Absolute continuity for operator valued completely positive maps on C*-algebras
- Department of Mathematics, Bilkent University, Bilkent, Ankara 06800, Turkey and Institutul de Matematica al Academiei Romane, C. P. 1-764, 014700 Bucuresti (Romania)
- Department of Mathematics, University of Houston, Houston, Texas 77204-3476 (United States)
Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C*-algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34-54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon-Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon-Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi's matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained.
- OSTI ID:
- 21175994
- Journal Information:
- Journal of Mathematical Physics, Vol. 50, Issue 2; Other Information: DOI: 10.1063/1.3072683; (c) 2009 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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