Uncertainty from the Aharonov–Vaidman identity

Authors

Matthew S. Leifer, Chapman UniversityFollow

Document Type

Article

Publication Date

4-24-2023

Abstract

In this article, I show how the Aharonov–Vaidman identity A|ψ>=<A⟩|ψ>+ΔA| ψ⊥A> 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 ΔAΔB≥12|< [A,B]>| 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 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.

Comments

This article was originally published in Quantum Studies: Mathematics and Foundations in 2023. https://doi.org/10.1007/s40509-023-00301-8

Recommended Citation

Leifer, M.S. Uncertainty from the Aharonov–Vaidman identity. Quantum Stud.: Math. Found. (2023). https://doi.org/10.1007/s40509-023-00301-8

Peer Reviewed

1

Copyright

The author

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.