Uncertainty from the Aharonov-Vaidman Identity

TL;DR

This paper demonstrates how the Aharonov-Vaidman identity can be used to derive and simplify various uncertainty relations in quantum mechanics, including proofs of the Robertson and Maccone-Pati inequalities, and extends these results to mixed states.

Contribution

It introduces a novel application of the Aharonov-Vaidman identity to prove and analyze quantum uncertainty relations more directly and extends the framework to mixed states.

Findings

Simplified proofs of the Robertson and Maccone-Pati uncertainty relations.

Derivation of the Cauchy-Schwarz inequality from the Aharonov-Vaidman identity.

Extension of the uncertainty analysis to mixed quantum states.

Abstract

In this article, I show how the Aharonov-Vaidman identity $A \left \vert \psi\right \rangle = \left \langle A \right \rangle \left \vert \psi\right \rangle + \Delta A \left \vert \psi^{\perp}_A \right \rangle$ 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 $\Delta A \Delta B \geq \frac{1}{2} \left \vert \left \langle [A,B] \right \rangle \right \vert$ 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.