Retrodiction beyond the Heisenberg uncertainty relation

TL;DR

This paper explores how quantum systems can be retrodicted beyond the traditional Heisenberg uncertainty limits by using measurements before and after a certain time, with implications for quantum state estimation.

Contribution

It demonstrates that precise knowledge of position and momentum at a given time can be achieved through a combination of quantum non-demolition measurements, surpassing the usual uncertainty bounds.

Findings

Retrodiction enables precise estimation of quantum observables beyond traditional limits.

Quantum non-demolition measurements before and after a time improve state knowledge.

Implications for quantum sensing and state estimation are significant.

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. We study what happens under such circumstances with an atomic ensemble containing $10^{11}$ $^{87}\text{Rb}$ atoms, initiated nearly in the ground state in presence of a magnetic field. The collective spin observables of the atoms are then well described by canonical position and momentum observables, $\hat{x}_A$ and $\hat{p}_A$ that satisfy $[\hat{x}_A,\hat{p}_A]=i\hbar$. Quantum non-demolition measurements of $\hat{p}_A$ before and of $\hat{x}_A$ after time $t$ allow precise estimates of both observables at time $t$. 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.