Hadamard Completely Positive Semigroups

2019 ◽  
Vol 26 (04) ◽  
pp. 1950020
Author(s):  
Fabio Benatti
Keyword(s):  
Continuous Semigroup ◽  
Hadamard Product ◽  
Product Rule ◽  
Density Matrices ◽  
Completely Positive ◽  
Positive Semigroups ◽  
Completely Positive Semigroups

We review the Gorini, Kossakowski, Sudarshan derivation of the generator of a completely positive norm-continuous semigroup when the constituent maps act on density matrices according to the Hadamard product rule.

scholarly journals ON THE INDEX AND DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS

1999 ◽  
Vol 10 (07) ◽  
pp. 791-823 ◽  
Author(s):  
WILLIAM ARVESON
Keyword(s):  
Bounded Operator ◽  
Positive Semigroup ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Semigroups ◽  
Matrix Algebras ◽  
Positive Maps ◽  
Completely Positive Semigroups ◽  
Numerical Index ◽  
Completely Positive Semigroup

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.

INDUCTION OF SEMIGROUPS OF ENDOMORPHISMS OF ${\mathfrak B} ({\mathfrak h})$ FROM COMPLETELY POSITIVE SEMIGROUPS OF (n × n) MATRIX ALGEBRAS

1999 ◽  
Vol 10 (07) ◽  
pp. 773-790 ◽  
Author(s):  
ROBERT T. POWERS
Keyword(s):  
Matrix Algebra ◽  
Completely Positive ◽  
Positive Semigroups ◽  
Matrix Algebras ◽  
Completely Positive Semigroups

The paper concerns Eo-semigroup of [Formula: see text] induced from unit preserving completely positive semigroups of mapping of an (n × n) matrix algebra into itself. It is shown that Eo-semigroups one obtains are completely spatial and the index of the induced semigroup can be easily computed.

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