Adding the environment to the description does not break the linearity of the Schrödinger equation, so superposition persists at the larger composite level.

Indeed, what happens if we include decoherence in the description? Decoherence tells us, among other things, that plenty of interactions are taking place all the time in which differently localised states of the apparatus registering, say, different \(x\)-spin values of an electron couple to different states of the environment. But now, by the same arguments as above, the composite of electron, apparatus and environment will be a superposition of (i) a state corresponding to the environment coup