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 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
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.
Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order α rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases, the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.