scholarly journals On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras

1993 ◽  
Vol 193 ◽  
pp. 107-127 ◽  
Author(s):  
L.J. Landau ◽  
R.F. Streater
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
Matrix Algebras ◽  
Doubly Stochastic ◽  
Positive Maps ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

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.

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

scholarly journals Quantum weakest preconditions

2006 ◽  
Vol 16 (3) ◽  
pp. 429-451 ◽  
Author(s):  
ELLIE D'HONDT ◽  
PRAKASH PANANGADEN
Keyword(s):  
Quantum Computation ◽  
Representation Theorem ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Weakest Precondition ◽  
Positive Maps ◽  
Weakest Preconditions ◽  
Predicate Transformer ◽  
Usual State ◽  
Stone Type

We develop a notion of predicate transformer and, in particular, the weakest precondition, appropriate for quantum computation. We show that there is a Stone-type duality between the usual state-transformer semantics and the weakest precondition semantics. Rather than trying to reduce quantum computation to probabilistic programming, we develop a notion that is directly taken from concepts used in quantum computation. The proof that weakest preconditions exist for completely positive maps follows immediately from the Kraus representation theorem. As an example, we give the semantics of Selinger's language in terms of our weakest preconditions. We also cover some specific situations and exhibit an interesting link with stabilisers.

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