  We introduce a diagnostic for quantum thermalization based on mixed-state
entanglement. Specifically, given a pure state on a tripartite system $ABC$, we
study the scaling of entanglement negativity between $A$ and $B$. For
representative states of self-thermalizing systems, either eigenstates or
states obtained by a long-time evolution of product states, negativity shows a
sharp transition from an area-law scaling to a volume-law scaling when the
subsystem volume fraction is tuned across a finite critical value. In contrast,
for a system with quasiparticles, it exhibits a volume-law scaling irrespective
of the subsystem fraction. For many-body localized systems, the same quantity
shows an area-law scaling for eigenstates, and volume-law scaling for long-time
evolved product states, irrespective of the subsystem fraction. We provide a
combination of numerical observations and analytical arguments in support of
our conjecture. Along the way, we prove and utilize a `continuity bound' for
negativity: we bound the difference in negativity for two density matrices in
terms of the Hilbert-Schmidt norm of their difference.
