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Additive uncertainty relations via pseudospectral methods
Abstract
Given Hermitian observable A, the variance of A is given by the difference of expectations ⟨ A2⟩ − ⟨ 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*-algebra, we prove that the sum of the variances for d observables A1, … , Ad is greater than some expression in the expectations ⟨ i[Aj,Ak]⟩ for j≠ 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.
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BibTeX
@misc{cerjan2026additive,
title = {{Additive uncertainty relations via pseudospectral methods}},
author = {Alexander Cerjan and Lara Ismert and Terry A. Loring},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32014},
eprint = {2610.32014},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32014},
}
RIS
TY - UNPB
TI - Additive uncertainty relations via pseudospectral methods
AU - Alexander Cerjan
AU - Lara Ismert
AU - Terry A. Loring
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32014
UR - https://arxiv.org/abs/2610.32014
AB - 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^*-algebra, we prove that the sum of the variances for d observables A_1, … , A_d is greater than some expression in the expectations ⟨ i[A_j,A_k]⟩ for j≠ 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.
ER -