 A stronger form of the decoherence condition, namely the vanishing of both the real and imaginary part of the decoherence functional, can be used to prove theorems on the existence of (later) ‘permanent records’ of (earlier) events in a history, which is a generalisation of the idea of ‘environmental  monitoring’.[8]  For instance, if the state \(\varrho\) is a pure state \(\lvert \psi \rangle\langle\psi\rvert\) this (strong) decoherence condition is equivalent, for all \(n\), to the orthogonality of the vectors