Constraining the non-Einsteinian polarizations of gravitational waves by pulsar timing array

Pulsar timing array (PTA) provides an excellent opportunity to detect the gravitational waves (GWs) in nanoHertz frequency band. In particular, due to the larger number of "arms" in PTA, it can be used to test gravity by probing the non-Einsteinian polarization modes of GWs, including two spin-1 shear modes labeled by "$sn$" and "$se$", the spin-0 transverse mode labeled by "$b$" and the longitudinal mode labeled by "$l$". In this paper, we investigate the capabilities of the current and potential future PTAs, which are quantified by the constraints on the amplitudes parameters $(c_b,c_{sn},c_{se},c_{l})$, by observing an individual supermassive black hole binary in Virgo cluster. We find that for binary with chirp mass $M_{c}=8.77 \times 10^{8} \mathrm{M_{\odot}}$ and GW frequency $f=10^{-9}\mathrm{Hz}$, the PTA at current level can detect these GW modes if $c_b > 0.00106$, $c_l > 0.00217$, $c_{se} > 0.00271$, $c_{sn} > 0.00141$, which will be improved by about two orders if considering the potential PTA in SKA era. Interesting enough, due to effects of the geometrical factors, we find that in SKA era, the constraints on the $l$, $sn$, $se$ modes of GWs are purely dominated by several pulsars, instead of the full pulsars in PTA.