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 basis of a preferred observable remain constant, then (i) the off-diagonal elements of the density matrix become smaller in magnitude, and (ii) the system gains information about the preferred observable in the sense that the uncertainty of the observable reduces. These results are independent of the details of how the system-environment interaction generates the transformation of the state of the system, and together imply that decoherence leads to information gain, not information loss.