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.

arXiv comment31 pages, 1 figure, pdfLaTeX

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Cited by in corpus (2)

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