Purity sieve for models with factorizable interactions

Evolution of purity in case of factorizable interaction between an open system and an environment is investigated. We derive a simple expression for purity decrease at the first instants of evolution (when purity is close to unity), which appears to be quadratic in time. According to this expression, purity at very small times is maximal when the initial state of an open system coincide with one of the eigenstates of the interaction operator, no matter how strong the interaction is. However, it is widely known that eigenstates of the interaction Hamiltonian are not pointer states when, for example, interaction is weak (compared to self-Hamiltonian). Therefore the procedure of selecting pointer states by purity maximization (known as "purity sieve") should not rely on short-time purity behavior. We propose a modification of the purity sieve criterion which takes into account purity evolution at longer times. As an example of its applicability we recover known results for pointer states of a particle undergoing quantum Brownian motion; we point out that the criterion is not applicable for some other models, however. It is argued that the proposed modified purity sieve may be used for selecting pointer states of a particle undergoing decoherence through colliions.