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Combinatorial laplacians and positivity under partial transpose

Published online by Cambridge University Press:  01 February 2008

ROLAND HILDEBRAND ,
STEFANO MANCINI  and
SIMONE SEVERINI
ROLAND HILDEBRAND
Affiliation:
LJK Université Joseph Fourier, Tour IRMA, 51 rue des Mathématiques, 38400 St. Martin d'Héres, France
STEFANO MANCINI
Affiliation:
Dipartimento di Fisica, Universitá di Camerino, Via Madonna delle Carceri 9, 62032 Camerino, Italy
SIMONE SEVERINI
Affiliation:
Department of Computer Science, University of York, Heslington, Y010 5DD York, United Kingdom
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Abstract

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The density matrices of graphs are combinatorial laplacians normalised to have trace one (Braunstein et al. 2006b). If the vertices of a graph are arranged as an array, its density matrix carries a block structure with respect to which properties such as separability can be considered. We prove that the so-called degree-criterion, which was conjectured to be necessary and sufficient for the separability of density matrices of graphs, is equivalent to the PPT-criterion. As such, it is not sufficient for testing the separability of density matrices of graphs (we provide an explicit example). Nonetheless, we prove the sufficiency when one of the array dimensions has length two (see Wu (2006) for an alternative proof). Finally, we derive a rational upper bound on the concurrence of density matrices of graphs and show that this bound is exact for graphs on four vertices.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 18 , Special Issue 1: in memory of Sauro Tulipani , February 2008 , pp. 205 - 219
Copyright
Copyright © Cambridge University Press2008

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