In this work, we prove that while all measures of mixedness can be used to
witness entanglement, no measure of mixedness is more sensitive than the
negativity of the partial transpose. However, computing either the negativity
or differences between von Neumann entropies to witness entanglement requires
complete knowledge of the joint density matrix (and is therefore not practical
at high dimension). In light of this, we examine joint vs marginal purities as
a witness of entanglement, (which can be obtained directly through interference
measurements) and find that comparing purities is actually more sensitive at
witnessing entanglement than using von Neumann entropies while also providing
tight upper and lower bounds to it in the high-entanglement limit.Comment: 4 pages, 1 figur