Optimal Uncertainty Relations for a Single Observable
Abstract
Uncertainty relations are usually formulated as trade-offs between two or more observables. Yet the uncertainty of a single observable can already contain an intrinsically quantum contribution arising from its noncommutativity with the state. In the sense of Luo, such a contribution can be quantified by a quantum uncertainty Qρ(A), with prominent examples including the Wigner--Yanase and Wigner--Yanase--Dyson skew informations and the quantum Fisher information. Here we ask how strong such single-observable uncertainty relations can be when the quantum state is fixed. We solve this fixed-state optimization problem by determining the largest coefficient copt(ρ) for which Vρ(A)≥ copt(ρ)Qρ(A) holds for every observable A, and show that it depends only on the smallest and largest eigenvalues of ρ. The same optimization principle extends to a new power-commutator family 1/2|[ρs,A]|2 (s≥ 1/2), providing a direct measure of state--observable noncommutativity beyond the standard quantum-uncertainty framework. More importantly, the optimal noncommutative contribution can be combined with the maximal classical contribution compatible with the state, yielding a sharpened variance decomposition without weakening the optimal bound. For states with exactly two distinct eigenvalues, this relation becomes an identity for every observable; in particular, qubit variances decompose exactly into classical and noncommutative parts. Finally, the corresponding variance-product bounds are no weaker than the Robertson relation for qubit spin observables and can be strictly stronger, showing that a substantial part of the uncertainty usually expressed as a trade-off between distinct observables is already encoded in their individual noncommutativity with the state.
Related
- Hiai--Petz Skew Informations and Sharp Uncertainty Relations for a Single Observable
- The Lévy wave packet and fractional uncertainty relations
- Additive uncertainty relations via pseudospectral methods
- Lie Algebraic Uncertainty Relations
- Receding Horizon Control and Dissipativity - Optimal Control, Games and Uncertainty
BibTeX
@misc{yamashita2026optimal,
title = {{Optimal Uncertainty Relations for a Single Observable}},
author = {Haruki Yamashita and Aina Mayumi and Gen Kimura},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.26331},
eprint = {2605.26331},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.26331},
}
RIS
TY - UNPB
TI - Optimal Uncertainty Relations for a Single Observable
AU - Haruki Yamashita
AU - Aina Mayumi
AU - Gen Kimura
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2605.26331
UR - https://arxiv.org/abs/2605.26331
AB - Uncertainty relations are usually formulated as trade-offs between two or more observables. Yet the uncertainty of a single observable can already contain an intrinsically quantum contribution arising from its noncommutativity with the state. In the sense of Luo, such a contribution can be quantified by a quantum uncertainty Q_ρ(A), with prominent examples including the Wigner--Yanase and Wigner--Yanase--Dyson skew informations and the quantum Fisher information. Here we ask how strong such single-observable uncertainty relations can be when the quantum state is fixed. We solve this fixed-state optimization problem by determining the largest coefficient c_opt(ρ) for which V_ρ(A)≥ c_opt(ρ)Q_ρ(A) holds for every observable A, and show that it depends only on the smallest and largest eigenvalues of ρ. The same optimization principle extends to a new power-commutator family 1/2|[ρ^s,A]|^2 (s≥ 1/2), providing a direct measure of state--observable noncommutativity beyond the standard quantum-uncertainty framework. More importantly, the optimal noncommutative contribution can be combined with the maximal classical contribution compatible with the state, yielding a sharpened variance decomposition without weakening the optimal bound. For states with exactly two distinct eigenvalues, this relation becomes an identity for every observable; in particular, qubit variances decompose exactly into classical and noncommutative parts. Finally, the corresponding variance-product bounds are no weaker than the Robertson relation for qubit spin observables and can be strictly stronger, showing that a substantial part of the uncertainty usually expressed as a trade-off between distinct observables is already encoded in their individual noncommutativity with the state.
ER -