Optimal Uncertainty Relations for a Single Observable

Abstract

Uncertainty relations are conventionally formulated as trade-off relations between two or more observables. Here we introduce a complementary viewpoint: even a single observable can possess unavoidable uncertainty originating from its noncommutativity with the quantum state. We formulate such bounds as single-observable preparation uncertainty relations and show that several familiar quantities, including the Wigner--Yanase and Wigner--Yanase--Dyson skew informations and the quantum Fisher information, naturally fit into this framework. We also introduce the power-commutator family $\Vert [{\rho }^s,A]{\Vert }^2$, $1/2\le s\le 1$, which directly quantifies state--observable noncommutativity. The first main aim of this work is to determine the strongest fixed-state refinements of these single-observable uncertainty relations. For all these quantities, we derive the optimal state-dependent coefficients for which the corresponding bounds remain valid for every observable. Remarkably, in all cases the optimal coefficient depends only on the smallest and largest eigenvalues of the state. Our second main aim is to strengthen these optimal lower bounds further by adding the maximal classical contribution to the variance, while preserving the same optimal coefficient for the noncommutative contribution. For qubit states, the resulting sharpened relations become exact identities, yielding an exact decomposition of the variance into classical and noncommutative contributions.

34 pages, 1 figure; substantially revised and expanded in V2, with stronger connections to quantum Fisher information and related previous work; reformatted for submission to Quantum; comments are welcome