Uncertainty Relation for a Single Observable

Abstract

Uncertainty relations are usually formulated as trade-off relations between two or more observables. Here we show that the uncertainty of a single observable already has a nontrivial lower bound originating from the noncommutativity between the observable and the quantum state. We prove sharp lower bounds on the variance of a single observable and then sharpen them further by introducing the classical uncertainty of the observable under a fixed state. The optimal coefficient is determined solely by the smallest and largest eigenvalues of the state. Our results include an optimal state-dependent improvement of Luo's Wigner--Yanase-type relation and a direct bound showing that coherence or asymmetry of the state with respect to the observable gives an unavoidable contribution to its uncertainty. For qubits, the sharpened bounds become exact identities, giving a complete decomposition of the variance into classical and noncommutative parts. These single-observable relations also yield improved product-form uncertainty relations for pairs of observables.