[Quantum Mechanics] Quantum Fisher Information for a Pure State

by Arpayon
Tags: fisher, information, mechanics, pure, quantum, state
 P: 2 Hi everyone. 1. The problem statement, all variables and given/known data 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 2. Relevant equations I have to use the given equation for Fisher Information with the fact that $\rho$ is pure. 3. 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 :(
 P: 52 With ρ = | ri >< ri | I get the result... (split J2 and insert Ʃj | rj >< rj |)
 P: 2 Problem is that I don't get how $r_i$ and $r_j$ behave with this particular $\rho$ What do you mean by splitting $J^2$?

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