  We demonstrate the following conclusion: If $|\Psi\rangle$ is a $1d$ or $2d$
nontrivial short range entangled state, and $|\Omega \rangle$ is a trivial
disordered state defined on the same Hilbert space, then the following quantity
(so called strange correlator) $C(r, r^\prime) = \frac{\langle \Omega|\phi(r)
\phi(r^\prime) | \Psi\rangle}{\langle \Omega| \Psi\rangle}$ either saturates to
a constant or decays as a power-law in the limit $|r - r^\prime| \rightarrow
+\infty$, even though both $| \Omega\rangle$ and $| \Psi\rangle$ are quantum
disordered states with short-range correlation. $\phi(r)$ is some local
operator in the Hilbert space. This result is obtained based on both field
theory analysis, and also an explicit computation of $C(r, r^\prime)$ for four
different examples: $1d$ Haldane phase of spin-1 chain, $2d$ quantum spin Hall
insulator with a strong Rashba spin-orbit coupling, $2d$ spin-2 AKLT state on
the square lattice, and the $2d$ bosonic symmetry protected topological phase
with $Z_2$ symmetry. This result can be used as a diagnosis for short range
entangled states in $1d$ and $2d$. A possible diagnosis for $3d$ short range
entangled states is also proposed.
