Erwin Brüning

Elements of Spectral Theory

The first part introduces special classes of bounded linear operator while the second part presents some results on Hamilton operators of quantum physics. Orthogonal projections or projectors are selfadjoint bounded linear operators P which are idempotent. They are characterized by their range. Isometric operators are bounded linear operators from one Hilbert space to another which do not change the length of vectors. If an isometric operator is surjective it is a unitary operator. A unitary operator \(U : \mathcal{H} \to \mathcal{K}\) is characterized by the relations \(U * U = id_{\mathcal{H}}\) and \(UU* = id_{\mathcal{K}}\). Next one parameter groups of unitary operators are introduced and under a mild continuity assumption the self-adjoint generator of such a group is determined in Stone’s theorem. Another important application of unitary operators is the mean ergodic theorem of von Neumann.We present it together with the early results of ergodic theory: Poincare recurrence theorem and Birkhoff’s strong ergodic theory. Given a free self-adjoint Hamilton operator \(H_{0} = \frac{1}{2m} p^{2}\) one often needs to knowunder which conditions on the potential the Hamilton H = H 0 + V (q) is selfadjoint in \(L^{2} = \mathbb{R}^{3}\). Sufficient condition are given in the Kato-Rellich theorem using the concept of a Kato perturbation. The from which it follows that H is self-adjoint whenever V is a real-valued function in \(L^{2} \mathbb{(R)}^{3} + L^{\infty} \mathbb{(R)}^{3}\).

Theoretical Foundations of Quantum Information Processing and Communication

An Introduction to Quantum Probability.- Covariant Mappings for the Description of Measurement, Dissipation and Decoherence in Quantum Mechanics.- Quantum Open Systems with Time-Dependent Control.- Five Lectures on Quantum Information Applications of Complex Many-Body Systems.- Non-Markovian Quantum Dynamics and the Method of Correlated Projection Super-Operators.- Testing Quantum Mechanics in High-Energy Physics.- Five Lectures on Optical Quantum Computing.- Quantum Information and Relativity: An Introduction.

Density Matrices and Their Time Evolution

Already in the case of finite dimensional Hilbert spaces the general form of density matrices ρ is not known. The main reason for this lack of knowledge is the nonlinear constraint for these matrices. We propose a representation of density matrices on finite dimensional Hilbert spaces in terms of finitely many independent parameters. For dimensions 2, 3, and 4 we write down this representation explicitly. As a further application of this representation we study the time dependence of density matrices ρ(t) which in our case is implemented through time dependence of the independent parameters. Under obvious differentiability assumptions the explicit form of is determined. As a special case we recover, for instance, the Lindblad form.

Boundary and Eigenvalue Problems

Here some basic applications of variational methods are presented. We begin by proving the projection theorem for closed convex subsets of a Hilbert space \(\mathcal{H}\) which is followed by the proof of the minimax principle for positive self-adjoint operators with discrete spectrum in a real Hilbert space. The next section explains the solution of the Dirichlet problem using the direct methods of the calculus of variations. As an illustration of the results on constrained minimization we prove the existence of eigenvalues for the Dirichlet Laplace operator. Next we explain how this strategy can be extended to general elliptic partial differential operators, first for the case of linear partial differential operators and then for certain classes of nonlinear partial differential operators. In each of these cases the existence of eigenvalues follows from the existence of a Lagrange multiplicator. In these applications we rely on some basic results about Sobolev spaces as explained in Part I. Finally we comment on some methods to determine critical points of functions which are not covered by our short introduction.

Parametrizing Density Matrices for Composite Quantum Systems

A parametrization of density operators for bipartite quantum systems is proposed. It is based on the particular parametrization of the unitary group found recently by Jarlskog. It is expected that this parametrization will find interesting applications in the study of quantum properties of multipartite systems.

Constrained Minimization Problems (Method of Lagrange Multipliers)

We consider the following type of minimization problem. Given a real valued function f on an open nonempty subset U of a real Banach space E we are looking for a minimum of f on the subset of U which is determined by the constraint condition \(g(x)=y\) where \(g: U \longrightarrow F\) is a given function on U with values in some Banach space F and \(y\in F\) is some selected point, i.e., we are looking for a minimum of f on the level set \([g=y]=\left\{x \in U; g(x)=y\right\}\) of g. If g is of class \(\mathcal{C}^1\) the level surfaces of g have a proper tangent space at regular points x 0 (i.e., \(g^{\prime}(x_0)\) is surjective) at which the nullspace of \(g^{\prime}(x_0)\) has a topological complement. This can be used to introduce local coordinates on the level surface \([g=y]\) in the neighborhood of such a regular point and the constrained minimization problem becomes a minimization problem on an open subset (in terms of the local coordinates) and the characterization of extremal points as derived earlier applies. The conclusion is that a Lagrange multiplier exists, i.e., a continuous linear mapping \(\ell: F \longrightarrow \mathbb{R}\) such that \(f^{\prime}(x_0)= \ell \circ g^{\prime}(x_0)\). The geometrical meaning of this condition is first explained heuristically in a two dimensional setting. In practical cases of interest the condition of the existence of a topological compliment is always met. The well known finite dimensional case is easily obtained. We conclude by some comments on the constrained minimization problem of Dido.

Hilbert Spaces: A Brief Historical Introduction

This introduction explains very briefly how and why the theory of Hilbert spaces and their operators emerged. In particular we mention which type of mathematical problems motivated the emergence of these spaces at the beginning of the 20th century as a generalization of finite linear vector spaces that have an inner product and therefore also metric and geometric properties. Then we present a short overview of the contents of this second part on Hilbert spaces and their operators. The final part of this introduction contains remarks on the particular and fundamental role which quantum physics has played in the development of the theory of Hilbert spaces and of modern functional analysis.

Positive Mappings in Quantum Physics

This chapter offers some results which will help to understand some foundational aspects of quantum mechanics. It relies on some results presented in the last chapter. The first section discusses the general form of σ-additive probability measures on the complete lattice of orthogonal projections on a Hilbert space (Gleason’s theorem) and its variations. In quantum mechanics and in quantum information theory quantum channels or quantum operations are defined mathematically as completely positive maps between density operators which do not increase the trace (see for instance the book “Quantum Computation and Quantum Information” by M.A. Nielsen and I. L. Chuang, Cambridge University Press 2000). Thus in the next section we determine the general form of quantum operations on a separable Hilbert space, i.e., we prove Kraus’ first representation theorem for operations. Usually quantum information theory studies systems of some finite dimension n and then density operators are just positive \(n \times n\) matrices with complex coefficients which have trace 1. In this context the relevant C\(^*\)-algebra is just the space \(M_n(\mathbb C)\) of all \(n \times n\) matrices with complex entries, for some \(n \in \mathbb{N}\). Therefore in the last section we determine the general form of completely positive maps for these algebras (Choi’s results). Of course, this is a special case of Stinespring factorization theorem, but some important aspects are added.

The Spectral Theorem

This chapter offers a proof of the spectral theorem for self-adjoint operators A by Hilbert space intrinsic methods. The starting point is the so-called geometric characterization of self-adjointness in terms of the subspaces of controlled growth \(F(A,r)\subset D^{\infty}(A)=\cap_n D(A^n)\) of the operator, i.e., for a closed symmetric operator A and \(0\leq r <s\) one has \(r\left\Vert{x}\right\Vert\leq \left\Vert{Ax}\right\Vert\leq s\left\Vert{x}\right\Vert\) for all \(x \in F(A,s)\cap F(A,r)^{\perp}\) and such an operator is self-adjoint if, and only if, \(\cup_n F(A,n)\) is dense in the Hilbert space \(\mathcal{H}\). Then spectral families \(E_t, t \in \mathbb{R}\) are introduced as monotone, right-continuous, normalized projection-valued functions on \(\mathbb{R}\).They are characterized by corresponding properties of the family of their ranges \(H_t,t\in \mathbb{R}\). Next we define integrals of continuous functions with respect to a spectral family and study their basic properties. For a self-adjoint operat or \(A\geq 0\) the spectral family E t is defined by \(\rm{ran}\, E_t =F(A,t)\) for \(t \geq 0\) and \(=\left\{0\right\}\) for \(t<0\). Through various approximations the spectral representation \(A=\int t \mathrm{d} E_t\) follows. The general case can be reduced to the case of positive A. As applications of this approach the maximal self-adjoint part of a closed symmetric operator is easily determined. Furthermore convenient sufficient conditions can be given under which a closed symmetric operator is essentially self-adjoint.

Hilbert–Schmidt and Trace Class Operators

This chapter introduces two subspaces of the space of compact operators and presents their basic theory in substantial detail. These spaces of operators are important in various areas of functional analysis and in applications of operator theory to quantum physics. Accordingly, after the characterization of Hilbert-Schmidt and trace class operators has been presented, the spectral representation for these operators is derived. Furthermore the dual spaces (spaces of continuous linear functionals) of these two spaces of operators are determined and their role in the description of locally convex topologies on the space \(\mathcal{B}(\mathcal{H})\) of all bounded linear operators on a Hilbert space \(\mathcal{H}\) is explained. Finally two results are included which are mainly used in quantum physics: Partial trace for trace class operators on tensor products of separable Hilbert spaces and Schmidt decomposition.