# Non-diagonal matrix element in density matrix

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?

• 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.) – Norbert Schuch Mar 13 at 0:07
• what does "determine" mean? Does looking at them suffice to "determine" them? – glS Mar 13 at 13:00

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.