Retrodiction beyond the Heisenberg uncertainty relation

Abstract In quantum mechanics, the Heisenberg uncertainty relation presents an ultimate limit to the precision by which one can predict the outcome of position and momentum measurements on a particle. Heisenberg explicitly stated this relation for the prediction of “hypothetical future measurements”, and it does not describe the situation where knowledge is available about the system both earlier and later than the time of the measurement. Here, we study what happens under such circumstances with an atomic ensemble containing 10 11 rubidium atoms, initiated nearly in the ground state in the presence of a magnetic field. The collective spin observables of the atoms are then well described by canonical position and momentum observables, $${\hat{x}}_{\text{A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̂</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $${\hat{p}}_{\text{A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̂</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:math> that satisfy $$[{\hat{x}}_{\text{A}},{\hat{p}}_{\text{A}}]=i\hslash$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow> <mml:msub> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̂</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̂</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>i</mml:mi> <mml:mi>ℏ</mml:mi> </mml:math> . Quantum non-demolition measurements of $${\hat{p}}_{\text{A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̂</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:math> before and of $${\hat{x}}_{\text{A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̂</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:math> after time t allow precise estimates of both observables at time t . By means of the past quantum state formalism, we demonstrate that outcomes of measurements of both the $${\hat{x}}_{\text{A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̂</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $${\hat{p}}_{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̂</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:math> observables can be inferred with errors below the standard quantum limit. The capability of assigning precise values to multiple observables and to observe their variation during physical processes may have implications in quantum state estimation and sensing.