Heisenberg Uncertainty Relation for Three Canonical Observables

Abstract

Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg-type uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state which saturates the triple uncertainty relation. Quantum optical experiments are proposed to verify our findings.

5 pages, 2 figures. Material rearranged to match published version

References in corpus (2)

• Towards higher precision and operational use of optical homodyne tomograms
• Mutually Unbiased Bases for Continuous Variables

Cited by in corpus (6)

• State-independent Uncertainty Relations and Entanglement Detection in Noisy Systems
• Multi-observable Uncertainty Relations in Product Form of Variances
• Experimental investigation of multi-observable uncertainty relations
• Uncertainties of genuinely incompatible triple measurements based on statistical distance
• Optimal approximations of available states and a triple uncertainty relation
• Optimizing incompatible triple quantum measurements