arXiv subjects

Sidiney B. Montanhano

Publications and source records attributed to Sidiney B. Montanhano.

4 recordsLinked to original sources

arXiv2024

Contextuality in the Bundle Approach, n-Contextuality, and the Role of Holonomy

Contextuality can be understood as the impossibility to construct a globally consistent description of a model even if there is local agreement. In particular, quantum models present this property. We can describe contextuality with the bundle approach, where the scenario is represented as a simplicial complex, the fibers being the sets of outcomes, and contextuality as the non-existence of global section in the measure bundle. Using the generalization to non-finite outcome fibers, we built in details the concept of measure bundle, demonstrating the Fine-Abramsky-Brandenburger theorem for the bundle formalism. We introduce a hierarchy called n-contextuality to explore the dependence of contextual behavior of a model to the topology of the scenario, following the construction of it as a simplicial complex. With it we exemplify the dependence on higher homology groups and show that GHZ models, thus quantum theory, has all levels of the hierarchy. Also, we give an example of how non-trivial topology of the scenario result an increase of contextual behavior. For the first level of the hierarchy, we construct the concept of connection through Markov operators for the measure bundle. Taking the case of equal fibers we can identify the outcomes as the basis of a vector space, that transform according to a group extracted from the connection. We thus show that contextuality has a relationship with the non-triviality of the holonomy group in the respective frame bundle.

quant-ph↗

arXiv2024

Differential Geometry of Contextuality

Contextuality has long been associated with topological properties. In this work, such a relationship is elevated to identification in the broader framework of generalized contextuality. We employ the usual identification of states, effects, and transformations as vectors in a vector space and encode them into a tangent space, rendering the noncontextual conditions the generic condition that discrete closed paths imply null phases in valuations, which are immediately extended to the continuous case. Contextual behavior admits two equivalent interpretations in this formalism. In the geometric or intrinsic-realistic view, termed "Schrödinger", flat space is imposed, leading to contextual behavior being expressed as non-trivial holonomy of probabilistic functions, analogous to the electromagnetic tensor. As a modification of the valuation function, we use the equivalent curvature to connect contextuality with interference, noncommutativity, and signed measures. In the topological or participatory-realistic view, termed "Heisenberg", valuation functions must satisfy classical measure axioms, resulting in contextual behavior needing to be expressed in topological defects in events, resulting in non-trivial monodromy. We utilize such defects to connect contextuality with non-embeddability and to construct a generalized Vorob'ev theorem, a result regarding the inevitability of noncontextuality. We identify in this formalism the contextual fraction for models with outcome-determinism, and a pathway to address disturbance in ontological models as non-trivial transition maps. We also discuss how the two views for encoding contextuality relate to interpretations of quantum theory.

quant-ph↗

arXiv2024

Wigner and friends, a map is not the territory! Contextuality in multi-agent paradoxes

Multi-agent scenarios, like Wigner's friend and Frauchiger-Renner scenarios, can show contradictory results when a non-classical formalism must deal with the knowledge between agents. Such paradoxes are described with multi-modal logic as violations of the structure in classical logic. Even if knowledge is treated in a relational way with the concept of trust, contradictory results can still be found in multi-agent scenarios. Contextuality deals with global inconsistencies in empirical models defined on measurement scenarios even when there is local consistency. In the present work, we take a step further to treat the scenarios in full relational language by using knowledge operators, thus showing that trust is equivalent to the Truth Axiom in these cases. A translation of measurement scenarios into multi-agent scenarios by using the topological semantics of multi-modal logic is constructed, demonstrating that logical contextuality can be understood as the violation of soundness by supposing mutual knowledge. To address the contradictions, assuming distributed knowledge is considered, which eliminates such violations but at the cost of lambda-dependence. We conclude by translating the main examples of multi-agent scenarios to their empirical model representation, contextuality is identified as the cause of their contradictory results.

quant-ph↗

arXiv2021

Characterization of Contextuality with Semi-Module Čech Cohomology and its Relation with Cohomology of Effect Algebras

I present a generalized notion of obstruction in Čech cohomology on semi-modules, which allows one to characterize non-disturbing contextual behaviors with any semi-field. This framework generalizes the usual Čech cohomology used in the sheaf approach to contextuality, avoiding false positives. I revise a similar work done in the framework of effect algebras with cyclic and order cohomologies and explore the relationship with the one presented here.

math.RA↗