the literature (). The fact that this subsumes our main question above, and in particular the one-body quantum marginal problem, while the existing literature does not, is the crucial second point of this work (). A third feature is our statement of the Abelianization procedure via the derivative principle for invariant measures (); though we have not seen this principle so formulated in the literature, there certainly are predecessors and the result should be equivalent to one of Harish-Chandra [@harishchandra57]. That a “quantized” version of our algorithm can be used to compute (efficiently, unlike existing algorithms, see ) multiplicities for the branching problem of representation theory may be seen as the fourth mathematical consequence of our approach.

The basic strategy we follow to address our main question above is a sequence of reductions. Firstly, our general quantum problem is replaced by an equivalent but more tractable one by “purification” of the quantum state (). This reduced problem is seen to satisfy a weak non-degeneracy assumption, so that the image of the moment map does not lie entirely on a wall in the relevant Weyl chamber. Under this assumption we can reduce via a *derivative principle* to the Duistermaat–Heckman measure for 