ORTHOGONAL REPRESENTATIONS, PROJECTIVE RANK, AND FRACTIONAL MINIMUM POSITIVE SEMIDEFINITE RANK: CONNECTIONS AND NEW DIRECTIONS

Fractional minimum positive semidefinite rank is defined from gamma-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 gamma-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 gamma-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.