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arXiv:2103.07810

Categories of Brègman operations and epistemic (co)monads

Ryszard Paweł Kostecki

Published Mar 13, 2021• 1 citation

Abstract

We construct a categorical framework for nonlinear postquantum inference, with embeddings of convex closed sets of suitable reflexive Banach spaces as objects and pullbacks of Brègman quasi-nonexpansive mappings (in particular, constrained maximisations of Brègman relative entropies) as morphisms. It provides a nonlinear convex analytic analogue of Chencov's programme of geometric study of categories of linear positive maps between spaces of states, a working model of Mielnik's nonlinear transmitters, and a setting for nonlinear resource theories (with monoids of Brègman quasi-nonexpansive maps as free operations, their asymptotic fixed point sets as free sets, and Brègman relative entropies as resource monotones). We construct a range of concrete examples for semi-finite JBW-algebras and any W*-algebras. Due to relative entropy's asymmetry, all constructions have left and right versions, with Legendre duality inducing categorical equivalence between their well-defined restrictions. Inner groupoids of these categories implement the notion of statistical equivalence. The hom-sets of a subcategory of morphisms given by entropic projections have the structure of partially ordered commutative monoids (so, they are resource theories in Fritz's sense). Further restriction of objects to affine sets turns Brègman relative entropy into a functor. Finally, following Lawvere's adjointness paradigm for deductive logic, but with a semantic twist representing Jaynes' and Chencov's views on statistical inference, we introduce a category-theoretic multi-(co)agent setting for inductive inference theories, implemented by families of monads and comonads. We show that the brègmanian approach provides some special cases of this setting.

DOI: 10.48550/arxiv.2103.07810

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We introduce the family of postquantum Bregman relative entropies, based on nonlinear embeddings into reflexive Banach spaces (with examples given by reflexive noncommutative Orlicz spaces over semi-finite W*-algebras, nonassociative L

_p

spaces over semi-finite JBW-algebras, and noncommutative L

_p

spaces over arbitrary W*-algebras). This allows us to define a class of geometric categories for nonlinear postquantum inference theory (providing an extension of Chencov's approach to foundations of statistical inference), with constrained maximisations of Bregman relative entropies as morphisms and nonlinear images of closed convex sets as objects. Further generalisation to a framework for nonlinear convex operational theories is developed using a larger class of morphisms, determined by Bregman nonexpansive operations (which provide a well-behaved family of Mielnik's nonlinear transmitters). As an application, we derive a range of nonlinear postquantum resource theories determined in terms of this class of operations.

We present some basic elements of the theory of generalised Brègman relative entropies over nonreflexive Banach spaces. Using nonlinear embeddings of Banach spaces together with the Euler--Legendre functions, this approach unifies two former approaches to Brègman relative entropy: one based on reflexive Banach spaces, another based on differential geometry. This construction allows to extend Brègman relative entropies, and related geometric and operator structures, to arbitrary-dimensional state spaces of probability, quantum, and postquantum theory. We give several examples, not considered previously in the literature.

From every pair of adjoint functors it is possible to produce a (possibly trivial) equivalence of categories by restricting to the subcategories where the unit and counit are isomorphisms. If we do this for the adjunction between effect algebras and abstract convex sets, we get the surprising result that the equivalent subcategories consist of reflexive order-unit spaces and reflexive base-norm spaces, respectively. These are the convex sets that can occur as state spaces in generalized probabilistic theories satisfying both the no-restriction hypothesis and its dual. The linearity of the morphisms is automatic. If we add a compact topology to either the states or the effects, we can obtain a duality for all Banach order-unit spaces or all Banach base-norm spaces, but not both at the same time.

We exhibit a functor from the category OUS of order unit spaces and positive, unit-preserving mappings into the category $\Prob$ of probabilistic models (test spaces with designated state spaces) and morphisms thereof. Restricted to any subcategory of OUS monoidal with respect to a positive, normalized, bilinear composition rule, our functor is also monoidal. This shows that the convex-operational approach to physical theories can be subsumed by the test-space approach, without resort to ``generalized test spaces''. A second construction, equipping a probabilistic model with tests representing ``weighted coins'', also sheds light on the nature of unsharp observables.

This paper charts a very direct path between the categorical approach to quantum mechanics, due to Abramsky and Coecke, and the older convex-operational approach based on ordered vector spaces (recently reincarnated as "generalized probabilistic theories"). In the former, the objects of a symmetric monoidal category C are understood to represent physical systems and morphisms, physical processes. Elements of the monoid C(I,I) are interpreted somewhat metaphorically as probabilities. Any monoid homomorphism from the scalars of a symmetric monoidal category C gives rise to a covariant functor V_o from C to a category of dual-pairs of ordered vector spaces. Specifying a natural transformation u from V_o to 1 (where 1 is the trivial such functor) allows us to identify normalized states, and, thus, to regard the image category V_o(C) as consisting of concrete operational models. In this case, if A and B are objects in C, then V_o(A x B) defines a non-signaling composite of V_o(A) and V_o(B). Provided either that C satisfies a "local tomography" condition, or that C is compact closed, this defines a symmetric monoidal structure on the image category, and makes V_o a (strict) monoidal functor.

We construct a 2-categorical extension of the relative entropy functor of Baez and Fritz, and show that our construction is functorial with respect to vertical morphisms. Moreover, we show such a ‘2-relative entropy’ satisfies natural 2-categorial analogues of convex linearity, vanishing under optimal hypotheses, and lower semicontinuity. While relative entropy is a relative measure of information between probability distributions, we view our construction as a relative measure of information between channels.