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

TL;DR

This paper proves that the maximum dimension of subspaces with all states having non-positive partial transpose matches the known maximum for entangled subspaces, and constructs states with the maximum number of negative eigenvalues in their partial transpose.

Contribution

It establishes the tight bound for subspaces with non-positive partial transpose and provides explicit constructions for states with maximal negative eigenvalues.

Findings

Maximum dimension of subspace with non-positive partial transpose equals (m-1)(n-1)

Explicit construction of states with maximal negative eigenvalues in partial transpose

Solves an open problem regarding the maximum number of negative eigenvalues

Abstract

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.