Non-positive-partial-transpose subspaces can be as large as any entangled subspace

Abstract

It is known that, in an $(m\ensuremath{\bigotimes}n)$-dimensional quantum system, the maximum dimension of a subspace that contains only entangled states is $(m\ensuremath{-}1)(n\ensuremath{-}1)$. 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 $(m\ensuremath{-}1)(n\ensuremath{-}1)$ negative eigenvalues, which is necessarily maximal, despite recent numerical evidence that suggested such states may not exist for large $m$ and $n$.