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, and (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 Jamiokowski 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 the 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.
Bibliographical noteFunding Information:
We enjoyed inspiring discussions with I. Bengtsson, F. Benatti, V. Cappellini, and H.-J. Sommers. We acknowledge financial support from the Polish Ministry of Science and Information Technology under the Grant No. DFG-SFB∕38∕2007, from the National Science Foundation (U.S.A.), and from the European Research Projects COCOS and PHD.
Copyright 2008 Elsevier B.V., All rights reserved.