Positive semidefinite rank
  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. 
Hamza Fawzi, Jo\~ao Gouveia, Pablo A. Parrilo, Richard Z. Robinson,   Rekha R. Thomas
math.OC cs.DM math.CO