On classical and quantum liftings

We analyze the procedure of lifting in classical stochastic and quantum systems. It enables one to `lift' a state of a system into a state of `system+reservoir'. This procedure is important both in quantum information theory and the theory of open systems. We illustrate the general theory of liftings by a particular class related to so called circulant states.Comment: 25 page

Fibre bundle formulation of relativistic quantum mechanics. I. Time-dependent approach

We propose a new fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in different directions. In the present first part of our investigation we consider the time-dependent or Hamiltonian approach to bundle description of relativistic quantum mechanics. In it the wavefunctions are replaced by (state) liftings of paths or sections along paths of a suitably chosen vector bundle over space-time whose (standard) fibre is the space of the wavefunctions. Now the quantum evolution is described as a linear transportation (by means of the evolution transport along paths in the space-time) of the state liftings/sections in the (total) bundle space. The equations of these transportations turn to be the bundle versions of the corresponding relativistic wave equations.Comment: 16 standard LaTeX pages. The packages AMS-LaTeX and amsfonts are required. The paper continuous the application of fibre bundle formalism to quantum physics began in the series of works quant-ph/9803083, quant-ph/9803084, quant-ph/9804062, quant-ph/9806046, quant-ph/9901039, quant-ph/9902068, and quant-ph/0004041. For related papers, view http://theo.inrne.bas.bg/~bozho

States of quantum systems and their liftings

Let H(1), H(2) be complex Hilbert spaces, H be their Hilbert tensor product and let tr2 be the operator of taking the partial trace of trace class operators in H with respect to the space H(2). The operation tr2 maps states in H (i.e. positive trace class operators in H with trace equal to one) into states in H(1). 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 H(1) into the states in H that is right inverse to tr2 is given by the tensor product of W with some state D in H(2). 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 Schroedinger 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.Comment: 14 page

The asymptotic lift of a completely positive map

Starting with a unit-preserving normal completely positive map L: M --> M acting on a von Neumann algebra - or more generally a dual operator system - we show that there is a unique reversible system \alpha: N --> N (i.e., a complete order automorphism \alpha of a dual operator system N) that captures all of the asymptotic behavior of L, called the {\em asymptotic lift} of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n x n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W*-dynamical system (N,\mathbb Z), whick we identify as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed point algebra of (N,\mathbb Z). In general, we show the action of the asymptotic lift is trivial iff L is {\em slowly oscillating} in the sense that $$\lim_{n\to\infty}\|\rho\circ L^{n+1}-\rho\circ L^n\|=0,\qquad \rho\in M_* .$$ Hence \alpha is often a nontrivial automorphism of N.Comment: New section added with an applicaton to the noncommutative Poisson boundary. Clarification of Sections 3 and 4. Additional references. 23 p