  We consider a quantum entangled state for two particles, each particle having
two basis states, which includes an entangled pair of spin 1/2 particles. We
show that, for any quantum entangled state vectors of such systems, one can
always find a pair of observable operators X, Y with zero-correlations <XY> =
<X><Y>. At the same time, if we consider the analogous classical system of a
"classically entangled" (statistically non-independent) pair of random
variables taking two values, one can never have zero correlations (zero
covariance, E[XY] - E[X]E[Y] = 0). We provide a general proof to illustrate the
different nature of entanglements in classical and quantum theories.
