Additive uncertainty relations via pseudospectral methods

Abstract

Given Hermitian observable $A$, the variance of $A$ is given by the difference of expectations $⟨A^2⟩-⟨A{⟩}^2$, where the expectation of an observable implicitly assumes a choice of state for the underlying system. As a sub-case of a general lower bound for the sum of variances for finitely many arbitrary elements in a $C^{\ast }$-algebra, we prove that the sum of the variances for $d$ observables $A_1,\dots ,A_d$ is greater than some expression in the expectations $⟨i[A_j,A_k]⟩$ for $j\ne k$ plus the square of the smallest singular value of the associated localizer. In special cases, this expression is just a root mean square average. Our technique also derives a general containment between the Clifford pseudospectra and quadratic pseudospectra for $d$ observables.

arXiv comment20 pages