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We investigate the set a) of positive, trace preserving maps acting ondensity matrices of size N, and a sequence of its nested subsets: the sets ofmaps which are b) decomposable, c) completely positive, d) extended by identityimpose positive partial transpose and e) are superpositive. Working with theHilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided boundsfor the volumes of all five sets. A sample consequence is the fact that, as Nincreases, a generic positive map becomes not decomposable and, a fortiori, notcompletely positive.  Due to the Jamiolkowski isomorphism, the results obtained for quantum mapsare closely connected to similar relations between the volume of the set ofquantum states and the volumes of its subsets (such as states with positivepartial transpose or separable states) or supersets. Our approach depends onsystematic use of duality to derive quantitative estimates, and on varioustools of classical convexity, high-dimensional probability and geometry ofBanach spaces, some of which are not standard.

Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive