We consider the low Mach number limit for the system of quantum Navier-Stokes in the whole three dimensional space. The QNS equations describe a barotropic compressible viscous fluid with a degenerate viscosity coefficient, augmented by a third-order (dispersive) stress tensor, called the quantum term. For general ill-prepared initial data of finite energy, we show strong convergence of finite energy weak solutions towards weak solutions to the incompressible Navier-Stokes equations. The main novelty is that our analysis only requires the initial data to be of finite energy, in particular no smallness, regularity or well-preparedness of the data is assumed. Our method is based on a precise analysis of acoustic waves based on some new Strichartz estimates for the linearised system. Further, we exploit a priori bounds derived from Bresch-Desjardins type inequalities. This is joint work with L.E. Hientzsch and P. Marcati.