 If the environment states \(\ket{\phi_{1}} _{e}\) are \(\ket{\phi_{2}}_{e}\) are distinguishable states, then this completely destroys the interference fringes: the particles interact with the screen as if they determinately went through one slit or the other, and the pattern that emerges is the result of overlaying the two single-slit patterns. That is, we can treat the particles as if they followed (approximately) definite trajectories, and apply probabilities in a classical manner.