- #1
Arpayon
- 2
- 0
Hi everyone.
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
I have to use the given equation for Fisher Information with the fact that \rho is pure.
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 :(
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 :(
