one) then we shall say that the two chambers are *adjacent* and call the common boundary a *singular wall*.

All Hilbert spaces which we consider in this article are complex and finite-dimensional. We write $P_\psi$ for the orthogonal projection onto a one-dimensional subspace ${\mathbb C}\psi$, and ${\lVertX\rVert}_2 = \sqrt{{\mathrm{tr}\left(X^*X\right)}} = \sum_j s_j^2$ for the Hilbert–Schmidt norm of an operator $X$ with singular values $(s_j)$. We use $\braket{-,-}$ to denote inner products as well as the pairing between measures (or more general distributions) and test functions. We write $\delta_p$ for the *Dirac measure* at $p$, i.e., the probability measure concentrated at the point $p$, and $H_\omega$ for the *Heaviside measure* which is defined by $\langle H_\omega, f \rangle = \int_0^\infty f(t \omega) dt$. We sometimes use the letter ${\mathbf P}$ for probability distributions.

Throughout the paper when we speak of the quantum marginal problem we always refer to its one-body version as described in .

Finally, we offer a word of caution for people acquainted with the theory of geometric quantization [@guilleminsternberg77; @woodhouse92]: Our quantum states do not arise via some quantization procedure from a classical symplectic phase space. In contrast, herein, as detailed in below, the spaces of quantum states 