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Is it possible to determine non-diagonal element of density matrix?

It will be cool, if somebody present few examples.

What is meaning of such element?

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  • $\begingroup$ What would an "example" be? -- In any case, your question feels to vague. (Also: If they have a meaning, you can determine them. If they don't, you can't.) $\endgroup$ – Norbert Schuch Mar 13 at 0:07
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    $\begingroup$ what does "determine" mean? Does looking at them suffice to "determine" them? $\endgroup$ – glS Mar 13 at 13:00
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It is possible, if the non-diagonal element corresponds to a measurable quantity. Indeed, the expectation value of any operator is \begin{equation} \langle A\rangle = \textrm{tr} (\hat{\rho}A). \end{equation} Let us look at a single spin with density matrix \begin{equation} \hat{\rho} = \begin{pmatrix} \rho_{11} & \rho_{12} \\ \rho_{21} & \rho_{22} \\ \end{pmatrix}, \end{equation} and calculate the expectation value of the spin projections on the x and y axes with operators \begin{equation} \hat{s}_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, \hat{s}_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -\imath \\ \imath & 0 \\ \end{pmatrix}, \end{equation} which come out to be \begin{equation} \langle s_x\rangle = \textrm{tr} (\hat{\rho}\hat{s}_x) = \frac{\hbar}{2}(\rho_{12} + \rho_{21}), \langle s_y\rangle = \textrm{tr} (\hat{\rho}\hat{s}_x) = \frac{-\imath\hbar}{2}(\rho_{12} - \rho_{21}). \end{equation} This gives us access to the non-diagonal density matrix elements.

One thing worth noting is that I could have simply rotated the whole system in x or y direction (or I could have chosen x or y axis as the quantization direction), in which case what seems like non-diagonal elements would be diagonal. This is an important point: which elements of the density matrix are diagonal depends on the representation that you work in.

Finally, it is worth noting that all kinds of interference experiments are designed specifically for detecting non-diagonal density matrix elements.

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