  We predict by first-principles calculations that thin films of Cr-doped
(Bi,Sb)$_2$Te$_3$ magnetic topological insulator have gapless non-chiral edge
states coexisting with the chiral edge state. Such gapless non-chiral states
are not immune to backscattering, which would explain dissipative transport in
the quantum anomalous Hall (QAH) state observed in this system experimentally.
Here we study the edge transport with both chiral and non-chiral states by
Landaur-B\"{u}ttiker formalism, and find that the longitudinal resistance is
nonzero whereas Hall resistance is quantized to $h/e^2$. In particular, the
longitudinal resistance can be greatly reduced by adding an extra floating
probe even if it is not used, while the Hall resistance remains at the
quantized value. We propose several transport experiments to detect the
dissipative non-chiral edge channels. These results will facilitate the
realization of pure dissipationless transport of QAH states in magnetic
topological insulators.

Mesoscale and Nanoscale Physics