I've confused myself about the following scenario:
Suppose you make a black hole out of states with spin aligned into one direction, say the positive x-direction, and let's call this "up". Then the so-created black hole has an non-zero angular momentum because the expectation value of the spin is nonzero. (Let's assume there's no other net contribution to the angular momentum, just for simplicity. I don't think this matters.) The so-created black hole evaporates in the semi-classical limit with an energy spectrum that depends on this expectation value of the total angular momentum. (Forget about what happens in the final stage where the semi-classical approximation breaks down, I don't think this matters.)
Next you make a black hole out of the same initial state with the only difference that all spins are down. Same thing: the emission spectrum - the distribution of outgoing particles over energy - depends on the expectation value of the angular momentum, at least in the semi-classical limit we're pretty sure of that.
Finally, you make a black hole out of a superposition of the two initial states. The expectation value of the total angular momentum is now zero, correspondingly the angular momentum in the semi-classical Kerr-background is zero. What is the decay spectrum (in the early radiation) in this case?
Is the outgoing radiation a) a superposition of the radiation that one obtained from the purely up and purely down case? But then it would depend on the angular momentum which the black hole doesn't have. Or does it b) not depend on the angular momentum of the superposed states (note that what is and what isn't a superposition is ambiguous anyway) but then the evolution doesn't seem to be linear? (In the sense that |psi_in> + |psi'_in> doesn't result in |psi_out> + |psi'_out> when |psi_in> resulted in |psi_out> and same for the dashed case).