themselves are Hamiltonian manifolds. Probability distributions can be realized as quantum states of a special form, and the passage from quantum to classical is related to passing from a non-Abelian group to its maximal torus (see for precise statements). The “semiclassical limit” well-known in geometric quantization does not have an analogous physical meaning in our setting; its significance is solely to connect the symplectic geometry with representation theory (see ).

Density Matrices and Purification {#math phys dictionary}
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The applicability of symplectic geometry to the quantum marginal problem relies on the close relation between the Lie algebra of $\operatorname{SU}(d)$ and the density matrices of quantum mechanics, and on the fact that restricting to certain subgroups has the physical meaning of passing to reduced density matrices, which describe the quantum state of subsystems. In this section we will describe this relationship in some detail (, ), and show how one can reduce the general problem of computing joint eigenvalue distributions of reduced density matrices to the case of globally pure quantum states, that is, to the Duistermaat–Heckman measure for a projective space (). We briefly discuss how probability distributions and the classical marginal problem are embedded in our setup () and 