The time evolution of fluctuation algebra in mean field theories

Fluctuation algebra is defined for equilibrium states of mean field theories. The time evolution is calculated; and, in contrast to interactions with finite range, the fluctuation algebra is not in a KMS state with respect to this time evolution though the underlying quasilocal state. If the system is coupled to an other system mimicking a laser then the evolution depends on the underlying mean field theory and shows varying large time behaviour, so that the fluctuations either rotate or increase linearly or exponentially in time, in correspondence to the stability of the underlying quasilocal state.     

On the Relation Between Strong Subadditivity and Entanglement

Strong subadditivity is used to improve the triangular inequality for the entropy of tensorproducts by the amount of entanglement.     

Time-ordering dependence of measurements in teleportation

We trace back the phenomenon of “delayed-choice entanglement swapping” as it was realized in a recent experiment to the commutativity of the projection operators that are involved in the corresponding measurement process. We also propose an experimental set-up which depends on the order of successive measurements corresponding to noncommutative projection operators. In this case entanglement swapping is used to teleport a quantum state from Alice to Bob, where Bob has now the possibility to examine the noncommutativity within the quantum history.     

Dynamical entropy of quasifree automorphisms

The dynamical entropy of quasifree automorphisms and quasifree states is caalculated and shown to be the entropy of the shift created by the group velocity. Some continuity properties are proved.     

A non-commutative version of the Arnold cat map

We provide a treatment of the ergodic properties of a noncommutative algebraic analogue of the dynamical system known as the Arnold cat map of the two-dimensional torus. Here, the algebra of functions on the torus is replaced by its noncommutative analogue, formulated by Connes and Rieffel, which arises in the quantum Hall effect. Our main results are that (a) the system is mixing and, as in the classical case, the unitary operator, representing its dynamical map, has countable Lebesgue spectrum; (b) for rational values of the noncommutativity parameter, , the model is a K-system, in the algebraic sense of Emch, Narnhofer, and Thirring, though not in the entropic sense of Narnhofer and Thirring; (c) for irrational values of , except possibly for a set of zero Lebesgue measures, it is neither an algebraic nor an entropic K-system.Supported in part by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P7101-PHY.     

The role of transposition and CPT operation for entanglement

The transposition is defined for infinite algebras with the help of the modular operator. Its effect as entanglement witness is discussed for quasifree states and in relativistic quantum field theory.     

Optimal Decompositions with Respect to Entropy and Symmetries

The entropy of a subalgebra, which has been used in quantum ergodic theory to construct a noncommutative dynamical entropy, coincides for N-level systems and Abelian subalgebras with the notion of maximal mutual information of quantum communication theory. The optimal decompositions of mixed quantum states singled out by the entropy of Abelian subalgebras correspond to optimal detection schemes at the receiving end of a quantum channel. It is then worthwhile studying in some detail the structure of the convex hull of quantum states brought about by the variational definition of the entropy of a subalgebra. In this Letter, we extend previous results on the optimal decompositions for 3-level systems.     

The pauli principle for a quantum theory on T with magnetic field

We discuss the reducible representation of n particles on a torus with magnetic field, where the gauge automorphisms are unitarily implementable. This representation is of type II_∞, if B is not integer. We show that there is a natural splitting into I_∞ ⊗ II₁ such that the Pauli exclusion principle acts only in the first factor. Under this condition we estimate the ground state energy for several particles and show that in the thermodynamic limit we obtain the correct result.     

Scattering theory for quasi-free time automorphisms of C1-algebras and von Neumann algebras

Relations between scattering theory in L² space, for quasi-free automorphism groups of C¹-algebras and for those of their representations are established and it is shown that wave operators as time limits do not exist in these representations. Finally, the connection of scattering theory and dynamical stability is discussed.