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arXiv:2610.32014

Additive uncertainty relations via pseudospectral methods

Alexander Cerjan, Lara Ismert, Terry A. Loring

quant-phmath.OASubmitted Oct 23, 2026• Announced Oct 26, 2026• 0 citations
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Abstract

Given Hermitian observable $A$, the variance of $A$ is given by the difference of expectations $\langle A^2\rangle - \langle A\rangle^2$, 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 $A_1, \dots , A_d$ is greater than some expression in the expectations $\langle i[A_j,A_k]\rangle$ for $j\neq k$ 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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Given Hermitian observable AAA, the variance of AAA is given by the difference of expectations ⟨A2⟩−⟨A⟩2\langle A^2\rangle - \langle A\rangle^2⟨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∗C^*C∗-algebra, we prove that the sum of the variances for ddd observables A1,…,AdA_1, \dots , A_dA1​,…,Ad​ is greater than some expression in the expectations ⟨i[Aj,Ak]⟩\langle i[A_j,A_k]\rangle⟨i[Aj​,Ak​]⟩ for j≠kj\neq kj=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 ddd observables.
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