State Evolution and Trace-Preserving Completely Positive Maps

2006 ◽  
pp. 117-143
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Maps ◽  
State Evolution

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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