  We report Atacama Large Millimeter/submillimeter Array (ALMA) observations of
CO $(8-7)$, $(9-8)$, $\rm H_{2}O (2_{0,2}-1_{1,1})$ and $\rm OH^{+}
(1_{1}-0_{1})$ and NOrthern Extended Millimeter Array (NOEMA) observations of
CO $(5-4)$, $(6-5)$, $(12-11)$ and $(13-12)$ towards the $z = 6.003$ quasar
SDSS J231038.88+185519.7, aiming to probe the physical conditions of the
molecular gas content of this source. We present the best sampled CO spectral
line energy distribution (SLED) at $z = 6.003$, and analyzed it with the
radiative transfer code MOLPOP-CEP. Fitting the CO SLED to a one-component
model indicates a kinetic temperature $T_{\rm kin} = 228 \ \rm K$, molecular
gas density $log (n(\rm H_{2})/\rm cm^{-3}$ )=4.75, and CO column density
$log(N(\rm CO)/\rm cm^{-2}) =17.5$, although a two-component model better fits
the data. In either case, the CO SLED is dominated by a "warm" and "dense"
component. Compared to samples of local (Ultra) Luminous Infrared Galaxies
((U)LIRGs), starburst galaxies and high redshift Submillimeter Galaxies (SMGs),
J2310+1855 exhibits higher CO excitation at ($J \geq 8$), like other high
redshift quasars. The high CO excitation, together with the enhanced $L_{\rm
H_{2}O}/ L_{IR} $, $L_{\rm H_{2}O}/ L_{CO} $ and $L_{OH^{+}}/L_{\rm H_{2}O} $
ratios, suggests that besides the UV radiation from young massive stars, other
mechanisms such as shocks, cosmic rays and X-rays might also be responsible for
the heating and ionization of the molecular gas. In the nuclear region probed
by the molecular emissions lines, any of these mechanisms might be present due
to the powerful quasar and the starburst activity.
