Full monitoring of phase-space trajectories with 10dB-sub-Heisenberg\n imprecision

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