I am currently trying to understand how to show that the Fubini-Study metric is the natural metric in the complex projective space $\mathbb{C}P^n$ in the context of Hilbert space corresponding to some quantum system. I've been trying to follow the explanation on a paper called "Connecting geometry and performance of two-qubit parameterized quantum circuits" but I'm having a hard time wrapping my head around it.
In short, the paper argues that the Fubini-Study metric on $\mathbb{C}P^n$ can be constructed by considering the tangent space $T_{[|\psi\rangle]}\mathbb{C}P^n$ on a point $[|\psi\rangle ]$, where $[|\psi\rangle ]$ denotes the equivalency class given by $[|\psi\rangle ]=\{a|\psi\rangle: a\in \mathbb{C}\}$. Given that tangent space, the Fubini-Study metric can then be defined in terms of the hermitian inner product $h$ of $\mathbb{C}^{n+1}$ by taking the orthogonal complement using the operator $P_{\psi}^{\perp}=I-|\psi\rangle\langle\psi|$ to evaluate the inner product on $T_{|\psi\rangle}\mathbb{C}^{n+1}$, relying on the natural isomorphism $T_{|\psi\rangle}\mathbb{C}^{n+1} \cong \mathbb{C}^{n+1}$, resulting in $$ h(\phi_1, \phi_2)=\langle\phi_1|P^{\perp}_\psi|\phi_2\rangle = \langle\phi_1|\phi_2\rangle-\langle\phi_1|\psi\rangle\langle\psi|\phi_2\rangle $$
This expression is also known as the quantum geometric tensor in the literature. The Fubini-Study metric is supposed to be the real part of this expression. The paper can be found here (pages 4 and 5) if anyone wishes to take a closer look.
In hopes to better understand this, I've been trying to come up with a more detailed derivation but I haven't had much success, so, starting from scratch:
Consider a Hilbert Space $\mathcal{H} = \mathbb{C}^{n+1}$ with inner product $h:\mathcal{H} \times\mathcal{H} \to \mathbb{C}$ defied by $$ h(\phi, \psi)= \langle\phi|\psi\rangle= G(\phi, \psi) + iF(\phi, \psi) $$ where $G(\phi, \psi)$ and $F(\phi, \psi)$ are the real and imaginary parts of $\langle\phi|\psi\rangle$, respectively. The projective state space $\mathcal{P}(\mathcal{H})=\mathbb{C}P^n$ can be constructed by taking th quotient $\mathcal{H}/\sim$ where the equivalency is given by $$ \psi \sim \phi \iff \psi=a\phi; \ \ \ a\in\mathbb{C}. $$
I've been advised to consider the canonical projection $$ \pi:\mathbb{C}^{n+1} \to \mathbb{C}P^{n}\\ z\mapsto[z] $$ and take its pushforward map $$ \pi_*:T_z\mathbb{C}^{n+1}\to T_{[z]}\mathbb{C}P^{n} $$ to show that, when restricted to $z^{\perp}$, $\pi_*$ defines an isomorphism, but I've been unable to do so. I belive it has to do with the fact that $T_z\mathbb{C}^{n+1}=[z] \oplus z^{\perp}$, as pointed out in this related question.
So I would like to know how can i prove this and, from there, how can I use this to arrive at the expression for the so called quantum geometric tensor and, thus, the Fubini-Study metric, showing that it is the natural metric in $\mathbb{C}P^n$.
I'm sorry if my reasoning is sloppy, I have a background in physics and only recently started taking an interest in learning math more seriously, specially geometry, so any help is much appreciated.