Full Quantum state tomography with Maximum Likelihood Estimation: Difference between revisions

Full Quantum state tomography with Maximum Likelihood Estimation (edit)

No change in size , 6 June 2023

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 * 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^n + 1} + it_{2^n+2} & t_2 & ... & 0 \\
+              t_{2^d + 1} + it_{2^d+2} & t_2 & ... & 0 \\
               ... & ... & ... & 0 \\
-              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}
+              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_{n^2})</math> using the formula  <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_{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}
+         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>.