[Banach] Abstract of a paper by Stanislaw J. Szarek, Elisabeth Werner and Karol Zyczkowski
alspach at www.math.okstate.edu
Sun Oct 14 09:30:12 CDT 2007
This is an announcement for the paper "Geometry of sets of quantum
maps: a generic positive map acting on a high-dimensional system
is not completely positive" by Stanislaw J. Szarek, Elisabeth Werner
and Karol Zyczkowski.
Abstract: We investigate the set a) of positive, trace preserving
maps acting on density matrices of size N, and a sequence of its
nested subsets: the sets of maps which are b) decomposable, c)
completely positive, d) extended by identity impose positive partial
transpose and e) are superpositive. Working with the Hilbert-Schmidt
(Euclidean) measure we derive tight explicit two-sided bounds for
the volumes of all five sets. A sample consequence is the fact that,
as N increases, a generic positive map becomes not decomposable
and, a fortiori, not completely positive.
Due to the Jamiolkowski isomorphism, the results obtained for
quantum maps are closely connected to similar relations between the volume of
the set of quantum states and the volumes of its subsets (such as
states with positive partial transpose or separable states) or
supersets. Our approach depends on systematic use of duality to
derive quantitative estimates, and on various tools of classical
convexity, high-dimensional probability and geometry of Banach
spaces, some of which are not standard.
Archive classification: quant-ph math.FA
Remarks: 34 pages in Latex including 3 figures in eps
The source file(s), , is(are) stored in gzipped form as with size
. The corresponding postcript file has gzipped size .
Submitted from: karol at tatry.if.uj.edu.pl
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