Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions
Palmowski, Kevin F
Roberson, David E
Date of Issue2017
School of Physical and Mathematical Sciences
Fractional minimum positive semidefinite rank is defined from r-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An r-fold version of the traditional definition of minimum positive semidefinite rank of a graph using Hermitian matrices that fit the graph is also presented. This paper also introduces r-fold orthogonal representations of graphs and formalizes the understanding of projective rank as fractional orthogonal rank. Connections of these concepts to quantum theory, including Tsirelson's problem, are discussed.
Electronic Journal of Linear Algebra
© 2017 The Author(s) (Published by International Linear Algebra Society). This paper was published in Electronic Journal of Linear Algebra and is made available as an electronic reprint (preprint) with permission of The Author(s) (Published by International Linear Algebra Society). The published version is available at: [http://dx.doi.org/10.13001/1081-3810.3102]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.