The Non-m-Positive Dimension of a Positive Linear Map. (arXiv:1906.04517v1 [quant-ph])

We introduce a property of a matrix-valued linear map $\Phi$ that we call its
"non-m-positive dimension" (or "non-mP dimension" for short), which measures
how large a subspace can be if every quantum state supported on the subspace is
non-positive under the action of $I_m \otimes \Phi$. Equivalently, the non-mP
dimension of $\Phi$ tells us the maximal number of negative eigenvalues that
the adjoint map $I_m \otimes \Phi^*$ can produce from a positive semidefinite
input. We explore the basic properties of this quantity and show that it can be
thought of as a measure of how good $\Phi$ is at detecting entanglement in
quantum states. We derive non-trivial bounds for this quantity for some
well-known positive maps of interest, including the transpose map, reduction
map, Choi map, and Breuer--Hall map. We also extend some of our results to the
case of higher Schmidt number as well as the multipartite case. In particular,
we construct the largest possible multipartite subspace with the property that
every state supported on that subspace has non-positive partial transpose
across at least one bipartite cut, and we use our results to construct
multipartite decomposable entanglement witnesses with the maximum number of
negative eigenvalues.

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