Zero-correlation entanglement
Abstract 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 $\mathcal{X}, \mathcal{Y}$ with zero correlations ($\langle{\psi}|\mathcal{X}\mathcal{Y}|{\psi}\rangle - \langle{\psi}|\mathcal{X}|{\psi}\rangle\langle{\psi}|\mathcal{Y}|{\psi}\rangle= 0$). 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.
