  The positive semidefinite (psd) rank of a polytope is the smallest $k$ for
which the cone of $k \times k$ 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.
