  We consider quantum correlations and quantum discord phenomena for two-qubit
states with $X$-type density matrices in the tomographic framework of quantum
mechanics. By introducing different measurements schemes, we establish the
relation between tomographic approach to quantum discord, symmetric discord,
and measurement-induced disturbance. In our consideration, $X$-states appear as
approximations of ground and low temperature thermal states of two coupled
harmonic oscillators realized by nanoelectric $LC$-circuits. Possibilities for
control the amounts of correlations and entropic asymmetry due to variation of
the frequency detuning and the coupling constant are also considered.
