  We report on the first ALMA observation of the CO(3$-$2) and rest-frame ~340
GHz continuum emission in PDS 456, which is the most luminous, radio-quiet QSO
in the local Universe ($z$~0.18), with a bolometric luminosity $L_{\rm
Bol}\sim10^{47}$ erg s$^{-1}$. ALMA angular resolution allowed us to map scales
as small as ~700 pc. The molecular gas reservoir, traced by the core of the
very bright CO(3$-$2) emission line, is distributed in a compact rotating disk,
with size of ~1.3 kpc, seen close to face-on ($i$~25 deg). Fast CO(3$-$2)
emission in the velocity range $v\in[-1000,+500]$ km s$^{-1}$ is also present.
Specifically, we detect several blue-shifted clumps out to ~5 kpc from the
nucleus, in addition to a compact ($R\lesssim1.2$ kpc), broad emission
component. These components reveal a galaxy-wide molecular outflow, with a
total mass $M_{\rm mol}^{\rm out}\sim2.5\times10^8$ $M_{\odot}$ and a mass
outflow rate $\dot{M}_{\rm mol}\sim290$ $M_{\odot}$ yr$^{-1}$. The
corresponding depletion time is ~8 Myr, shorter than the rate at which the
molecular gas is converted into stars, indicating that the detected outflow is
potentially able to quench star-formation in the host. The momentum flux of the
molecular outflow normalised to the radiative momentum output (i.e. $L_{\rm
Bol}/c$) is $\lesssim1$, comparable to that of the X-ray ultra-fast outflow
(UFO) detected in PDS 456. This is at odds with the expectations for an
energy-conserving expansion suggested for most of the large-scale outflows
detected in low-luminosity AGN so far. We suggest three possible scenarios that
may explain this observation: (i) in very luminous AGN such as our target the
molecular gas phase is tracing only a fraction of the total outflowing mass;
(ii) a small coupling between the shocked gas by the UFO and the host-galaxy
ISM (iii) AGN radiation pressure may play an important role in driving the
outflow.
