Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras

We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the...

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Bibliographic Details
Published in:Communications in mathematical physics Vol. 303; no. 2; pp. 555 - 594
Main Authors: Haagerup, Uffe, Musat, Magdalena
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.04.2011
Springer
Subjects:
Classical and Quantum Gravitation
Complex Systems
Exact sciences and technology
Functional analysis
Group theory
Group theory and generalizations
Mathematical analysis
Mathematical and Computational Physics
Mathematical methods in physics
Mathematical Physics
Mathematics
Operator theory
Other topics in mathematical methods in physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Sciences and techniques of general use
Theoretical
Embedding Problem
Asymptotic Quantum
Normal Faithful State
Noncommutative Setting
Tracial State
Lower bound
Semigroup
Quantum information
Asymptotic approximation
Mathematical physics
Quantum theory
Information theory
ISSN:0010-3616, 1432-0916
Online Access:Get full text
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Summary:We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on for all n ≥ 3, as well as an example of a one-parameter semigroup ( T ( t )) t ≥0 of Markov maps on such that T ( t ) fails to be factorizable for all small values of t > 0. As applications, we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little Grothendieck inequality.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-011-1216-y