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.
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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 -