## Non-Positive Partial Transpose Subspaces Can be as Large as Any Entangled Subspace

**Abstract:**

It is known that, in an (m ⊗ 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.

**Authors:**

- Nathaniel Johnston

**Download:**

- Official publication from Physical Review A
- Preprint from arXiv:1305.0257 [quant-ph]
- Local preprint [pdf]

**Cite as:**

- N. Johnston. Non-positive partial transpose subspaces can be as large as any entangled subspace.
*Physical Review A*, 87:064302, 2013.

**Supplementary material:**

- Code for constructing a state whose partial transpose has the maximum number of negative eigenvalues – A MATLAB script for explicitly constructing a density matrix whose partial transpose has (m-1)(n-1) negative eigenvalues.
- The spectrum of the partial transpose of a density matrix – A blog post that discusses what is and isn’t known about the eigenvalues of the partial transpose of a density matrix.

**Related Papers:**

- The non-m-positive dimension of a positive linear map – a follow-up paper that builds on many of the ideas in this one