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Dear Rainer (if I may),
it is not completely clear to me what SQM is. In the operational account of the theory (let’s call that OQM) the only requirement on a preparation process is that the preparing system is dynamically decoupled from the ES and from whatever measuring device you plan to use. Entanglement between preparing system and ES is fine, in which case, of course, the prepared state would be mixed. That is not a conceptual issue. If the artful experimenter has achieved a pure state then, of course, the initial state was a product.
If you insist that every system (including your ES) ought to be in an ontologically interpreted pure state (some people, but not OQM, consider that part of SQM), then you would perhaps demand some particular choice of basis on the preparing system, and the outgoing pure states would be conditional on that. The prepared density matrix would thereby represented as a particular ensemble of pure states. But any choice of basis will do (This is one of the ways to discuss the statistics of a CHSH-type correlation experiment). In particular, each of the outgoing pure states of the ensemble will satisfy the Schrödinger equation for that subsystem (which makes sense, because we assumed dynamical decoupling).
Now in the Bohmian case you need more, as this theory rarely restricts to subsystems gracefully. In fact, they will probably sell this bug to you as a feature (the discovery of nonlocality) and say that anyhow the theory is not meant to work for any ES, but only for the whole Universe. If you want BM to hold effectively for the subsystem, you should take the “effective wave function” as described by Roderich. In an entangled situation the prepared system’s trajectory usually depends on the Y of the preparing device, and thus, for example on the dynamics of the preparing system, even long after, in SQM or OQM you have traced out the preparing system. That is, in BM you can usually not “trace out”. In particular, your conditional wave functions have no inclination to satisfy the Schrödinger equation of the ES. This is why you need an extra condition on the position space supports of the wave functions, as Roderich explained, which effectively puts you in a product situation, basically the only case where it works. There is no general physical reason I can see why that support condition should hold, and it is probably easy to come up with quite ordinary preparing processes where it does not.
Bohmians usually argue that it “obviously” holds when the selection criterion on the preparing device is macroscopic. I have no idea what they would say if what is selected on such a pointer criterion is still mixed. Maybe call in an exorcist. Or do some more hand waving and wishful thinking. Or denounce such cases as artificial. Maybe somebody will even answer here.
With best regards, Reinhard