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

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.

9 pages, 2 figures

References in corpus (6)

• Steering, Entanglement, Nonlocality, and the EPR Paradox
• Evading quantum mechanics
• Correlations in local measurements on a quantum state, and complementarity as an explanation of nonclassicality
• Prediction and retrodiction with continuously monitored Gaussian states
• Retrodiction beyond the Heisenberg uncertainty relation
• The "mean king's problem" with continuous variables

Cited by in corpus (6)

• Continuous Gaussian Measurements of the Free Boson CFT: A model for Exactly Solvable and Detectable Measurement-Induced Dynamics
• Effects of the free evolution in the Arthurs-Kelly model of simultaneous measurement and in the retrodictive predictions of the Heisenberg uncertainty relations
• Retrodiction of measurement outcomes on a single quantum system reveals entanglement with its environment
• Uncertainty Relations in Pre- and Post-Selected Systems
• Causal State Estimation and the Heisenberg Uncertainty Principle
• Quantum backaction effects in sequential measurements