Decoherence Implies Information Gain

It is shown that if the wave function of a quantum system undergoes an arbitrary random transformation such that the diagonal elements of the density matrix in the decoherence basis associated with a preferred observable remain constant, then (i) the off-diagonal elements of the density matrix become smaller in magnitude, and (ii) the state of the system gains information about the preferred observable from its environment in the sense that the uncertainty of the observable is reduced in the transformed state. These results do not depend on the details of how the system-environment interaction generates the random state transformation, and together imply that decoherence leads in general to information gain, not information loss.