Decoherence is often presented as a program to solve the measurement problem using only the bare bones quantum mechanics framework of a Hilbert space and unitary Schrödinger time evolution. As often stated, there is no need to postulate a collapse or splittings into many worlds for decoherence to work.
However, I have my misgivings. What do you guys think of the following example? The system is an electron and the environment contains a Stern-Gerlach apparatus. In configuration A, the apparatus measures the spin of the electron in the z-direction, and in configuration B, the x-direction. A standard decoherence analysis will pick out the z-basis as the pointer states for configuration A, and the x-basis for B. So far, so good.
Now instead, set up a control qubit $\left( |0\rangle + |1\rangle \right)/\sqrt{2}$ using a Hadamard gate. A value of 0 sets up the apparatus in configuration A, while a value of 1 sets it up in configuration B. The crucial detail here is the environment is in a superposition. Now as far as I know, all the standard criteria for picking out system pointer states like for instance, diagonalizing the reduced density matrix, the predictability sieve, purification time, efficiency threshold, etc. , all give out crap for the pointer basis in this example. None of them can capture the fact that for a value of 0, the pointer states lie in the z-direction, but for a value of 1, they lie in the x-direction instead.
Would it be fair to claim we need something extra like a collapse of the environment into configuration A or B, or a splitting into the worlds of configuration A and configuration B with each world treated in a different manner, to actually make decoherence work? Or are there loopholes allowing us to stick to a bare bones quantum mechanical framework?
Most treatments of decoherence assume the system is in a superposition right after preparation but hardly any consider the scenario where the environment is also in a superposition during preparation. This, I think, is incomplete. And this is no idle matter either because the environment is always in a superposition. Structure formation leading to the condensation of matter into superclusters, galaxies, stars and planets have their origins in quantum fluctuations. Even major patterns here on earth like large scale weather events depend sensitively upon quantum fluctuations via the butterfly effect. The unpredictable, or at least not fully predictable, behavior of experimenters are also sensitive to quantum fluctuations in their brain. Without presupposing either collapse or a many worlds split, how can one get down to a proper decoherence analysis?