Abstract: Abstract
Using the Grothendieck approach to the tensor product of locally convex spaces, we review a characterization of positive maps as well as Belavkin-Ohya characterization of PPT states. Moreover, within this scheme, a generalization of the idea of Choi matrices for genuine quantum systems will be presented. PubDate: 2014-07-01

Abstract: Abstract
In this note we describe algorithms for obtaining formulae for transformations of measures on infinite dimensional topological vector spaces or manifolds, generated by transformations of the domains of the measures and by transformations of the range. PubDate: 2014-07-01

Abstract: Abstract
In classical information theory, one of the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov-Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes-Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy are done [10, 3, 8, 39, 15, 44, 9, 27, 28, 2, 19, 45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S- mixing entropy by Ohya [26, 27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S- mixing entropy. In this paper, we discuss the efficiency of information transmission to calculate the mean mutual entropy with respect to the modulated initial states and the connected channel for the quantum dynamical systems. PubDate: 2014-07-01

Abstract: Abstract
In this paper, we formulate limit Zeno dynamics of general open systems as the adiabatic elimination of fast components. We are able to exploit previous work on adiabatic elimination of quantum stochastic models to give explicitly the conditions under which open Zeno dynamics will exist. The open systems formulation is further developed as a framework for Zeno master equations, and Zeno filtering (that is, quantum trajectories based on a limit Zeno dynamical model). We discuss several models from the point of view of quantum control. For the case of linear quantum stochastic systems, we present a condition for stability of the asymptotic Zeno dynamics. PubDate: 2014-07-01

Abstract: Abstract
Boyle temperature is interpreted as the temperature at which the formation of dimers becomes impossible. To Irving Fisher’s correspondence principle we assign two more quantities: the number of degrees of freedom, and credit. We determine the danger level of the mass of money M when the mutual trust between economic agents begins to fall. PubDate: 2014-07-01

Abstract: Abstract
We analyze the unravelling of the quantum optical master equation at finite temperature due to direct, continuous, general-dyne detection of the environment. We first express the general-dyne Positive Operator Valued Measure (POVM) in terms of the eigenstates of a non-Hermitian operator associated to the general-dyne measurement. Then we derive the stochastic master equation obtained by considering the interaction between the system and a reservoir at thermal equilibrium, which is measured according to the POVM previously determined. Finally, we present a feasible measurement scheme, which reproduces general-dyne detection for any value of the parameter characterizing the stochastic master equation. PubDate: 2014-07-01

Abstract: Abstract
Theorem 4.1 of the author’s paper “Quantum Yang-Mills-Weyl dynamics in the Schroedinger paradigm”, RJMP 21 (2), 169–188 (2014) claims the relative ellipticity of cutoff Yang-Mills quantum energy-mass operators in von Neumann algebras with regular traces. This implies that the spectra of cutoff self-adjoint Yang-Mills energy-mass operators in a nonperturbative quantum Yang-Mills theory (with an arbitrary compact simple gauge group) are nonnegative sequences of the eigenvalues converging to +∞. The spectra are self-similar in the inverse proportion to the running coupling constant. In particular, they have self-similar positive spectral mass gaps. Presumably, this is a solution of the Yang-Mills Millennium problem.
The present note shows that the fundamental spectral value of a cutoff quantum Yang-Mills energy-mass operator is the simple zero eigenvalue with the vacuum eigenvector. The direct proof (without von Neumann algebras) is based on the domination over the number operator (with simple fundamental eigenvalue) and the standard spectral variational principle. PubDate: 2014-07-01

Abstract: Abstract
We show that there is a unique C*-algebra for the transverse quantum electromagnetic field obeying the Maxwell equations with any classical charge-current. For nonzero charge, the representation of the C*-algebra differs from the representation with zero charge. PubDate: 2014-07-01

Abstract: Abstract
We show that iterated stochastic integrals can be described equivalently either by the conventional forward adapted, or by backward adapted quantum stochastic calculus. By using this equivalence, we establish two properties of triangular (causal) and rectangular double quantum stochastic product integrals, namely a necessary and sufficient condition for their unitarity, and the coboundary relation between the former and the latter. PubDate: 2014-07-01

Abstract: Abstract
In this paper we shall re-visit the well-known Schrödinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt. And so the unitary dynamics of a quantum object, described by the Schrödinger equation, may be obtained as the expectation of a counting process of object-clock interactions. PubDate: 2014-07-01

Abstract: Abstract
In this paper, we treat the quantum filtering problem for multiple input multiple output (MIMO) Markovian open quantum systems coupled to multiple boson fields in an arbitrary zero-mean jointly Gaussian state, using the reference probability approach formulated by Bouten and van Handel as a quantum version of a well-known method of the same name from classical nonlinear filtering theory, and exploiting the generalized Araki-Woods representation of Gough. This includes Gaussian field states such as vacuum, squeezed vacuum, thermal, and squeezed thermal states as special cases. The contribution is a derivation of the general quantum filtering equation (or stochastic master equation as they are known in the quantum optics community) in the full MIMO setup for any zero-mean jointly Gaussian input field states, up to some mild rank assumptions on certain matrices relating to the measurement vector. PubDate: 2014-07-01

Abstract: Abstract
The fidelity between the state of a continuously observed quantum system and the state of its associated quantum filter, is shown to be always a submartingale. The observed system is assumed to be governed by a continuous-time Stochastic Master Equation (SME), driven simultaneously by Wiener and Poisson processes and that takes into account incompleteness and errors in measurements. This stability result is the continuous-time counterpart of a similar stability result already established for discrete-time quantum systems and where the measurement imperfections are modelled by a left stochastic matrix. PubDate: 2014-07-01

Abstract: Abstract
We suggest a mathematical construction to define an individual state of a quantum particle which enables one to eliminate a contradiction between quantum mechanics of correlated particles and the Relativity Theory. PubDate: 2014-04-01

Abstract: Abstract
In this paper, we consider the q-analog of the Laplace transform and investigate some properties of the q-Laplace transform. In our investigation, we derive some interesting formulas related to the q-Laplace transform. PubDate: 2014-04-01

Abstract: Abstract
We consider problems of the linearized theory of hydrodynamic stability for the case in which the unperturbed plane-parallel-flow of a viscous incompressible fluid in a layer is substantially unsteady. We analyze the Orr-Sommerfeld equation, which is generalized for this case, with different combinations of the four boundary conditions specified on the straight parts of the boundaries of the layer. Using the apparatus of integral relations, including, in particular, the analysis of the minimization problem for quadratic functionals, we derive upper bounds for the growth or decay of kinematic perturbations with respect to the integral measure. A special attention is paid to the longitudinal oscillation mode of the layer, to the power-law acceleration or deceleration, and also to the process similar to the diffusion of the vortex layer. An investigation of the reducibility of the three-dimensional picture of perturbations imposed on a plane-parallel unsteady shift to a two-dimensional picture in the plane of this shift is carried out. Generalizations of the Squire theorem are established. PubDate: 2014-04-01

Abstract: Abstract
We obtain a representation, using a Feynman formula, for the operator semigroup generated by a second-order parabolic differential equation with respect to functions defined on the Cartesian product of the line ℝ and a graph consisting of n rays issuing from a common vertex. PubDate: 2014-04-01

Abstract: Abstract
We present a version of Maslov’s canonical operator to be used when constructing asymptotic solutions of hyperbolic equations degenerating in a special way on the boundary of the spatial domain. PubDate: 2014-04-01

Abstract: Abstract
The purpose of this paper is to investigate the connection between context-free grammars and normal ordered problem, and then to explore various extensions of the Stirling grammar. We present grammatical characterizations of several well known combinatorial sequences, including the generalized Stirling numbers of the second kind related to the normal ordered problem and the r-Dowling polynomials. Also, possible avenues for future research are described. PubDate: 2014-04-01