Editing Full Quantum state tomography with Maximum Likelihood Estimation
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 76: | Line 76: | ||
* Formula for <math>\hat{\rho_p}</math> is <math>\hat{\rho_p}(t) = \hat{T^{\dagger}}(t) \hat{T}(t) / tr\{ \hat{T}^{\dagger}(t) \hat{T}(t)\}</math>. Here <math>\hat{T}(t)</math> is,<math> \hat{T}(t) = \begin{bmatrix} | * Formula for <math>\hat{\rho_p}</math> is <math>\hat{\rho_p}(t) = \hat{T^{\dagger}}(t) \hat{T}(t) / tr\{ \hat{T}^{\dagger}(t) \hat{T}(t)\}</math>. Here <math>\hat{T}(t)</math> is,<math> \hat{T}(t) = \begin{bmatrix} | ||
t_1 & 0 & ... & 0 \\ | t_1 & 0 & ... & 0 \\ | ||
t_{2^ | t_{2^n + 1} + it_{2^n+2} & t_2 & ... & 0 \\ | ||
... & ... & ... & 0 \\ | ... & ... & ... & 0 \\ | ||
t_{4^ | t_{4^n -1} + it_{4^n} & t_{4^n -3} + it_{4^n - 2} & t_{4^n - 5} + it_{4^n - 4} & t_{2^n} | ||
\end{bmatrix} | \end{bmatrix} | ||
</math> | </math> | ||
* Find the minimum of the <math>L(t_1, t_2, ..., t_{ | * Find the minimum of the <math>L(t_1, t_2, ..., t_{n^2})</math> using the formula <math> | ||
L(t_1, t_2, ..., t_{ | L(t_1, t_2, ..., t_{n^2}) = \sum_j \frac{(N\langle E_j|\hat{\rho_p}(t_1, t_2, ..., t_{n^2})|E_j\rangle - p_j)^2}{2N\langle E_j|\hat{\rho_p}(t_1, t_2, ..., t_{n^2})|E_j\rangle} | ||
</math> | </math> | ||
* <math>\hat{\rho_p}</math> can be reconstructed from the values of <math>t_i</math>. | * <math>\hat{\rho_p}</math> can be reconstructed from the values of <math>t_i</math>. |