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Postquantum Br\`{e}gman relative entropies and nonlinear resource theories

Ryszard Paweł Kostecki

Published Oct 5, 2017• 2 citations
OpenAlex

Abstract

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.

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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 Lp_pp​ spaces over semi-finite JBW-algebras, and noncommutative Lp_pp​ 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 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.
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.
In this paper we: 1) show how the smooth geometry of spaces of normal quantum states over W*-algebras (generalised spaces of density matrices) may be used to substantially enrich the description of quantum dynamics in the algebraic and path integral approaches; 2) discuss a framework for postquantum information theories, such that quantum mechanical linearity holds only locally, while the nonlocal multi-user dynamics exhibits some similarity with general relativity. In the algebraic setting, we propose a method of incorporating nonlinear Poisson and relative entropic local dynamics, as well as local gauge and local source structures, into an effective description of local temporal evolution of quantum states by using fibrewise perturbations of liouvilleans in the fibre bundle of Hilbert spaces. In the path integral setting, we incorporate local geometry by a generalisation of the Daubechies--Klauder coherent state phase space propagator formula. Finally, we discuss the role of Bregman relative entropy in the Jaynes--Mitchell--Favretti renormalisation scheme. As a result, we show that: 1) the propagation of quantum particles (in Wigner's sense) can be naturally explained as a free fall along the trajectories locally minimising the quantum relative entropy; 2) the contribution of particular trajectories to the global path integral is weighted by the local quantum entropic prior, measuring user's lack of information; 3) the presence of nonlinear quantum control variables results in the change of the curvature of the global quantum state space; 4) the behaviour of zero-point energy under renormalisation of local entropic dynamics is maintained by local redefinition of information mass (prior), which encodes the curvature change. This leads us to a new framework for nonequilibrium quantum statistical mechanics based on quantum Orlicz spaces, quantum Bregman distances and Banach Lie algebras.
Introduction The aim of information geometry is to introduce a suitable geometrical structure on families of probability distributions or quantum states. For parametrised statistical models, such structure is based on two fundamental notions: the Fisher information and the exponential family with its dual mixed parametrisation, see for example (Amari 1985, Amari and Nagaoka 2000). For the non-parametric situation, the solution was given by Pistone and Sempi (Pistone and Sempi 1995, Pistone and Rogantin 1999), who introduced a Banach manifold structure on the set P of probability distributions, equivalent to a given one. For each µ ∈ ρ, the authors considered the non-parametric exponential family at µ. As it turned out, this provides a C ∞ -atlas on ρ, with the exponential Orlicz spaces Lφ(µ) as the underlying Banach spaces, here φ is the Young function of the form φ(x) = cosh(x) — 1. The present contribution deals with the case of quantum states: we want to introduce a similar manifold structure on the set of faithful normal states of a von Neumann algebra M. Since there is no suitable definition of a non-commutative Orlicz space with respect to a state φ, it is not clear how to choose the Banach space for the manifold. Of course, there is a natural Banach space structure, inherited from the predualM. But, as it was already pointed out in (Streater 2004), this structure is not suitable to define the geometry of states: for example, any neighbourhood of a state φ contains states such that the relative entropy with respect to φ is infinite.
Dynamic quantum resource theories study the manipulation of quantum channels by means of a restricted set of free superoperations. In this paper, we formulate general dynamic resource theories using a "top-down" framework, and we provide systematic characterizations for closed and convex resource theories from both information-theoretic and operational perspectives. Our results are summarized as follows. First, we propose and investigate a branch of resource-induced measures of uncertainty, called the free conditional min-entropy (FCME), generalizing the conditional min-entropy and its dynamic extension to scenarios where information processing is subject to variable operational restriction. We provide a complete set of entropic conditions in terms of the FCME for characterizing channel convertibility via free superoperations in any closed and convex resource theory. We also find that the resource global robustness of channels can be equivalently cast as a mutual-information-like quantity derived from the FCME, thereby offering the resource global robustness an information-theoretic interpretation. Apart from the entropic approach, we also study closed and convex resource theories in the contexts of various operational tasks. These tasks are formulated such that each of them induces a complete set of operationally meaningful resource monotones, and therefore they can be used to faithfully test free convertibility between channels. We also systematically study the quantitative relations between the operational advantage of channels in these tasks and the resource robustness measures of channels. In particular, we prove that every well-defined robustness-based measure can be operationally interpreted as some kind of advantage in a task called semiquantum partial preprocessing. Ultimately, our results provide both entropic and operational characterizations for general dynamic quantum resources with a closed and convex structure.
We discuss new approach to mathematical foundations of quantum theory, which is completely independent of Hilbert spaces and measure spaces. New kinematics is defined by nonlinear geometry of spaces of integrals on abstract non-commutative algebras. New dynamics is defined by constrained maximisation of quantum relative entropy. We recover Hilbert space based approach (including unitary evolution and the von Neumann-Lüders rule) and measure theoretic approach to probability theory (including Bayes' rule) as special cases of our approach.