This paper is intended to: 1) show how the local 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) provide a framework for construction of quantum information theories beyond quantum mechanics, such that 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 over the quantum state manifold. We apply this method to construct also an exact algebraic generalisation of Savvidou's action operator. In the path integral setting, motivated by the Savvidou--Anastopoulous analysis of the role of Kähler space geometry in the Isham--Linden quantum histories, we propose to incorporate local geometry by means of a generalisation of the Daubechies--Klauder coherent state phase space propagator formula. Finally, we discuss the role of Brègman relative entropy in the Jaynes--Mitchell--Favretti renormalisation scheme. Using these tools 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. We conclude this work with a proposal of a new framework for nonequilibrium quantum statistical mechanics based on quantum Orlicz spaces, quantum Brègman distances and Banach Lie algebras.
Using the Falcone--Takesaki theory of noncommutative integration, we construct a family of noncommutative Orlicz spaces that are canonically 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.
We prove that the standard quantum mechanical description of a quantum state change due to measurement, given by Lüders' 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.
We show that the von Neumann--Lüders 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.
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--Nikodým 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.
This text is a detailed overview of the quantum and classical information geometry, containing several new concepts and some new results. The key role played by the relative entropy is exposed, and the interconnections between various structures are analysed. We consider the convex/variational nonsmooth part of the theory on the equal footing with the smooth/infinitesimal part, and we also consider the duality principles (as embodied in the concepts of Brègman relative entropy and dually flat smooth geometries) on the equal footing with monotonicity under quantum channels. All results are spelled out with the maximal available generality, so the functional analytic setting of W*-algebras and Banach spaces is widely used.
The purpose of this text is to equip the reader with an intuitive but precise understanding of elementary structures of category and topos theory. In order to achieve this goal, we provide a guided tour through category theory, leading to the definition of an elementary (Lawvere--Tierney) topos. Then we turn to the investigation of consequences of this definition. In particular, we analyse in detail the topos Set2op, the internal structure of its subobject classifier and its variation over stages. Next we turn to the discussion of the interpretation of a logic and language in topos, viewed as a model of higher order intuitionistic multisort type theory, as well as the geometric perspective on a topos, viewed as a category of set-valued sheaves over base category equipped with a Grothendieck topology. This text is designed as an elementary introduction, written in a self-contained way, with no previous knowledge required.
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.
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.
On principles of inductive inference
in: Goyal P. et al (eds.), Proceedings of the 31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 9–16 July 2011, Waterloo, Canada, AIP Conf. Proc. 1443 (2012), 22-31.
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.
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 Jenčová--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.
Quantum theory as inductive inference
extended version of a paper published in: Mohammad-Djafari A., Bercher J.-F., Bessière P. (eds.), Proceedings of the 30th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, AIP Conf. Proc. 1305 (2010), 28-35.
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.
Cykl artykułów Na przestrzeni dziejów: Jak liczyli Babilończycy?, Jak liczyli Chińczycy?, Jak liczyli Majowie?, Jak liczyli starożytni Hindusi?, Jak liczyli starożytni Egipcjanie?, Matematyka375, 29-30; 376, 40-41; 377, 38-39; 378, 25-27; 379, 12-14 (2011) (są to framenty tekstu Krótka historia matematyki)
04.05.2016 - Towards (post)quantum information relativity, Quantum Foundations Seminar, Perimeter Institute for Theoretical Physics, Waterloo.
30.06.2015 - Operator-algebraic quantum foundations revisited, 34 Workshop on Geometric Methods in Physics, University of Białystok, Białowieża.
24.06.2015 - Towards geometric (and nonlinear) quantum information theory, National Quantum Information Centre, Sopot.
22.06.2015 - Is mathematics the vanguard of culture? - Some sketches on anthropology of mathematics in the context of category theory, Polish Academy of Arts and Sciences, Commission for Philosophy of Science, Kraków.
21.05.2015 - Quantum information geometric approach to quantum foundations, Underground seminar on open problems in quantum foundations, Perimeter Institute, Waterloo.
18.07.2011 - Categories of quantum theoretic models, International Category Theory Conference CT2011, University of British Columbia, Vancouver.
12.07.2011 - Information geometric foundations of quantum theory, Quantum Foundations Seminar, Perimeter Institute, Waterloo.
11.07.2011 - Information geometric foundations of quantum theory (poster), 31th International Workshop on Bayesian Inference and Maximum Entropy Methods, Perimeter Institute, Waterloo.
07.07.2011 - The emergence of space-time geometry from quantum theory, Quantum Gravity Group Meeting, Perimeter Institute, Waterloo.
29.06.2011 - Quantum entropy and information geometry based on non-commutative integration, 30th Workshop of Geometric Methods in Physics, University of Białystok, Białowieża.
24.06.2011 - Quantum theory and space-time from quantum information geometry, Underground seminar on open problems in quantum foundations, University of Warsaw, Warszawa.
22.06.2011 - New results on quantum relative entropy and quantum information geometry, 43th Symposium on Mathematical Physics, Institute of Physics, Uniwersytet Toruński, Toruń.
16.06.2011 - Information geometric foundations of quantum theory, Foundations of Probability and Physics 6, Linnaeus University, Växjö.
31.03.2011 - Algebraic approach to quantum theory, Underground seminar on open problems in quantum foundations, University of Warsaw, Warszawa.
17.02.2011 - Esquisse d'un programme, Underground seminar on open problems in quantum foundations, University of Warsaw, Warszawa.
08.10.2010 - Riemannian geometry on the spaces of quantum states, Exact Results in Quantum and Gravity Seminar, Institute for Theoretical Physics, Warszawa.
02.08.2010 - Quantum information geometry and non-commutative flow of weights (poster), Information Geometry and Applications III, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig.
04.07.2010 - Quantum theory as inductive inference, 30th International Workshop on Bayesian Inference and Maximum Entropy Methods, Centre National de la Recherche Scientifique, Chamonix.
22.06.2010 - Information dynamics and geometric foundations of quantum theory, 42th Symposium on Mathematical Physics, Institute of Physics, Uniwersytet Toruński, Toruń.
03.04.2009 - Riemannian and dual geometries on the spaces of probabilistic measures, Exact Results in Quantum and Gravity Seminar, Institute for Theoretical Physics, Warszawa.
22.07.2008 - Differential geometry and smooth analysis in toposes, Underground seminar on open problems in quantum foundations, Institute for Theoretical Physics, Warszawa.
07.05.2008 - The physical meaning of the modular theory and the measurement problem, 4th Quantum Gravity Colloquium, Albert-Einstein-Institute, Golm.
Jacek Kaczmarski - Stalker // translated to English by Jadwiga Smulko & Ryszard P. Kostecki (2014)
Black Hole Stalker(II.2016) // The world's first shamanic meditation music based on the real sound of merging black holes feat.: Jessy Cerritos on bass & LIGO's registration of gravitational wave signal of a black hole binary system
(check out also the paper)