TY - JOUR

T1 - How often is a random quantum state k-entangled?

AU - Szarek, Stanislaw J.

AU - Werner, Elisabeth

AU - Zyczkowski, Karol

PY - 2011/1/28

Y1 - 2011/1/28

N2 - 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 kpositive 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 × d system.

AB - 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 kpositive 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 × d system.

UR - http://www.scopus.com/inward/record.url?scp=78751604499&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78751604499&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/44/4/045303

DO - 10.1088/1751-8113/44/4/045303

M3 - Article

AN - SCOPUS:78751604499

SN - 1751-8113

VL - 44

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

IS - 4

M1 - 045303

ER -