Negativity vs Entropy in Entanglement Witnessing

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.