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The Bloch sphere is homeomorphic to the Riemann sphere, and there exists a stereographic projection $\Bbb S^2\to \Bbb C_\infty$. But this only holds for pure states. To quote Wikipedia:

Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space. The space of pure states of a quantum system is given by the one-dimensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space). For a two-dimensional Hilbert space, this is simply the complex projective line $\Bbb C P^1$. This is the Bloch sphere.

Essentially the set of all possible pure states of a qubit is homeomorphic to $\Bbb C P^1$.

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Questions:

  1. Is the set of mixed states homeomorphic to any geometrical structure (or complex projective space) in particular? Is there any similar stereographic projection corresponding to the mixed states?

  2. Can we assign a manifold structure to the set of mixed states? For instance, $\Bbb C P^1$ is basically the manifold constructed by quotient as Lie group ($\Bbb C^\times$) action on $\Bbb C^2 \setminus \{0\}$. But that only corresponds to pure states.


Edit:

Moretti says in his Physics SE answer that:

In finite dimension, barring the trivial case $\text{dim}({\cal H})=2$ where the structure of the space of the states is pictured by the Poincaré-Bloch ball as a manifold with boundary, $S(\cal H)$ has a structure which generalizes that of a manifold with boundary.

So, I suppose, in this context of qubits, the boundary is $\Bbb C P^1$, and the manifold without the boundary is the set of mixed states with some additional structure (?).

How do we describe this manifold without boundary in mathematical terms? I guess it would trivially be a 3-ball.

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Almost. You get a manifold with boundary with the Bloch ball. The radius from the origin parameterizing how pure it is. The origin being maximally mixed. This isn't a manifold because a point on the boundary has a neighborhood that is homeomorphic to a half space but not one homeomorphic to $\Bbb R^n$.

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  • $\begingroup$ "The radius from the origin parameterizing how pure it is." Do you have any reference on this and its proof? This is something I didn't know about but it certainly looks interesting. I've asked it here. $\endgroup$ – Sanchayan Dutta Mar 3 at 0:29
  • $\begingroup$ I can just edit answer here with the proof. No need for new question. $\endgroup$ – AHusain Mar 3 at 1:02
  • $\begingroup$ I think a new thread with a specific title will be useful for future visitors with the same question. Descriptive titles are helpful for searchability purposes. You certainly can edit your answer to add it; I'll link it to the new thread. :) $\endgroup$ – Sanchayan Dutta Mar 3 at 8:06

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