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Multi-observable uncertainty bounds via matrix inequalities

Part of: Basic linear algebra

Published online by Cambridge University Press:  24 June 2026

Saptak Bhattacharya
Saptak Bhattacharya*
Affiliation:
Indian Statistical Institute, Delhi Centre , India
*
e-mail: saptak21r@isid.ac.in
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Abstract

We establish new uncertainty bounds for commutators and anti-commutators of observables in both finite- and infinite-dimensional quantum systems. Beginning with a short, elementary proof of Robertson’s standard uncertainty principle, we then derive inequalities whose lower bounds remain nonzero, regardless of the number of observables. The main result is a novel trace inequality comparing the block matrices of commutators and anti-commutators of several observables.

Keywords

Uncertainty principlesmatrix inequalitiescommutator bounds

MSC classification

Primary: 15A45: Miscellaneous inequalities involving matrices
Secondary: 81P16: Quantum state spaces, operational and probabilistic concepts

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 19
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Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

I thank the Indian Statistical Institute for the financial support during my ongoing Ph.D. through their Institute Fellowship Programme.

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