  The 21 cm linear polarization due to Thomson scattering off free electrons
can probe the distribution of neutral hydrogen in the intergalactic medium
during the epoch of reionization, complementary to the 21 cm temperature
fluctuations. Previous study (Babich & Loeb 2005) estimated the strength of
polarization with a toy model and claimed that it can be detected with 1-month
observation of the Square Kilometre Array (SKA). Here we revisit this
investigation with account of nonlinear terms due to inhomogeneous
reionization, using seminumerical reionization simulations to provide the
realistic estimation of the 21 cm TE and EE angular power spectra ($C^{\rm
TE}_\ell$ and $C^{\rm EE}_\ell$). We find that (1) both power spectra are
enhanced on sub-bubble scales but suppressed on super-bubble scales, compared
with previous results; (2) $C^{\rm TE}_\ell$ flips its sign at $\ell \simeq
50$, which can probe the H I bias at large scales; (3) the ratios of the power
spectrum to its maximum value during reionization at a given $\ell$, i.e.
$C^{\rm TE}_\ell / C^{\rm TE}_{\ell,{\rm max}} $ and $C^{\rm EE}_{\ell}/C^{\rm
EE}_{\ell,{\rm max}}$, show robust correlations with the global ionized
fraction. However, measurement of this signal will be very challenging not only
because the overall strength is weaker than the sensitivity of SKA, but also
because Faraday rotation due to Galactic and extragalactic magnetic fields
significantly modifies the observed polarization. Nevertheless, it may still be
possible that the 21 cm linear polarization signal may be detected through
other approaches, e.g. its cross-correlation with other probes.
