Mathematics – Operator Algebras
Scientific paper
2010-10-07
J. Funct. Anal. 260, 2407-2423 (2011)
Mathematics
Operator Algebras
17 pages, to appear in JFA
Scientific paper
10.1016/j.jfa.2010.10.003
We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.
Johnston Nathaniel
Kribs David W.
Paulsen Vern I.
Pereira Rajesh
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