Let M be a p-by-q matrix with nonnegative entries. The positive semidefinite rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite matrices $A_i, B_j$ of size $k \times k$ such that $M_{ij} = \text{trace}(A_i B_j)$. The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedra and information-theoretic applications. In this paper we develop and survey the main mathematical properties of psd rank, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.Comment: 35 page

The positive semidefinite (psd) rank of a polytope is the smallest k for which the cone of k × k real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound. We give several classes of polytopes that achieve the minimum possible psd rank including a complete characterization in dimensions two and three.

We present various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv) 1 4 improving on previous lower bounds. For polygons with v vertices, we show that psd rank cannot exceed 4 v/6 which in turn shows that the psd rank of a p × q matrix of rank three is at most 4 min{p, q}/6 . In general, a nonnegative matrix of rank k+1 2 has psd rank at least k and we pose the problem of deciding whether the psd rank is exactly k. Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when k is fixed.