  The change of a quantum state can generally only be fully monitored through
simultaneous measurements of two non-commuting observables X and Y spanning a
phase space. A measurement device that is coupled to the thermal environment
provides at a time a pair of values that have a minimal uncertainty product set
by the Heisenberg uncertainty relation, which limits the precision of the
monitoring. Here we report on an optical measurement setup that is able to
monitor the time dependent change of the quantum state's displacement in phase
space (< X (t)>; < Y (t)>) with an imprecision 10\,dB below the Heisenberg
uncertainty limit. Our setup provides pairs of values (X(t_i); Y(t_i)) from
simultaneous measurements at subsequent times t_i. The measurement references
are not coupled to the thermal environment but are established by an entangled
quantum state. Our achievement of a tenfold reduced quantum imprecision in
monitoring arbitrary time-dependent displacements supports the potential of the
quantum technology required for entanglement-enhanced metrology and sensing as
well as measurement-based quantum computing.
