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


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.


  • Nathaniel Johnston


Cite as:

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

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