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The Fokker-Planck equation is a partial differential equation which is a key ingredient in many models in physics. This paper aims to obtain a quantum counterpart of Fokker-Planck dynamics, as a means to describing quantum Fokker-Planck dynamics. Given that relevant models relate to the description of large systems, the quantization of the Fokker-Planck equation should be done in a manner that respects this fact, and is therefore carried out within the setting of non-commutative analysis based on general von Neumann algebras. Within this framework we present a quantization of the generalized Laplace operator, and then go on to incorporate a potential term conditioned to noncommutative analysis. In closing we then construct and examine the asymptotic behaviour of the corresponding Markov semigroups. We also present a noncommutative Csiszar-Kullback inequality formulated in terms of a notion of relative entropy, and show that for more general systems, good behaviour with respect to this notion of entropy ensures similar asymptotic behaviour of the relevant dynamics., Comment: The final version submitted to AHP. A brief account of applied quantization as well as the comprehensive description of closability of quantum Laplacian is added

In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac's formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theory., Comment: Presentation improved. In particular: the new Theorem 5.6, new Section 6, and some references are added

The Radon-Nikodym formalism is used to study the structure of the set of positive maps from $\mathcal{B}(\mathcal{H})$ into itself, where $\mathcal{H}$ is a finite dimensional Hilbert space. In particular, this formalism was employed to formulate simple criteria which ensure that certain maps are non decomposable. In that way, a recipe for construction of non decomposable maps was obtained., Comment: Incorrect example given in subsection 5.5 was removed

The aim of this work is to firstly demonstrate the efficacy of the recently proposed Orlicz space formalism for Quantum theory \cite{ML}, and secondly to show how noncommutative differential structures may naturally be incorporated into this framework. To start off with we specifically propose regularity conditions which in the context of local algebras corresponding to Minkowski space, ensure good behaviour of field operators as observables, and then show that fields obtained by the Osterwalder-Schrader reconstruction theorem are regular in this sense. The pair of Orlicz spaces we explicitly use for this purpose, are respectively built on the exponential function and on an entropic type function. This formalism has been shown to be well suited to a description of quantum statistical mechanics, and in the present work we show that it is also a very useful and elegant tool for Quantum Field Theory. We then introduce the class of tangentially conditioned algebras, which is a large class of local algebras corresponding to globally hyperbolic Lorentzian manifolds that locally ``look like'' the local algebras of Minkowski space. On the one hand this ensures that at a local level, the Orlicz space formalism discussed above is also relevant for a much more general class of local algebras. On the other hand, the structure of this class of algebras, allows for the development of a non-commutative differential geometric structure along the lines of the du Bois-Violette approach to such a theory. In this way we obtain a complete depiction: integrable structures based on local algebras provide a static setting for an analysis of Quantum Field Theory and an effective tool for describing regular behaviour of field operators, whereas differentiable structures posit indispensable tools for a description of equations of motion., Comment: Amended version. In particular, the statement that fields obtained by the Osterwalder-Schrader reconstruction are regular has been studied in detail

These notes are intended as an introduction to a study of applications of noncommutative calculus to quantum statistical Physics. Centered on noncommutative calculus we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems, and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite dimensional structures. As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory and quantum correlations are discussed., Comment: Improved presentation. Few changes

Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [W. A. Majewski, L.E. Labuschagne, Ann. H. Poincare. 15, 1197-1221, (2014)] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces $\langle L^{\cosh - 1}, L\log(L+1)\rangle$. The present paper therefore in some sense "completes" the picture by showing that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition, canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation technique, specifically suited to the above context, for extending the action of such maps to the appropriate intermediate spaces of the pair $\langle L^\infty,L^1\rangle$. Moreover, it is shown that quantum dynamics in the form of Markov semigroups described by some Dirichlet forms naturally extends to the context proposed in [W. A. Majewski, L.E. Labuschagne, Ann. H. Poincare. 15, 1197-1221, (2014)].

We review a new formalism based on Orlicz spaces for the description of large regular statistical systems. Our presentation includes both classical and quantum systems. The presented approach has the advantage that statistical mechanics is much better settled., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1302.3460

This survey gives a comprehensive account of quantum correlations understood as a phenomenon stemming from the rules of quantization. Centered on quantum probability it describes the physical concepts related to correlations (both classical and quantum), mathematical structures, and their consequences. These include the canonical form of classical correlation functionals, general definitions of separable (entangled) states, definition and analysis of quantumness of correlations, description of entanglement of formation, and PPT states. This work is intended both for physicists interested not only in collection of results but also in the mathematical methods justifying them, and mathematicians looking for an application of quantum probability to concrete new problems of quantum theory., Comment: Revised version, Minor improvements. Typos fixed

We present a new rigorous approach based on Orlicz spaces for the description of the statistics of large regular statistical systems, both classical and quantum. This approach has the advantage that statistical mechanics is much better settled. In particular, a new kind of renormalization leading to states having a well defined entropy function is presented., Comment: 20 pages

A detailed characterization of PPT states, both in the Heisenberg and in the Schr\"odinger picture, is given. Measures of entanglement are defined and discussed in details. Illustrative examples are provided., Comment: We would be grateful for remarks

Let $\A$ ($\cM$) be a $C^*$-algebra (a von Neumann algebra respectively). By a quantum dynamical system we shall understand the pair $({\A}, T)$ ($({\cM}, T)$) where $T : {\A} \to {\A}$ ($T : {\cM} \to {\cM}$) is a linear, positive (normal respectively), and identity preserving map. In our lecture, we discuss how the techniques of quantum Orlicz spaces may be used to study quantum dynamical systems. To this end, we firstly give a brief exposition of the theory of quantum dynamical systems in quantum $L_p$ spaces. Secondly, we describe the Banach space approach to quantization of classical Orlicz spaces. We will discuss the necessity of the generalization of $L_p$-space techniques. Some emphasis will be put on the construction of non-commutative Orlicz spaces. The question of lifting dynamical systems defined on von Neumann algebra to a dynamical system defined in terms of quantum Orlicz space will be discussed., Comment: 11 pages

We present a partial characterization of matrices in $M_n(\cA)^+$ satisfying the St{\o}rmer condition., Comment: 5 pages

We present two different descriptions of positive partially transposed (PPT) states. One is based on the theory of positive maps while the second description provides a characterization of PPT states in terms of Hilbert space vectors. Our note is based on our previous results., Comment: a report based on previous results

The new arguments based on Majorana fermions indicating that non-completely positive maps can describe open quantum evolution are presented., Comment: published; small changes

We extend the concept of implementability of semigroups of evolution operators associated with dynamical systems to quantum case. We show that such an extension can be properly formulated in terms of Jordan morphisms and isometries on non-commutative $L^p$ spaces. We focus our attention on a non-commutative analog of the Banach-Lamperti theorem.

We outline the scheme for quantization of classical Banach space results associated with some prototypes of dynamical maps and describe the quantization of correlations as well. A relation between these two areas is discussed.

We present a discussion on local quantum correlations and their relations with entanglement. We prove that vanishing coefficient of quantum correlations implies separability. The new results on locally decomposable maps which we obtain in the course of proof also seem to be of independent interest., Comment: The paper completes the proof of Proposition 3 given in: W. A. Majewski, On entanglement of states and quantum correlations, in Operator algebras and Mathematical Physics, Eds. J.M. Combes, J. Cuntz, G.A. Elliott, G. Nenciu, H. Siedentop, S. Stratila; Theta, Bucharest, 2003. e-print, LANL math-ph/0202030. The paper will be published in Lett. Math. Phys

In the paper we present the new approach to Kolmogorov-Sinai entropy and its quantization. Our presentation stems from an application of the Choquet theory to the theory of decompositions of states and therefore, it resembles our rigorous description of entanglement of formation, Comment: 10 pages

In this paper we present the novel qualities of entanglement of formation for general (so also infinite dimensional) quantum systems and we introduce the notion of coefficient of quantum correlations. Our presentation stems from rigorous description of entanglement of formation., Comment: 12 pages

A model of evolution of bipartite quantum state entanglement is studied. It involves recently introduced quantum block spin-flip dynamics on a lattice. We find that for initially separable states the considered evolution leads, in general, to entangled states. We also present a complete characterization of two-point correlation functions for that type of dynamics to confirm enhancement of quantum correlations for the considered system., Comment: 15 pages. No figures. Amended version with simplifications. Comments welcome

We shortly review the progress in the domain of deterministic chaos for quantum dynamical systems. With the appropriately extended definition of quantum Lyapunov exponent we analyze various quantum dynamical maps. It is argued that, within Quantum Mechanics, irregular evolution for properly chosen observables can coexist with regular and predictable evolution of states., Comment: Latex, 28 pages