[Quantum Mechanics] Quantum Fisher Information for a Pure State

[Arpayon]

Hi everyone.

Homework Statement

We are given N spins 1/2. A rotation is defined as
$\rho_\theta=e^{-i\theta J_n}\rho_\theta e^{i\theta J_n}$
on an Hilbert Space H, with
$J_n=n_xJ_x+n_yJ_y+n_zJ_z\:,\quad n_x^2+n_y^2+n_z^2=1$,
and $\theta$ isn't related to any observable.
Given a quantum state $\rho=\sum_ir_i|r_i\rangle\langle r_i|$,
the Formula for the Quantum Fisher Information I've come to is
$F[\rho,J]=2\sum_{i,j}\frac{(r_i-r_j)^2}{r_i+r_j}|\langle r_i|J|r_j\rangle|^2$ (which is indeed right).
Problem is that I have to calculate the Quantum Fisher Information for a Pure state $\rho=|\psi\rangle\langle\psi|$.
The solution should be $F[\rho,J]=4\Delta_\psi^2J$,
where $\Delta_\psi^2J=\langle\psi|J^2|\psi\rangle-(\langle\psi|J|\psi\rangle)^2$ is the variance of J, but I can't come to it

Homework Equations

I have to use the given equation for Fisher Information with the fact that $\rho$ is pure.

The Attempt at a Solution

I have difficulties in how to procede. In pure states all the coefficient $r_i$ should be 0, except for one of the, which should be 1.
Any idea?
Many thanks, this is quite urgent :(

 

[bloby]

With ρ = | ri >< ri | I get the result... (split J² and insert Ʃj | rj >< rj |)

 

[Arpayon]

Problem is that I don't get how $r_i$ and $r_j$ behave with this particular
$\rho$
What do you mean by splitting $J^2$?