The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States |
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Authors: | Samuel L Braunstein Sibasish Ghosh Simone Severini |
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Institution: | (1) Department of Computer Science, University of York, Heslington, York, YO10 5DD, UK;(2) Department of Mathematics and Department of Computer Science, University of York, Heslington, York, YO10 5DD, UK |
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Abstract: | We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing
the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density
matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann
entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss
the problem of separability by pointing out that separability of density matrices of graphs does not always depend on the
labelling of the vertices. We consider graphs with a tensor product structure and simple cases for which combinatorial properties
are linked to the entanglement of the state. We calculate the concurrence of all graphs on four vertices representing entangled
states. It turns out that for these graphs the value of the concurrence is exactly fractional.
Received July 28, 2004 |
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Keywords: | 05C50 81P68 |
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