The uncertainty principle is considered to be one of the most striking features in quantum mechanics. In the textbook literature, uncertainty relations usually refer to the preparation uncertainty which imposes a limitation on the spread of measurement outcomes for a pair of noncommuting observables. In this work, we study the preparation uncertainty for the angular momentum, especially for spin-1/2. We derive uncertainty relations encompassing the triple components of angular momentum and show that, compared with the relations involving only two components, a triple constant 2/sqrt[3] often arises. Intriguingly, this constant is the same for the position and momentum case. Experimental verification is carried out on a single spin in diamond, and the results confirm the triple constant in a wide range of experimental parameters.The uncertainty principle is considered to be one of the most striking features in quantum mechanics. In the textbook literature, uncertainty relations usually refer to the preparation uncertainty which imposes a limitation on the spread of measurement outcomes for a pair of noncommuting observables. In this work, we study the preparation uncertainty for the angular momentum, especially for spin-1/2. We derive uncertainty relations encompassing the triple components of angular momentum and show that, compared with the relations involving only two components, a triple constant 2/sqrt[3] often arises. Intriguingly, this constant is the same for the position and momentum case. Experimental verification is carried out on a single spin in diamond, and the results confirm the triple constant in a wide range of experimental parameters.

arXiv:1611.04775Physical Review LettersMay 4, 201747 citations

Equality Conditions for an Additive Three-Observable Uncertainty Relation

Yao-Yi Zeng, Zhi-Jie Liu, Jie Zhou +1 more

Uncertainty relations play a fundamental role in quantum mechanics by quantifying the intrinsic limitations on the simultaneous sharpness of incompatible observables. Beyond the standard two-observable product form, additive uncertainty relations for triples of observables provide a natural framework for describing collective constraints among three noncommuting components. In this work, we study an additive uncertainty relation for three Hermitian observables from the viewpoint of rotational symmetry and covariance geometry. We give a short rotational derivation by rotating the observable triple and applying the Robertson uncertainty relation to the two transverse observables. This derivation makes the saturation mechanism transparent and leads to a necessary and sufficient condition for equality for general density operators. In the nontrivial equality case, the covariance ellipsoid of the observable triple degenerates into a disk perpendicular to the expectation value of the commutator vector. We also discuss an inverse construction based on finite-dimensional representations of the Lie algebra $\mathfrak{su}(2)$ , which provides a systematic way to construct observable triples with prescribed saturating states. These results clarify the geometric and representation-theoretic structure underlying the tightness of additive three-observable uncertainty relations.Uncertainty relations play a fundamental role in quantum mechanics by quantifying the intrinsic limitations on the simultaneous sharpness of incompatible observables. Beyond the standard two-observable product form, additive uncertainty relations for triples of observables provide a natural framework for describing collective constraints among three noncommuting components. In this work, we study an additive uncertainty relation for three Hermitian observables from the viewpoint of rotational symmetry and covariance geometry. We give a short rotational derivation by rotating the observable triple and applying the Robertson uncertainty relation to the two transverse observables. This derivation makes the saturation mechanism transparent and leads to a necessary and sufficient condition for equality for general density operators. In the nontrivial equality case, the covariance ellipsoid of the observable triple degenerates into a disk perpendicular to the expectation value of the commutator vector. We also discuss an inverse construction based on finite-dimensional representations of the Lie algebra $\mathfrak{su}(2)$, which provides a systematic way to construct observable triples with prescribed saturating states. These results clarify the geometric and representation-theoretic structure underlying the tightness of additive three-observable uncertainty relations.

quant-pharXiv:2606.29992Jun 30, 20260 citations

Uncertainty relations in simultaneous measurements for arbitrary observables

Shiro Ishikawa

Reports on Mathematical PhysicsJun 1, 199165 citations

Observables and Uncertainty Relations

Günter Ludyk

Springer eBooksOct 25, 20170 citations

Heisenberg uncertainty relation for three canonical observables

Spiros Kechrimparis, Stefan Weigert

Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state which saturates the triple uncertainty relation. Quantum optical experiments are proposed to verify our findings.Uncertainty relations provide fundamental limits on what can be said about the properties of quantum systems. For a quantum particle, the commutation relation of position and momentum observables entails Heisenberg's uncertainty relation. A third observable is presented which satisfies canonical commutation relations with both position and momentum. The resulting triple of pairwise canonical observables gives rise to a Heisenberg uncertainty relation for the product of three standard deviations. We derive the smallest possible value of this bound and determine the specific squeezed state which saturates the triple uncertainty relation. Quantum optical experiments are proposed to verify our findings.

arXiv:1407.0083Physical Review ADec 15, 201451 citations

Uncertainty regions of observables and state-independent uncertainty relations

Lin Zhang, Shunlong Luo, Shao-Ming Fei +1 more

arXiv:2110.14134Quantum Information ProcessingOct 26, 20217 citations

\mathfrak{su}(2)

, which provides a systematic way to construct observable triples with prescribed saturating states. These results clarify the geometric and representation-theoretic structure underlying the tightness of additive three-observable uncertainty relations.