  Einstein's weak equivalence principle (WEP) states that any freely falling,
uncharged test particle follows the same identical trajectory independent of
its internal structure and composition. Since the polarization of a photon is
considered to be part of its internal structure, we propose that polarized
photons from astrophysical transients, such as gamma-ray bursts (GRBs) and fast
radio bursts (FRBs), can be used to constrain the accuracy of the WEP through
the Shapiro time delay effect. Assuming that the arrival time delays of photons
with different polarizations are mainly attributed to the gravitational
potential of the Laniakea supercluster of galaxies, we show that a strict upper
limit on the differences of the parametrized post-Newtonian parameter $\gamma$
value for the polarized optical emission of GRB 120308A is
$\Delta\gamma<1.2\times10^{-10}$, for the polarized gamma-ray emission of GRB
100826A is $\Delta\gamma<1.2\times10^{-10}$, and for the polarized radio
emission of FRB 150807 is $\Delta\gamma<2.2\times10^{-16}$. These are the first
direct verifications of the WEP for multiband photons with different
polarizations. In particular, the result from FRB 150807 provides the most
stringent limit to date on a deviation from the WEP, improving by one order of
magnitude the previous best result based on Crab pulsar photons with different
energies.
