Editing Full Quantum state tomography with Maximum Likelihood Estimation (section)

==Procedure Description==

'''Input''': copies of the unknown quantum state

'''Output''': Density matrix of the quantum state, <math>\rho</math>

* Pick the measurement basis <math>A</math> where, <math>
       A = \begin{bmatrix}
        \vec{E_1} \\ \vec{E_2} \\ . \\. \\ \vec{E_{d^2}}
       \end{bmatrix}
    </math>
* For <math>j = 1, 2, ..., d^2</math>:
** For <math>i = 1, 2, ..., n</math>:
*** Measure <math>\rho</math> with measurement operator <math>E_j</math>
*** Get measurement result <math>m_{ij}</math>
** Calculate <math>m_{j} = \sum^{n}_{i=1} m_{ij}/n</math>
** Estimate <math>p_j, p_j = E(m_j) = \sum^{n}_{i=1} E_(m_{ij})/n</math>
* 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_{2^d + 1} + it_{2^d+2} & t_2 & ... & 0 \\
             ... & ... & ... & 0 \\ 
             t_{4^d -1} + it_{4^d} &  t_{4^d -3} + it_{4^d - 2} &  t_{4^d - 5} + it_{4^d - 4} &  t_{2^d}
           \end{bmatrix}
     </math>
* Find the minimum of the <math>L(t_1, t_2, ..., t_{d^2})</math> using the formula  <math>
        L(t_1, t_2, ..., t_{d^2}) = \sum_j \frac{(N\langle E_j|\hat{\rho_p}(t_1, t_2, ..., t_{d^2})|E_j\rangle - p_j)^2}{2N\langle E_j|\hat{\rho_p}(t_1, t_2, ..., t_{d^2})|E_j\rangle}
    </math>
* <math>\hat{\rho_p}</math> can be reconstructed from the values of <math>t_i</math>.