\(\newcommand{\bbA}{\mathbb{A}}\)
\(\newcommand{\bbB}{\mathbb{B}}\)
\(\newcommand{\bbC}{\mathbb{C}}\)
\(\newcommand{\bbD}{\mathbb{D}}\)
\(\newcommand{\bbE}{\mathbb{E}}\)
\(\newcommand{\bbF}{\mathbb{F}}\)
\(\newcommand{\bbG}{\mathbb{G}}\)
\(\newcommand{\bbH}{\mathbb{H}}\)
\(\newcommand{\bbI}{\mathbb{I}}\)
\(\newcommand{\bbJ}{\mathbb{J}}\)
\(\newcommand{\bbK}{\mathbb{K}}\)
\(\newcommand{\bbL}{\mathbb{L}}\)
\(\newcommand{\bbM}{\mathbb{M}}\)
\(\newcommand{\bbN}{\mathbb{N}}\)
\(\newcommand{\bbO}{\mathbb{O}}\)
\(\newcommand{\bbP}{\mathbb{P}}\)
\(\newcommand{\bbQ}{\mathbb{Q}}\)
\(\newcommand{\bbR}{\mathbb{R}}\)
\(\newcommand{\bbS}{\mathbb{S}}\)
\(\newcommand{\bbT}{\mathbb{T}}\)
\(\newcommand{\bbU}{\mathbb{U}}\)
\(\newcommand{\bbV}{\mathbb{V}}\)
\(\newcommand{\bbW}{\mathbb{W}}\)
\(\newcommand{\bbX}{\mathbb{X}}\)
\(\newcommand{\bbY}{\mathbb{Y}}\)
\(\newcommand{\bbZ}{\mathbb{Z}}\)
\(\newcommand{\calA}{\mathcal{A}}\)
\(\newcommand{\calB}{\mathcal{B}}\)
\(\newcommand{\calC}{\mathcal{C}}\)
\(\newcommand{\calD}{\mathcal{D}}\)
\(\newcommand{\calE}{\mathcal{E}}\)
\(\newcommand{\calF}{\mathcal{F}}\)
\(\newcommand{\calG}{\mathcal{G}}\)
\(\newcommand{\calH}{\mathcal{H}}\)
\(\newcommand{\calI}{\mathcal{I}}\)
\(\newcommand{\calJ}{\mathcal{J}}\)
\(\newcommand{\calK}{\mathcal{K}}\)
\(\newcommand{\calL}{\mathcal{L}}\)
\(\newcommand{\calM}{\mathcal{M}}\)
\(\newcommand{\calN}{\mathcal{N}}\)
\(\newcommand{\calO}{\mathcal{O}}\)
\(\newcommand{\calP}{\mathcal{P}}\)
\(\newcommand{\calQ}{\mathcal{Q}}\)
\(\newcommand{\calR}{\mathcal{R}}\)
\(\newcommand{\calS}{\mathcal{S}}\)
\(\newcommand{\calT}{\mathcal{T}}\)
\(\newcommand{\calU}{\mathcal{U}}\)
\(\newcommand{\calV}{\mathcal{V}}\)
\(\newcommand{\calW}{\mathcal{W}}\)
\(\newcommand{\calX}{\mathcal{X}}\)
\(\newcommand{\calY}{\mathcal{Y}}\)
\(\newcommand{\calZ}{\mathcal{Z}}\)
\(\newcommand{\dag}{\dagger}\)
\(\newcommand{\tr}{\operatorname{tr}}\)
\(\newcommand{\Tr}{\operatorname{Tr}}\)
\(\newcommand{\det}{\operatorname{det}}\)
\(\newcommand{\perm}{\operatorname{perm}}\)
\(\newcommand{\coloneqq}{\mathrel{\vcenter{:}}=}\)
\(\newcommand{\eqqcolon}{\mathrel{=\!\vcenter{:}}}\)
\(\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\)
\(\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\)
\(\DeclareMathOperator*{\argmax}{arg\!\max}\)
\(\DeclareMathOperator*{\argmin}{arg\!\min}\)
\(\newcommand{\binom}[2]\)
\(\newcommand{\bm}[1]{\boldsymbol{#1}}\)
\(\newcommand{\Im}[1]{\operatorname{Im}\pargs{#1}}\)
\(\newcommand{\Re}[1]{\operatorname{Re}\pargs{#1}}\)
\(\newcommand{\sgn}[1]{\operatorname{sgn}\pargs{#1}}\)
\(\newcommand{\ket}[1]{\left\lvert #1 \right\rangle}\)
\(\newcommand{\bra}[1]{\left\langle #1 \right\rvert}\)
\(\newcommand{\braket}[2]{\left\langle #1 \vert #2 \right\rangle}\)
\(\newcommand{\ketbra}[2]{\left\lvert #1 \right\rangle\!\left\langle #2 \right\rvert}\)
\(\newcommand{\parentheses}[1]{\left(#1\right)}\)
\(\newcommand{\brackets}[1]{\left[#1\right]}\)
\(\newcommand{\curlybrackets}[1]{\left\{#1\right\}}\)
\(\newcommand{\angles}[1]{\left\langle #1\right\rangle}\)
\(\newcommand{\ceil}[1]{\left\lceil #1\right\rceil}\)
\(\newcommand{\floor}[1]{\left\lfloor #1\right\rfloor}\)
\(\newcommand{\set}{\curlybrackets}\)
\(\newcommand{\expval}{\angles}\)
\(\newcommand{\comm}[2]{\brackets{#1, #2}}\)
\(\newcommand{\pargs}[1]{\!\parentheses{#1}}\)
\(\newcommand{\bargs}[1]{\!\brackets{#1}}\)
\(\newcommand{\cbargs}[1]{\!\curlybrackets{#1}}\)
\(\newcommand{\Dom}[1]{\text{Dom}\pargs{#1}}\)
\(\DeclareMathOperator*{\Expval}{\mathbb{E}}\)
\(\newcommand{\EX}[2]{\Expval_{#1}\bargs{#2}}\)
\(\newcommand{\Pr}{\operatorname{Pr}}\)
\(\newcommand{\Hom}[1]{\operatorname{Hom}\pargs{#1}}\)
\(\newcommand{\End}[1]{\operatorname{End}\pargs{#1}}\)
\(\newcommand{\Aut}[1]{\operatorname{Aut}\pargs{#1}}\)
\(\newcommand{\bigO}[1]{\calO\pargs{#1}}\)
\(\newcommand{\littleo}[1]{o\pargs{#1}}\)
\(\newcommand{\bigOmega}[1]{\Omega\pargs{#1}}\)
\(\newcommand{\littleomega}[1]{\omega\pargs{#1}}\)
\(\newcommand{\bigTheta}[1]{\Theta\pargs{#1}}\)
\(\newcommand{\poly}[1]{\operatorname{poly}\pargs{#1}}\)
\(\newcommand{\polylog}[1]{\operatorname{polylog}\pargs{#1}}\)
\(\newcommand{\dd}{\mathop{}\!\mathrm{d}}\)
\(\newcommand{\Dd}[1]{\mathop{}\!\mathrm{d^#1}}\)
\(\newcommand{\e}{\mathrm{e}}\)
\(\newcommand{\i}{\mathrm{i}}\)
\(\newcommand{\U}{\mathrm{U}}\)
\(\newcommand{\O}{\mathrm{O}}\)
\(\newcommand{\SU}{\mathrm{SU}}\)
\(\newcommand{\SO}{\mathrm{SO}}\)
\(\newcommand{\Sp}{\mathrm{Sp}}\)
\(\newcommand{\GL}{\mathrm{GL}}\)
(07 Jul 2021) Denote the set of all pure quantum states on an $n$-dimensional Hilbert space as $S$. Every pure quantum state is a unit vector in $\bbC^n$. As a real vector space, $\bbC^n \cong \bbR^{2n}$. A unit vector in $\bbR^{2n}$ lives on the sphere $\text S^{2n-1}$. Therefore, we are tempted to say that $S \cong \text S^{2n-1}$. However, pure states come along with the equivalence relation $\sim$ denoting the irrelevance of a global phase factor; $\ket \psi \sim e^{i \theta} \ket \psi$. Thus, $S \cong \text S^{2n-1} / \mathord \sim$. Equivalently, $S \cong \text S^{2n-1} / \text U(1)$ where $\text U(1)$ is the one-dimensional unitary group. This is exactly the definition of the complex projective space $\bbC \bbP^{n-1}$. So we finally conclude that
\[S \cong \bbC \bbP^{n-1}.\]
Denote the set of all mixed quantum states (i.e. density matrices) on an $n$-dimensional Hilbert space as $D$. Any mixed state can be written as $\rho = \sum_i p_i \ketbra{i}{i}$ for an orthonormal basis $\set{\ket i}$. Such a $\rho$ can be purified as $\ket \psi = \sum_i \sqrt{p_i}\ket i \otimes \ket j$ for an orthonormal basis $\set{\ket j}$. $\psi$ is an $n^2$-dimensional pure state, so we are tempted to say that $D \cong \bbC\bbP^{n^2-1}$. However, we also know that $\rho$ could have just as well been purified as $\ket \psi = \sum_i \sqrt{p_i}\ket i \otimes U\ket j$ for any unitary $U \in \text U(n)$. Therefore, $D \cong \bbC\bbP^{n^2-1}/ \text U(n)$. But we already divided out by $\text U(1)$ in the definition of $\bbC\bbP^{n^2-1}$. Therefore,
\[D \cong \bbC\bbP^{n^2-1}/ \text{SU}(n).\]