NON-POSITIVE SEMIGROUP DYNAMICS IN CONTINUOUS VARIABLE MODELS

Non-positive, Markovian semigroups are sometimes used to describe the time evolution of subsystems immersed in an external environment. A widely adopted prescription to avoid the appearance of negative probabilities is to eliminate from the admissible initial conditions those density matrices that would not remain positive by the action of the semigroup dynamics. Using a continuous variable model, we show that this procedure leads to physical inconsistencies when two subsystems are considered and their initial state is entangled.

RANDOM FINITE SUBSETS WITH EXPONENTIAL DISTRIBUTIONS

Let S denote the collection of all finite subsets of . We define an operation on S that makes S into a positive semigroup with set inclusion as the associated partial order. Positive semigroups are the natural home for probability distributions with exponential properties, such as the memoryless and constant rate properties. We show that there are no exponential distributions on S, but that S can be partitioned into subsemigroups, each of which supports a one-parameter family of exponential distributions. We then find the distribution on S that is closest to exponential, in a certain sense. This work might have applications to the problem of selecting a finite sample from a countably infinite population in the most random way.

COCYCLE PERTURBATION OF QUASIFREE ALGEBRAIC K-FLOW LEADS TO REQUIRED ASYMPTOTIC DYNAMICS OF ASSOCIATED COMPLETELY POSITIVE SEMIGROUP

We study the quasifree algebraic K-flow τ on the hyperfinite factor ℳ with the expanding subfactor [Formula: see text] generated by representations π of the C*-algebra of the canonical anticommutation relations (CAR) [Formula: see text] over separable Hilbert space ℋ. The type of ℳ and [Formula: see text] can be II1 or IIIλ, 0<λ<1, depending on π. The K-flow τ is obtained by the quasifree lifting of one-parameter group ST consisting of shifts in ℋ with the discrete parameter T=Z or the continuous one T=R. We prove that acting on τ by a quasifree inner Markovian cocycle, one can get the required asymptotic behavior of the perturbed group restriction on [Formula: see text].

ON THE INDEX AND DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS

It is known that every semigroup of normal completely positive maps P = {Pt:t≥ 0} of ℬ(H), satisfying Pt(1) = 1 for every t ≥ 0, has a minimal dilation to an E0 acting on ℬ(K) for some Hilbert space K⊇H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator [Formula: see text] in terms of natural structures associated with the generator. This includes all unital CP semigroups acting on matrix algebras. We also show that the minimal dilation of the semigroup P={ exp tL:t≥ 0} to an E0-semigroup is is cocycle conjugate to a CAR/CCR flow.