Quantum theory: Concepts and methods

Lectures on stochastic differential equations and Malliavin calculus

The aim of this article is to extend linear quantum dynamical network theory to include static Bogoliubov components (such as squeezers). Within this integrated quantum network theory, we provide general methods for cascade or series connections, as well as feedback interconnections using linear fractional transformations. In addition, we define input-output maps and transfer functions for representing components and describing convergence. We also discuss the underlying group structure in this theory arising from series interconnection. Several examples illustrate the theory.

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Squeezing Components in Linear Quantum Feedback Networks

Linear maps of matrices describing the evolution of density matrices for a quantum system initially entangled with another are identified and found to be not always completely positive. They can even map a positive matrix to a matrix that is not positive, unless we restrict the domain on which the map acts. Nevertheless, their form is similar to that of completely positive maps. Only some minus signs are inserted in the operator-sum representation. Each map is the difference of two completely positive maps. The maps are first obtained as maps of mean values and then as maps of basis matrices. These forms also prove to be useful. An example for two entangled qubits is worked out in detail. The relation to earlier work is discussed.

Dynamics of initially entangled open quantum systems

When the structures are analyzed by classical method, several governing differential equations are needed if a beam is loaded by many loads But if delta function and its derivatives as generalized functions are used, one governing differential equation is enough In Lw(x) = q(x), where L is a linear differential operator, the particular solution Wp(x) can be obtained as a integral from using the method of variation of parameters Since the external load term, q(x), can be described by using these generalized functions, the filtration property of delta function in a integral makes the form of Wp(x) simpler one The usage of these generalized functions is shown in several sample cases and we can see the convenience of the generalized functions

구조해석에서의 Generalized Functions의 응용

Continuous and discrete superselection rules induced by the interaction with the environment are investigated for a class of exactly soluble Hamiltonian models. The environment is given by a boson field. Stable superselection sectors can only emerge if the low frequences dominate and the ground state of the boson field disappears due to infrared divergence. The models allow uniform estimates of all transition matrix elements between different superselection sectors.

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The role of infrared divergence for decoherence

The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.

https://kluedo.ub.uni-kl.de/files/1168/no_series_139.pdf

Functional Representations for Fock Superalgebras

Canonical transformations are defined and discussed along with the coherent and the ultracoherent vectors. It is shown that the single-mode and the n-mode squeezing operators are elements of the group of canonical transformations. An application of canonical transformations is made, in the context of open quantum systems, by studying the effect of squeezing of the bath on the decoherence properties of the system. Two cases are analysed. In the first case, the bath consists of a massless bosonic field with the bath reference states being the squeezed vacuum states and squeezed thermal states while in the second case a system consisting of a harmonic oscillator interacting with a bath of harmonic oscillators is analysed with the bath being initially in a squeezed thermal state.

Applications of canonical transformations

Let H1, H2 be complex Hilbert spaces, H be their Hilbert tensor product and let tr2 be the operator of taking partial trace, with respect to the space H2, of trace class operators in H The operation tr2 maps states in H (=positive trace class operators in H with trace equal to 1) into states in H1 In this paper we give the full description of mappings that are linear right inverse to tr2 More precisely, we prove that any affine mapping F(W) of the convex set of states in H1 into the states in H that is right inverse to tr2 is given by W⟼W⊗D for some state D in H2 In addition we investigate a representation of the quantum mechanical state space by probability measures on the set of pure states and a representation—used in the theory of stochastic Schrodinger equations—by probability measures on the Hilbert space We prove that there are no affine mappings from the state space of quantum mechanics into these spaces of probability measures

https://kluedo.ub.uni-kl.de/files/1188/no_series_174.pdf

States of quantum systems and their liftings

This paper presents a solution to a problem from superanalysis about the existence of Hilbert-Banach superalgebras. Two main results are derived: 1) There exist Hilbert norms on some graded algebras (infinite-dimensional superalgebras included) with respect to which the multiplication is continuous. 2) Such norms cannot be chosen to be submultiplicative and equal to one on the unit of the algebra.

Joachim Kupsch

Papers

The role of infrared divergence for decoherence

Functional Representations for Fock Superalgebras

Applications of canonical transformations

States of quantum systems and their liftings

Hilbert norms for graded algebras