of simplicial GPTs of the same dimensionality. Finally, we establish as a corollary the necessary and sufficient condition for a single resourceful measurement or state to promote an ontologically noncontextual (i.e. classical) general probabilistic theory to an ontologically contextual (i.e. nonclassical) one under the no-restriction hypothesis.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Generalized contextuality refers to our inability of explaining measurement statistics using a context-independent probabilistic and ontological model.	background 1.00	0.07	0.04	asserted 1.00
s1	On the other hand, measurement statistics can also be modeled using the framework of general probabilistic theories (GPTs).	background 1.00	0.09	0.02	hedged possibility 0.58
s2	Here, starting from a construction of GPTs based on a Gleason-type theorem, we fully characterize these structures with respect to their permission and rejection of generalized (non)contextual ontological models.	contribution 0.89	0.92	0.10	asserted 1.00
s3	It follows that in any GPT construction the three insistence of (i) the no-restriction hypothesis, (ii) the ontological noncontextuality, and (iii) multiple nonrefinable measurements for any fixed number of outcomes are incompatible.	result 0.98	0.85	0.45	asserted 1.00
s4	Hence, any GPT satisfying the no-restriction hypothesis is ontologically noncontextual if and only if it is simplicity.	result 0.99	0.85	0.34	asserted 1.00
s5	We give a detailed discussion of GPTs for which the no-restriction hypothesis is violated, and show that they can always be considered as subtheories (subGPTs) of GPTs satisfying the hypothesis.	contribution 0.78	0.90	0.05	asserted 1.00
s6	It is shown that subGPTs are ontologically noncontextual if and only if they are subtheories of simplicial GPTs of the same dimensionality.	result 1.00	0.88	0.12	asserted 1.00
s7	Finally, we establish as a corollary the necessary and sufficient condition for a single resourceful measurement or state to promote an ontologically noncontextual (i.e. classical) general probabilistic theory to an ontologically contextual (i.e. nonclassical) one under the no-restriction hypothesis.	result 0.97	0.92	0.24	asserted 1.00

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Original abstract

Generalized contextuality refers to our inability of explaining measurement statistics using a context-independent probabilistic and ontological model. On the other hand, measurement statistics can also be modeled using the framework of general probabilistic theories (GPTs). Here, starting from a construction of GPTs based on a Gleason-type theorem, we fully characterize these structures with respect to their permission and rejection of generalized (non)contextual ontological models. It follows that in any GPT construction the three insistence of (i) the no-restriction hypothesis, (ii) the ontological noncontextuality, and (iii) multiple nonrefinable measurements for any fixed number of outcomes are incompatible. Hence, any GPT satisfying the no-restriction hypothesis is ontologically noncontextual if and only if it is simplicity. We give a detailed discussion of GPTs for which the no-restriction hypothesis is violated, and show that they can always be considered as subtheories (subGPTs) of GPTs satisfying the hypothesis. It is shown that subGPTs are ontologically noncontextual if and only if they are subtheories of simplicial GPTs of the same dimensionality. Finally, we establish as a corollary the necessary and sufficient condition for a single resourceful measurement or state to promote an ontologically noncontextual (i.e. classical) general probabilistic theory to an ontologically contextual (i.e. nonclassical) one under the no-restriction hypothesis.

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7 arXiv papers cite this in 13 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 7 sentences without a confident label

Builds on it 1 paper

This allows us to extend the result of Refs. [ref] —that every noncontextual ontological model of an operational theory is in one-to-one correspondence with an ontological model of the associated GPT—to arbitrary compositional scenarios.

A structure theorem for generalized-noncontextual ontological models · Introduction · 1.00

Compares with it 1 paper

We discuss the relation between Ref. [ref] and this work further in the Supplementary Material, and also demonstrate that it is only for GPTs satisfying the no-restriction hypothesis that simplex embeddability and simpliciality coincide.

The Characterization of Noncontextuality in the Framework of Generalized Probabilistic Theories · Introduction · 0.96

Cites it as background 1 paper

The framework of generalised probabilistic theories [ref] (GPTs) has become the preferred tool for such studies, for example, shedding light on matters pertaining to: cryptography [ref] ; computation [ref] ; interference [ref] ; thermodynamics [ref] ; contextuality [ref] ; nonlocality [ref] ; steering [ref] ; decoherence [ref] ; information processing [ref] ; incompatibility [ref] ; uncertainty [ref] ; as well as providing a foundational view of the primitive structures of physical theories [ref] .

Simulating all multipartite non-signalling channels via quasiprobabilistic mixtures of local channels in generalised probabilistic theories · Introduction · 0.99

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Cites as background 5 papers

Their constructions begin with assuming a vector space $$ whose elements can (at least partially) be ordered and on which an inner-product $··$ can be defined [ref] .

Reformulating and Reconstructing Quantum Theory · General Probabilistic Theories · 0.99
Their constructions begin with assuming a vector space $$ whose elements can (at least partially) be ordered and on which an inner-product $··$ can be defined [ref] .

Quantum Theory From Five Reasonable Axioms · General Probabilistic Theories · 0.99
Contextuality [ref] , and more recently, generalized contextuality [ref] , refer to no-go theorems dismissing context-independent classical models for measured statistics.

(Non-)Contextuality of Physical Theories as an Axiom · Introduction · 1.00

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