  We revisit the criterion of multi-particle entanglement based on the overlaps
of a given quantum state $\rho$ with maximally entangled states. For a system
of $m$ particles, each with $N$ distinct states, we prove that $\rho$ is
$m$-particle negative partial transpose (NPT) entangled, if there exists a
maximally entangled state $|{\rm MES}>$, such that $<{\rm MES}|\rho|{\rm
MES}>>{1}/{N}$. While this sufficiency condition is weaker than the
Peres-Horodecki criterion in all cases, it applies to multi-particle systems,
and becomes especially useful when the number of particles ($m$) is large. We
also consider the converse of this criterion and illustrate its invalidity with
counter examples.
