Projective rank, orthogonal representation, minimum positive semidefinite rank, fractional, Tsirelson’s problem, graph, matrix
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.
Hogben, Leslie; Palmowski, Kevin F.; Roberson, David E.; and Severini, Simone.
"Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions",
Electronic Journal of Linear Algebra,
Volume 32, pp. 98-115.