  The three-dimensional magneto-conductivity tensor was derived in a gauge
invariant form based on the Kubo formula considering the quantum effect under a
magnetic field, such as the Landau quantization and the quantum oscillations.
We analytically demonstrated that the quantum formula of the
magneto-conductivity can be obtained by adding a quantum oscillation factor to
the classical formula. This result establishes the quantum--classical
correspondence, which has long been missing in magnetotransport phenomena.
Moreover, we found dissipative-to-dissipationless crossover in the Hall
conductivity by paying special attention to the analytic properties of thermal
Green's function. Finally, by calculating the magnetoresistance of semimetals,
we identified a phase shift in quantum oscillation originating from the
dissipationless transport predominant at high fields.
