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 X, Y with zero correlations (ψ|XY|ψ-ψ|X|ψ (ψ|Y|ψ= 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.