[Quantum Mechanics] Quantum Fisher Information for a Pure State

Arpayon

Hi everyone.

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


2. Relevant equations
I have to use the given equation for Fisher Information with the fact that [itex]\rho[/itex] is pure.


3. The attempt at a solution
I have difficulties in how to procede. In pure states all the coefficient [itex]r_i[/itex] 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 [itex]r_i[/itex] and [itex]r_j[/itex] behave with this particular
[itex]\rho[/itex]
What do you mean by splitting [itex]J^2[/itex]?