Uncertainty from the Aharonov-Vaidman Identity

Abstract

In this article, I show how the Aharonov-Vaidman identity $A|\psi ⟩=⟨A⟩|\psi ⟩+\Delta A|{\psi }_A^{\perp }⟩$ can be used to prove relations between the standard deviations of observables in quantum mechanics. In particular, I review how it leads to a more direct and less abstract proof of the Robertson uncertainty relation $\Delta A\Delta B\ge \frac{1}{2}\left|⟨[A,B]⟩\right|$ than the textbook proof. I discuss the relationship between these two proofs and show how the Cauchy-Schwarz inequality can be derived from the Aharonov-Vaidman identity. I give Aharonov-Vaidman based proofs of the Maccone-Pati uncertainty relations and I show how the Aharonov-Vaidman identity can be used to handle propagation of uncertainty in quantum mechanics. Finally, I show how the Aharonov-Vaidman identity can be extended to mixed states and discuss how to generalize the results to the mixed case.

31 pages, 1 figure, pdfLaTeX

References in corpus (9)

• Stronger uncertainty relations for the sum of variances
• Sum Uncertainty Relation in Quantum Theory
• What is nonclassical about uncertainty relations?
• Variance-based uncertainty relations for incompatible observables
• Multi-observable Uncertainty Relations in Product Form of Variances
• Implications and applications of the variance-based uncertainty equalities
• Uncertainty regions of observables and state-independent uncertainty relations
• On Quantum Uncertainty Relations and Uncertainty Regions
• General and Stronger Uncertainty Relation

Cited by in corpus (2)

• Quantum Uncertainty Equalities and Inequalities for Unitary Operators
• Uncertainty Relations in Pre- and Post-Selected Systems