Zero-correlation entanglement. (13th January 2020) - Ohira, Toru
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 ($\langle{ψ}|\mathcal{X}\mathcal{Y}|ψ\rangle - \langle{ψ}|\mathcal{X}|ψ\rangle\langle{ψ}|\mathcal{Y}|ψ\rangle= 0$ ). 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.