  It is known that, in an $(m \otimes n)$-dimensional quantum system, the
maximum dimension of a subspace that contains only entangled states is
(m-1)(n-1). We show that the exact same bound is tight if we require the
stronger condition that every state with range in the subspace has non-positive
partial transpose. As an immediate corollary of our result, we solve an open
question that asks for the maximum number of negative eigenvalues of the
partial transpose of a quantum state. In particular, we give an explicit method
of construction of a bipartite state whose partial transpose has (m-1)(n-1)
negative eigenvalues, which is necessarily maximal, despite recent numerical
evidence that suggested such states may not exist for large m and n.
