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A density matrix is a matrix $\rho \in \mathscr{D}:=\{A \in \mathbb{C}^{n \times n}; A^*=A; \operatorname{tr}(A)=1; A \ge 0\}.$

In Quantum Mechanics it is natural to look at a group action

$\Phi: U(n) \times \mathscr{D} \rightarrow \mathscr{D}, (U,\rho) \mapsto U\rho U^*.$

Now, my quetion is: What is the kind of nicest structure that $\mathscr{D}$ can have and why?

First, I observed that $\mathscr{D}$ without the positivity condition is just something like a affine space, but $\mathscr{D}$ does not seem to be something like a manifold, as $\rho \ge 0$ apparently kind of destroys it.

So is there any interesting structure that $\mathscr{D}$ naturally posesses?

I am curious about the answers

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    $\begingroup$ The interior, where $A>0$, is a manifold, since positive definiteness is an open condition. If you want to talk about the boundary as well, it is a spectrahedron math.berkeley.edu/~bernd/WhatIsSpec.pdf . $\endgroup$
    – David E Speyer
    Commented Sep 3, 2015 at 13:10
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    $\begingroup$ It the intersection (let's call it slice) of the convex cone of positive semidefinite matrices with a hyperplane; the open part of the PSD cone may also be viewed as a nice Riemannian manifold. (seems like I was typing my comment while David was also typing...) $\endgroup$
    – Suvrit
    Commented Sep 3, 2015 at 13:11

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This is a symmetric space when equipped by the metric discussed in this question (it is Riemannian, when $p=2$). The metric is clearly invariant under the action; its other properties can be checked (see the references in Suvrit's nice answer to the linked question, or see one of the papers by Freitas and Friedland, e.g. MR2014882 (2004j:30086) Reviewed Friedland, Shmuel(1-ILCC-MS); Freitas, Pedro J.(P-LISBS) Revisiting the Siegel upper half plane. I. (English summary) Linear Algebra Appl. 376 (2004), 19–44. )

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