the maximal torus action (), which we evaluate by a “single-summand” algorithm along the lines of Boysal–Vergne [@boysalvergne09] ().

This derivative principle holds for more general $K$-invariant measures on the dual of the Lie algebra, $\mathfrak k^*$: Every invariant measure can be reconstructed from its projection onto the dual of the Lie algebra of the maximal torus by taking partial derivatives in the direction of negative roots (). Again, we remark that such a reduction is not possible on the level of the supports of the Duistermaat–Heckman measures, i.e., on the moment polytopes. We also remark that in the case of the one-body quantum marginal problem, the Duistermaat–Heckman measure for the maximal torus action is the joint distribution of the diagonal entries of the one-body reduced density matrices (as opposed to their eigenvalues). Amusingly, this distribution can be viewed as the solution to the quantitative version of a classical marginal problem ().

The single-summand algorithm alluded to above is an effective method for computing Duistermaat–Heckman measures for torus actions on projective spaces (). Its name stems from the fact that it amounts to evaluating a single summand of the kind that occurs in the well-known Heckman formula of Guillemin–Lerman–Sternberg [@guilleminlermansternberg88], 