  In this paper we analyse the implications of the latest cosmological data
sets to test the Etherington's distance duality relation (DDR), which connects
the luminosity distance $D_L$ and angular diameter distance $D_A$ at the same
redshift. For $D_L$ we consider the simulated data of gravitational waves from
the third-generation gravitational wave detector (the Einstein Telescope, ET),
which can be considered as standard candles (or standard siren), while the
angular diameter distances $D_A$ are derived from the newly-compiled sample of
compact radio quasars observed by very-long-baseline interferometry (VLBI),
which represents a type of new cosmological standard ruler. Alleviating the
absorption and scattering effects of dust in the Universe, this will create a
valuable opportunity to directly test DDR at much higher precision with the
combination of gravitational wave (GW) and electromagnetic (EM) signals. Our
results show that, with the combination of the current radio quasar
observations, the duality-distance relation can be verified at the precision of
$10^{-2}$. Moreover, the Einstein Telescope ET would produce more robust
constraints on the validity of such distance duality relation (at the precision
of $10^{-3}$), with a larger sample of compact milliarcsecond radio quasars
detected in future VLBI surveys.
