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arXiv:1709.07807 · Theory and applications of categories

Information structures and their cohomology

Juan Pablo Vigneaux

Published Jan 1, 2020• 3 citations
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Abstract

We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings.Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement.We extend Baudot and Bennequin's definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature.Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions.The only 1-cocycles are Shannon entropy or Tsallis α-entropy, depending on the value of the parameter.

DOI
10.70930/tac/o4v3tux5

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We introduce the category of information structures, whose objects are suitable diagrams of measurable sets that encode the possible outputs of a given family of observables and their mutual relationships of refinement; they serve as mathematical models of contextuality in classical and quantum settings.Each information structure can be regarded as a ringed site with trivial topology; the structure ring is generated by the observables themselves and its multiplication corresponds to joint measurement.We extend Baudot and Bennequin's definition of information cohomology to this setting, as a derived functor in the category of modules over the structure ring, and show explicitly that the bar construction gives a projective resolution in that category, recovering in this way the cochain complexes previously considered in the literature.Finally, we study the particular case of a one-parameter family of coefficients made of functions of probability distributions.The only 1-cocycles are Shannon entropy or Tsallis α-entropy, depending on the value of the parameter.
Baudot and Bennequin introduced a cohomological construction adapted to information theory, called \emph{information cohomology}; it characterizes Shannon entropy as the only nontrivial cohomology class in degree one, for suitable coefficients. This text develops the relation between information cohomology and topos theory. To this end, we redefine information structures, as categories of random variables related by a notion of extension or refinement; classical and quantum probability spaces appear as models (or representations) of them. These generalized information structures form a category with finite products and coproducts. Information cohomology is a derived functor in the abelian category of modules over an algebra of random variables, a subcategory of the presheaf topos associated with the information structure. The cohomology is invariant under isomorphisms of generalized structures. We prove that the relatively free bar construction gives a projective object for the computation of this invariant. We provide detailed computations of H0H^0H0 and H1H^1H1 when the coefficients are functionals of classical probabilities, including the degenerate cases. In the process, we also establish the homological nature of Tsallis entropies, as the only cocycles in degree one for certain one-parameter family of modules.
We propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways. The article is divided into two parts, that can be read independently. In the first part, the introduction, we provide an overview of the results, some open questions, future results and lines of research, and discuss briefly the application to complex data. In the second part we give the complete definitions and proofs of the theorems A, C and E in the introduction, which show why entropy is the first homological invariant of a structure of information in four contexts: static classical or quantum probability, dynamics of classical or quantum strategies of observation of a finite system.
This paper is an attempt to give a Quantum theory of Mathematical integrated information theory which is mathematical version of integrated information theory by Masafumi Oizumi, Larissa Albantakis, Giulio Tononi [2]. Using the definitions of Classical Mathematical Integrated Information Theory [1]. And considering that the Quantum theory is given by the functor which maps from a category whose objects is topology to a linear category whose objects are Hilbert spaces indexed with the objects from previous category. Also, we will be using the definition of conditional density matrix to define repertoire. We will be also discussing the relationship between consciousness and entanglement.
The data of a physical experiment can be represented as a presheaf of probability distributions.A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a joint probability distribution, a celebrated observation due to Bell.Such distributions are called contextual.Simplicial distributions are combinatorial models that extend presheaves of probability distributions by elevating sets of measurements and outcomes to spaces.Contextuality can be defined in this generalized setting.This paper introduces the notion of convex categories to study simplicial distributions from a categorical perspective.Simplicial distributions can be given the structure of a convex monoid, a convex category with a single object, when the outcome space has the structure of a group.We describe contextuality as a monoid-theoretic notion by introducing a weak version of invertibility for monoids.Our main result is that a simplicial distribution is noncontextual if and only if it is weakly invertible.Similarly, strong contextuality and contextual fraction can be characterized in terms of invertibility in monoids.Finally, we show that simplicial homotopy can be used to detect extremal simplicial distributions refining the earlier methods based on Čech cohomology and the cohomology of groups.The authors would like to thank the anonymous referee for useful comments that significantly improved the exposition of the paper, and Walker H. Stern and Redi Haderi for enlightening discussions.
Quantum mechanics exhibits many counter-intuitive features, which challenge our classical understanding of the world. This makes the theory simultaneously difficult to grasp, and a source of great potential for new technologies. We study various characteristics of classical and quantum mechanics and their respective roles in computation. This is interesting from a philosophical perspective, as it brings us a small step closer to understanding the nature of quantum mechanics. Furthermore, it leads us to discover scenarios where quantum resources provide a genuine advantage over classical resources, as well as more intuitive ways to model quantum protocols. Firstly, we explore the concept of quantum contextuality, which distinguishes uniquely-quantum experimental outcomes from outcomes that can be realised classically. In pursuit of a universal theory of contextuality, we unify two abstract frameworks: the sheaf-theoretic approach to contextuality by Abramsky and Brandenburger and the equivalence-based approach by Spekkens. We apply the sheaf-theoretic formalism to the field of communication complexity. We derive a bound on the success probability of a communication complexity protocol, based on the amount of contextuality present in its quantum resource. We provide a set of necessary conditions for distributed functions that allow a contextual advantage. Secondly, we consider a family of bicategorical models to describe the interaction of classical and quantum information in quantum protocols. We prove that this family is symmetric monoidal, and therefore suitable for describing compound and parallel protocols. This family serves as a mathematical foundation for the planar language for quantum protocols by Vi- cary. Important examples of this family are the bicategories 2Hilb and 2Vect. Additionally, we introduce the structure 2[CP∗[FHilb]], which contains a symmetric monoidal bicategory of mixed states and completely positive maps. This allows a unified description of quantum teleportation and classical encryption in a single 2-category, as well as a universal security proof applicable simultaneously to both scenarios. As part of the process, we develop extensive categorical machinery for constructing symmetric monoidal structures of bicategories, pseudo functors, pseudo transformations and modifications. We consider a method of constructing symmetric monoidal bicategories from symmetric monoidal double categories. We show that this method can be generalised to a functor from the locally cubical bicategory of monoidal double categories to the locally cubical bicategory of monoidal bicategories.
We introduce a new framework for contextuality based on simplicial sets, combinatorial models of topological spaces that play a prominent role in modern homotopy theory. Our approach extends measurement scenarios to consist of spaces (rather than sets) of measurements and outcomes, and thereby generalizes nonsignaling distributions to simplicial distributions, which are distributions on spaces modeled by simplicial sets. Using this formalism we present a topologically inspired new proof of Fine's theorem for characterizing noncontextuality in Bell scenarios. Strong contextuality is generalized suitably for simplicial distributions, allowing us to define cohomological witnesses that extend the earlier topological constructions restricted to algebraic relations among quantum observables to the level of probability distributions. Foundational theorems of quantum theory such as the Gleason's theorem and Kochen--Specker theorem can be expressed naturally within this new language.