observed by Archimedes, any two zones of the same height on a sphere have the same area. Hence this latter measure is proportional to Lebesgue measure on the interval $[-r,r]$. An analogous statement holds for arbitrary projective spaces ().

![A Bloch sphere, its height function and the induced measure: a generic coadjoint $\operatorname{SU}(2)$-orbit, its Abelian moment map and the Abelian Duistermaat–Heckman measure.[]{data-label="bloch ball figure"}](blochsphere.png){width="6cm"}

Observe that the components $\braket{X, \Phi_{\operatorname{SU}(\mathcal H)}} \colon \mathcal O_\lambda \rightarrow {\mathbb R}$ of the moment map send a quantum state to the expectation value of the corresponding observable $-i X$. Without loss of generality, we may assume that $X$ generates a one-dimensional torus and that $X$ has one-dimensional eigenspaces. Then $\braket{X, \Phi_{\operatorname{SU}(\mathcal H)}}$ is just the moment map for the action of the torus generated by $X$ and its distribution can be computed immediately by using the Abelian Heckman formula (). This gives a short and conceptual proof of the formula derived recently in [@venutizanardi12].

Reduced Density Matrices {#reduced density matrices}
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Composite quantum systems are modeled by the tensor product of the Hilbert spaces describing their constituents. It is useful to think of these subsystems as individual particles, although they can be of more 