How well can we guess the outcome of measurements of non-commuting observables?

TL;DR

This paper explores the limits of predicting past measurement outcomes of non-commuting observables in quantum systems, revealing scenarios where traditional uncertainty bounds can be surpassed using entanglement and ancillary systems.

Contribution

It demonstrates that the Heisenberg uncertainty limit can be violated for retrodicting past measurement outcomes using entanglement and ancillary systems.

Findings

Retrodiction of past measurement outcomes can surpass traditional uncertainty limits.

Entanglement enables accurate retrodiction of linear combinations of observables.

No formal lower limit exists for guessing outcomes of joint measurements with ancillary probes.

Abstract

According to Heisenberg's uncertainty relation, there is an ultimate limit to how precisely we may predict the outcome of position and momentum measurements on a quantum system. We show that this limit may be violated by an arbitrarily large factor if one aims, instead, to guess the unknown value of a past measurement. For experiments on a single quantum system, the precise assignment of past position and momentum measurement outcomes is accompanied by large uncertainty about their linear combinations, while we show that entanglement with an ancillary system permits accurate retrodiction of any such linear combination. Finally, we show that the outcomes of experiments that jointly measure multiple linear combinations of position and momentum observables by means of ancillary probe particles can also be guessed with no formal lower limit. We present quantitative results for projective measurements and for generalized measurements where all components are prepared in Gaussian states.