  We compute the electric conductivity of quark matter at finite temperature
$T$ and quark chemical potential $\mu$ under a magnetic field $B$ beyond the
Lowest Landau level approximation. The electric conductivity transverse to $B$
is dominated by the Hall conductivity $\sigma_H$. For the longitudinal
conductivity $\sigma_\parallel$, we need to solve kinetic equations. Then, we
numerically find that $\sigma_\parallel$ has only mild dependence on $\mu$ and
the quark mass $m_q$. Moreover, $\sigma_\parallel$ first decreases and then
linearly increases as a function of $B$, leading to an intermediate $B$ region
which looks consistent with the experimental signature for the chiral magnetic
effect. We also point out that $\sigma_\parallel$ at nonzero $B$ remains within
the range of the lattice-QCD estimate at $B=0$.
