How often is a random quantum state k-entangled?

Abstract

The set of trace preserving, positive maps acting on density matrices of size d forms a convex body. We investigate its nested subsets consisting of k-positive maps, where k=2,...,d. Working with the measure induced by the Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes of these sets. Our results strongly suggest that the inner set of (k+1)-positive maps forms a small fraction of the outer set of k-positive maps. These results are related to analogous bounds for the relative volume of the sets of k-entangled states describing a bipartite d X d system.

19 pages in latex, 1 figure included

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• The absolute positive partial transpose property for random induced states
• There are many more positive maps than completely positive maps
• A class of genuinely high-dimensionally entangled states with a positive partial transpose
• Interpolating between positive and completely positive maps: a new hierarchy of entangled states
• $L_p$ geominimal surface areas and their inequalities
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• Random constructions in Bell inequalities: A survey
• Convex bodies of states and maps
• Completely Bounded Norms of $k$-positive Maps
• The Loewner function of a log-concave function
• Quantifying entanglement in multipartite conditional states of open quantum systems by measurements of their photonic environment
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