
#1
Mar812, 05:43 AM

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 [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_ir_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_ir_j)^2}{r_i+r_j}\langle r_iJr_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\psiJ^2\psi\rangle(\langle\psiJ\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 :( 



#2
Mar812, 06:20 AM

P: 52

With ρ =  ri >< ri  I get the result... (split J^{2} and insert Ʃj  rj >< rj )




#3
Mar812, 06:43 AM

P: 2

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]? 


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