- #1

Arpayon

- 2

- 0

## Homework Statement

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

## Homework Equations

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

## 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 :(