How to obtain the Fubini-Study metric for $SU(2)$ coherent states, using bra-ket notation? I know that the Fubini Study metric for the $SU(2)$ coherent state is the metric on $CP^1$. The $SU(2)$ un-normalised coherent state is given by
$$
\mid z\rangle=\sum_{m=-j}^{m=+j}\sqrt{\left(
\begin{array}{c}
 2 j \\
 j+m \\
\end{array}
\right)}z^{j+m}\mid j,m\rangle
$$
The normalisation is
$$
    \langle z\mid z\rangle =(1+\mid z\mid^2)^{2j}
$$
I want to obtain the Fubini Study metric using the above state vector. Where I want to use the formula given below
$$
ds^2=\frac{\langle dz\mid dz\rangle}{\langle z\mid z\rangle}-\frac{\langle dz\mid z\rangle\langle z\mid dz\rangle}{\langle z\mid z\rangle^2}
$$
PS: After Prof. Mike Stone's answer, I figured it out myself. Answer is posted below.
 A: To get the metric for ${\mathbb C}P^1\equiv S^2$ you only need the unnormalized  $j=1/2$ coherent state
$$
\psi(z) = |z\rangle= \left[\matrix {1\cr z}\right], \quad 
\psi^\dagger(z)= \langle z|= \left[\matrix{ 1 &  \bar z}\right],
$$
so $\langle z|z\rangle= 1+|z|^2$ and
$$
\langle z |dz\rangle = \left[\matrix{1 & \bar z}\right]\left[\matrix{0\cr dz}\right]=\bar z dz.
$$
Similarly
$$
 \langle dz |z\rangle=  \left[\matrix{0 &  d\bar  z}\right]\left[\matrix{1\cr z}\right] z d\bar z
$$
and
$$
\langle dz |dz\rangle= \left[\matrix{0 &  d\bar  z}\right]\left[\matrix{0\cr dz}\right]= d\bar z dz.
$$
The Fubini-Study metric becomes
$$
ds^2 = \frac{\langle dz|dz\rangle}{\langle z|z\rangle} - \frac{\langle dz|z\rangle\langle z|dz\rangle}{\langle z|z\rangle^2}\\=\frac{d\bar z dz}{(1+|z^2|)} - \frac{ \bar z z  d\bar z dz}{(1+|z|^2)^2}\\
\frac{(1+|z|^2) d\bar z dz}{(1+|z|^2)^2} -  \frac{ |z|^2  d\bar z dz}{(1+|z|^2)^2}\\
= \frac {d\bar z dz}{(1+|z|^2)^2} 
$$
which is (1/4)  the usual metric on the 2-sphere in stereographic coordinates.
PS: I think your $j$ coherent state needs $(1+|z|^2)^j$ to normaize it.   Similar but more compicated algebra gives the metric for a sphere of area $2j$ times larger than that for $j=1/2$
A: I know that the Fubini Study metric for the SU(2) coherent state is the metric on $CP^1$. The SU(2) un normalised coherent state is given by
$$
\mid z\rangle=\sum_{m=-j}^{m=+j}\sqrt{\left(
\begin{array}{c}
 2 j \\
 j+m \\
\end{array}
\right)}z^{j+m}\mid j,m\rangle
$$
The normalisation is
$$
    \langle z\mid z\rangle =(1+\mid z\mid^2)^{2j}
$$
We want to obtain the Fubini Study metric using the above state vector. After Prof. Mike's answer I calculated this
$$
\mid dz\rangle=\sum_{m=-j}^{m=+j}\sqrt{\left(
\begin{array}{c}
 2 j \\
 j+m \\
\end{array}
\right)}(j+m)z^{j+m-1}dz\mid j,m\rangle
$$
Thus
$$
\langle z\mid dz\rangle=\sum_{m=-j}^{m=+j}\left(
\begin{array}{c}
 2 j \\
 j+m \\
\end{array}
\right)(j+m)(z^*z)^{j+m-1}z^*dz
$$
After some manipulation we get
$$
\langle z\mid dz\rangle\\
=(z^*dz)\sum_{k=0}^{n}\left(
\begin{array}{c}
 n \\
  k \\
\end{array}
\right)k(z^*z)^{k-1}=2j(z^*dz)(1+\mid z\mid^2)^{2j-1}
$$
Where, $n=2j$ and $k=j+m$. Similarly, we get
$$
\langle dz\mid z\rangle=2j(zdz^*)(1+\mid z\mid^2)^{2j-1}
$$
and finally we get
$$
\frac{\langle dz\mid z\rangle\langle z\mid dz\rangle}{\langle z\mid z\rangle^2}=4j^2\frac{\mid z\mid^2 dz dz^*}{(1+\mid z\mid^2)^2}
$$
Now let us proceed to the next term
$$
\langle dz\mid dz\rangle=dz^*dz\sum_{m=-j}^{m=+j}\left(
\begin{array}{c}
 2 j \\
 j+m \\
\end{array}
\right)(j+m)^2(z^*z)^{j+m-1}\\
=dz^*dz\sum_{k=0}^{n}\left(
\begin{array}{c}
 n \\
 k \\
\end{array}
\right)k^2(z^*z)^{k-1}\\
=dzdz^*\left(2j(1+\mid z\mid^2)^{2j-1}+2j(2j-1)\mid z\mid^2(1+\mid z\mid^2)^{2j-2}\right)
$$
Thus, finally we get
$$
\frac{\langle dz\mid dz\rangle}{\langle z\mid z\rangle}=dzdz^*\left(\frac{2j}{(1+\mid z\mid^2)}+\frac{(4j^2-2j)\mid z\mid^2}{(1+\mid z\mid^2)^2}\right)
$$
Now finally we plug these expressions back into the expression for the FS metric. The $4j^2$ term cancels out
$$
ds^2=\frac{\langle dz\mid dz\rangle}{\langle z\mid z\rangle}-\frac{\langle dz\mid z\rangle\langle z\mid dz\rangle}{\langle z\mid z\rangle^2}\\
=dzdz^*\left(\frac{2j}{(1+\mid z\mid^2)}-\frac{2j\mid z\mid^2}{(1+\mid z\mid^2)^2}\right)\\
=\frac{2j~~dz dz^*}{(1+\mid z\mid^2)^2}
$$
This is the standard metric on $CP^1$ in affine coordinates. This result matches with the metric obtained using the Kahler form $\phi(z)=2j\log(1+\mid z\mid^2)$
