scholarly journals Extremal problems for completely positive maps

1994 ◽  
Vol 17 (3) ◽  
pp. 607-608
Author(s):  
Mingze Yang
Keyword(s):  
Extremal Problems ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Maps ◽  
Linear Maps ◽  
Positive Linear Maps ◽  
Convex Subsets

In this note, we study the faces of some convex subsets ofCPc(A,B(ℋ))(the continuous completely positive linear maps from pro-C*-algebraAtoB(ℋ)).

scholarly journals On the convergence of a sequence of completely positive maps to the identity

2000 ◽  
Vol 68 (3) ◽  
pp. 340-348 ◽  
Author(s):  
George A. Elliott
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Maps ◽  
Linear Maps ◽  
Positive Linear Maps

AbstractIt is shown that a sequence of completely positive linear maps on a W*-algebra that converges pointwise in norm to the identity converges uniformly.

Dilations and Hahn Decompositions for Linear Maps

1981 ◽  
Vol 33 (4) ◽  
pp. 826-839 ◽  
Author(s):  
D. W. Hadwin
Keyword(s):  
Linear Operators ◽  
Completely Positive Maps ◽  
Linear Combinations ◽  
Completely Positive ◽  
Bounded Linear ◽  
Positive Maps ◽  
Linear Maps ◽  
Positive Linear Map ◽  
Finite Linear ◽  
Analogous Class

Suppose is a C*-algebra and H is a Hilbert space. Let denote the set of completely positive maps from into the set B(H) of (bounded linear) operators on H. This paper studies the vector space spanned by , i.e., the linear maps that are finite linear combinations of completely positive maps. From another viewpoint, a map ϕ is in precisely when it has a decomposition ϕ = (ϕ1 – ϕ2) + i(ϕ3 – ϕ4) with ϕ1, ϕ2, ϕ3, ϕ4 in CP ; this decomposition is analogous to the Hahn decomposition for measures [8, 111.4.10] (see also Theorem 20). The analogous class of maps with “completely positive” replaced by “positive” was studied by R. I. Loebl [11] and S.-K. Tsui [17], and when is commutative, this latter class coincides withi , since every positive linear map on a commutative C*-algebra is completely positive [16].

Positive Linear Maps on C*-Algebras

1972 ◽  
Vol 24 (3) ◽  
pp. 520-529 ◽  
Author(s):  
Man-Duen Choi
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Complex Matrices ◽  
Completely Positive ◽  
C Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

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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