Spin-squeezing by prediction and retrodiction measurements: beyond the Heisenberg uncertainty relation

If one has access to information about a quantum system both before and after a measurement, the usual remit of Heisenberg uncertainty relation (HUR) does not apply. Here, we theoretically and experimentally demonstrate that, in such a scenario, one can retrodict position and momentum measurements without being limited by HUR. The experiment uses a large ensemble of 10^11 atoms in a macroscopic vapor cell, and the results may be applicable for quantum state estimation and sensing, especially for sensitive magnetometers equally capable of measuring the amplitude and phase of an RF magnetic field both below the standard quantum limit.