scholarly journals $$C^*$$-Extreme Points of Positive Operator Valued Measures and Unital Completely Positive Maps

2021 ◽  
Author(s):  
Tathagata Banerjee ◽  
B. V. Rajarama Bhat ◽  
Manish Kumar
Keyword(s):  
Positive Operator ◽  
Extreme Points ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Operator Valued Measures ◽  
Positive Maps

On Completely Positive Maps Defined by an Irreducible Correspondence

1990 ◽  
Vol 33 (4) ◽  
pp. 434-441 ◽  
Author(s):  
C. Anantharaman-Delaroche
Keyword(s):  
Von Neumann Algebras ◽  
Convex Set ◽  
Extreme Points ◽  
Completely Positive Maps ◽  
Complex Matrices ◽  
Completely Positive ◽  
Von Neumann ◽  
Positive Maps

AbstractCompletely positive maps defined by an irreducible correspondence between two von Neumann algebras M and N are introduced. We give results about their structure and characterize, among them, those which are extreme points in the convex set of all unital completely positive maps from M to N. As particular cases we obtain known results of M. D. Choi [4] on completely positive maps between complex matrices and of J. A. Mingo [8] on inner completely positive maps.

scholarly journals Natural Quantum Operational Semantics with Predicates

2008 ◽  
Vol 18 (3) ◽  
pp. 341-359 ◽  
Author(s):  
Marek Sawerwain ◽  
Roman Gielerak
Keyword(s):  
Positive Operator ◽  
Operational Semantics ◽  
General Definition ◽  
Completely Positive Maps ◽  
Transition Systems ◽  
Positive Operator Valued Measures ◽  
Labelled Transition Systems ◽  
Positive Maps ◽  
Definition Of ◽  
Positive Operator Valued Measure

Natural Quantum Operational Semantics with PredicatesA general definition of a quantum predicate and quantum labelled transition systems for finite quantum computation systems is presented. The notion of a quantum predicate as a positive operator-valued measure is developed. The main results of this paper are a theorem about the existence of generalised predicates for quantum programs defined as completely positive maps and a theorem about the existence of a GSOS format for quantum labelled transition systems. The first theorem is a slight generalisation of D'Hondt and Panagaden's theorem about the quantum weakest precondition in terms of discrete support positive operator-valued measures.

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.

MODULES OVER MONOTONE COMPLETE C*-ALGEBRAS

1992 ◽  
Vol 03 (02) ◽  
pp. 185-204 ◽  
Author(s):  
MASAMICHI HAMANA
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
C Algebra ◽  
C Algebras ◽  
Positive Maps

The main result asserts that given two monotone complete C*-algebras A and B, B is faithfully represented as a monotone closed C*-subalgebra of the monotone complete C*-algebra End A(X) consisting of all bounded module endomorphisms of some self-dual Hilbert A-module X if and only if there are sufficiently many normal completely positive maps of B into A. The key to the proof is the fact that each pre-Hilbert A-module can be completed uniquely to a self-dual Hilbert A-module.

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