Uncertainty from the Aharonov-Vaidman Identity

In this article, I show how the Aharonov-Vaidman identity $A\mid \psi ⟩=⟨A⟩\mid \psi ⟩+\mathrm{\Delta }A\mid {\psi }_A^{\perp }⟩$ 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 $\mathrm{\Delta }A\mathrm{\Delta }B\ge \frac{1}{2}\mid ⟨[A,B]⟩\mid$ 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.