arXiv subjects

Ryszard Paweł Kostecki

Publications and source records attributed to Ryszard Paweł Kostecki.

13 recordsLinked to original sources

arXiv2024

Generalised Brègman relative entropies: a brief introduction

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.

math-ph↗

arXiv2021

Categories of Brègman operations and epistemic (co)monads

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.

math-ph↗

arXiv2018

Noncommutative Orlicz spaces over W*-algebras

Using the Falcone--Takesaki theory of noncommutative integration and Kosaki's canonical representation, we construct a family of noncommutative Orlicz spaces that are associated to an arbitrary W*-algebra without any choice of weight involved, and we show that this construction is functorial over the category of W*-algebras with *-isomorphisms as arrows. Under a choice of representation, these spaces are isometrically isomorphic to Kunze's noncommutative Orlicz spaces over crossed products.

math.OA↗

arXiv2017

Equivalence of tensor products over a category of W*-algebras

We prove the equivalence of two tensor products over a category of W*-algebras with normal (not necessarily unital) *-homomorphisms, defined by Guichardet and Dauns, respectively. This structure differs from the standard tensor product construction by Misonou--Takeda--Turumaru, which is based on weak topological completion, and does not have a categorical universality property.

math-ph↗

arXiv2016

Local quantum information dynamics

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.

quant-ph↗

arXiv2016

Quantum collapse rules from the maximum relative entropy principle

We show that the von Neumann--Lueders collapse rules in quantum mechanics always select the unique state that maximises the quantum relative entropy with respect to the premeasurement state, subject to the constraint that the postmeasurement state has to be compatible with the knowledge gained in the measurement. This way we provide an information theoretic characterisation of quantum collapse rules by means of the maximum relative entropy principle.

quant-ph↗

arXiv2014

W*-algebras and noncommutative integration

This text is a detailed overview of the theories of W*-algebras and noncommutative integration, up to the Falcone-Takesaki theory of noncommutative Lp spaces over arbitrary W*-algebras, and its extension to noncommutative Orlicz spaces. The topics under consideration include the Tomita-Takesaki modular theory, the relative modular theory (featuring bimodules, spatial quotients, and canonical representation), the theory of W*-dynamical systems (featuring derivations, liouvilleans, and crossed products), noncommutative Radon-Nikodym type theorems, and operator valued weights. We pay special attention to abstract algebraic formulation of all properties (avoiding the dependence on Hilbert spaces wherever it is possible), to functoriality of canonical structures arising in the theory, and to the relationship between commutative and noncommutative integration theories. Several new results are proved.

math.OA↗

arXiv2014

Lüders' and quantum Jeffrey's rules as entropic projections

We prove that the standard quantum mechanical description of a quantum state change due to measurement, given by Lueders' rules, is a special case of the constrained maximisation of a quantum relative entropy functional. This result is a quantum analogue of the derivation of the Bayes--Laplace rule as a special case of the constrained maximisation of relative entropy. The proof is provided for the Umegaki relative entropy of density operators over a Hilbert space as well as for the Araki relative entropy of normal states over a W*-algebra. We also introduce a quantum analogue of Jeffrey's rule, derive it in the same way as above, and discuss the meaning of these results for quantum bayesianism.

quant-ph↗

arXiv2012

On principles of inductive inference

We propose an intersubjective epistemic approach to foundations of probability theory and statistical inference, based on relative entropy and category theory, and aimed to bypass the mathematical and conceptual problems of existing foundational approaches.

math.ST↗

arXiv2012

Quantum Schwarzschild space-time

Using new approach to construction of space-times emerging from quantum information theory, we identify the space of quantum states that generates the Schwarzschild space-time. No quantisation procedure is used. The emergent space-time is obtained by the Poincaré-Wick rotation and Fronsdal embedding of certain submanifold of the riemanian manifold of six-dimensional strictly positive matrices with the Bogolyubov-Kubo-Mori metric.

gr-qc↗

arXiv2011

Information dynamics and new geometric foundations of quantum theory

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 non-linear 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.

quant-ph↗

arXiv2011

Quantum theory as inductive inference

We present the elements of a new approach to the foundations of quantum theory and probability theory which is based on the algebraic approach to integration, information geometry, and maximum relative entropy methods. It enables us to deal with conceptual and mathematical problems of quantum theory without any appeal to frameworks of Hilbert spaces and measure spaces.

math-ph↗

arXiv2011

The general form of gamma-family of quantum relative entropies

We use the Falcone-Takesaki non-commutative flow of weights and the resulting theory of non-commutative Lp spaces in order to define the family of relative entropy functionals that naturally generalise the quantum relative entropies of Jencova-Ojima and the classical relative entropies of Zhu-Rohwer, and belong to an intersection of families of Petz relative entropies with Bregman relative entropies. For the purpose of this task, we generalise the notion of Bregman entropy to the infinite-dimensional non-commutative case using the Legendre-Fenchel duality. In addition, we use the Falcone-Takesaki duality to extend the duality between coarse-grainings and Markov maps to the infinite-dimensional non-commutative case. Following the recent result of Amari for the Zhu-Rohwer entropies, we conjecture that the proposed family of relative entropies is uniquely characterised by the Markov monotonicity and the Legendre-Fenchel duality. The role of these results in the foundations and applications of quantum information theory is discussed.

quant-ph↗