A. Neumaier said:
A theoretical mixed state, but not a mixed state realized in Nature according to the tracing out rule given - unless the state of the big system from which this state was obtained by tracing out the environment was already very weird.
Well, part of the difficulty here is that we really can't do quantum mechanics with 10^{23} particles except in heuristic ways. So the weirdness is perhaps lost in the complexity. But it seems to me that you could set up a situation in which a microscopic difference (whether an electron is spin-up or spin-down) is magnified to make a macroscopic difference. That's what Schrodinger's cat is about. For that matter, that's what any measurement does. So if you consider it weird for a microscopic difference to be magnified to become a macroscopic difference, then such weirdness is an inherent part of the empirical content of QM.
Suppose you set things up so that:
- The detection of a spin-up electron leads to a dead cat.
- The detection of a spin-down electron leads to a live cat.
Then you create an electron that is in a superposition \alpha |up\rangle + \beta |down\rangle, and you send it to the detector. What happens? Well, the Copenhagen interpretation would tell us that macroscopic objects like cats are classical, not quantum. So rather than leading to a superposition of a dead cat and a live cat, what we would get is EITHER a dead cat, with probability |\alpha|^2, or a live cat, with probability |\beta|^2. But that seems inconsistent to me. Why, for small systems, do we get superpositions, rather than alternatives, but for large systems, we get alternatives? That's the weirdness, if not outright inconsistency, of standard quantum mechanics.
Of course, some people claim that decoherence explains why we get alternatives, rather than superpositions, but I don't think it actually does that. What it explains is that superpositions rapidly spread with time: You start off with a single particle in a superposition of states, and then it interacts with more particles putting that composite system into a superposition, and that composite system interacts with the environment (the electromagnetic field) putting it into a superposition of states. The superposition doesn't go away, but it spreads to infect the whole universe (or our little part of it, anyway). But then a trace over everything other than the system of interest gives us what looks like a mixed state, where we can interpret the components of the mixture as alternatives, rather than superpositions.