  Given a nonnegative matrix M with rational entries, we consider two
quantities: the usual positive semidefinite (psd) rank, where the matrix is
factored through the cone of real symmetric psd matrices, and the
rational-restricted psd rank, where the matrix factors are required to be
rational symmetric psd matrices. It is clear that the rational-restricted psd
rank is always an upper bound to the usual psd rank. We show that this
inequality may be strict by exhibiting a matrix with psd rank four whose
rational-restricted psd rank is strictly greater than four.
