Uncertainty from the Aharonov–Vaidman identity

Abstract

Abstract In this article, I show how the Aharonov–Vaidman identity $$A\left| \psi \right\rangle = \left\langle A \right\rangle \left| \psi \right\rangle + \Delta A \left| \psi ^{\perp }_A \right\rangle $$ A ψ = A ψ + Δ A 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 $$\Delta A \Delta B \ge \frac{1}{2} \left| \left\langle [A,B] \right\rangle \right| $$ Δ A Δ B ≥ 1 2 [ 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.