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arXiv:2605.26331

Optimal Uncertainty Relations for a Single Observable

Haruki Yamashita, Aina Mayumi, Gen Kimura

quant-phmath-phSubmitted May 25, 2026• Announced May 27, 2026• 0 citations• Revised Sep 14, 2026 (v2)
v1 · May 25, 2026v2 · Sep 14, 2026v3 · Sep 24, 2026
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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 $\|[ρ^s,A]\|^2$, $1/2\le s\le1$, 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.

Comments
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

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•Oct 5, 2016•0 citations
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 ∥[ρs,A]∥2\|[ρ^s,A]\|^2∥[ρs,A]∥2, 1/2≤s≤11/2\le s\le11/2≤s≤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.
Abstract We report a universal improvement to the standard Robertson–Schrödinger uncertainty relation. Our result shows that the Robertson–Schrödinger lower bound can be supplemented by a new noncommutativity-induced term. This term represents a previously overlooked quantum contribution and becomes more pronounced as the state becomes more mixed. Moreover, it is expressed as the expectation value of a positive observable, namely the squared modulus of the commutator, and therefore preserves the direct, experimentally accessible character of the Robertson–Schrödinger relation. For two-level quantum systems, our relation becomes an exact equality for any state and any pair of observables, thereby ensuring the tightness of the bound in the strongest possible sense. The relation also yields, as a corollary, a complete proof of a general uncertainty bound that had previously been supported only by numerical evidence.
We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalues of the quantum state. The conventional constant $ \tfrac{1}{4} $ is replaced by the state-dependent coefficient (λmax⁡+λmin⁡)24(λmax⁡−λmin⁡)2, \frac{(λ_{\max} + λ_{\min})^2}{4(λ_{\max} - λ_{\min})^2}, 4(λmax​−λmin​)2(λmax​+λmin​)2​, where $ λ_{\max} $ and λmin⁡λ_{\min}λmin​ denote the largest and smallest eigenvalues of the density operator $ ρ$, respectively. We prove that this coefficient is optimal among all Robertson-type generalizations and does not admit further improvement. We also obtain a complete characterization of all nontrivial equality cases. Our relation becomes more pronounced as the quantum state becomes more mixed, capturing a trade-off in quantum uncertainty that the conventional Robertson's relation fails to detect. In addition, our result also provides a strict generalization of the Schrödinger's uncertainty relation, showing that the uncertainty trade-off is governed by the sum of the covariance term and a state-dependent improvement over the Robertson bound. As applications, we also refine error-disturbance trade-offs by incorporating spectral information of both the system and the measuring apparatus, thereby generalizing the Arthurs-Goodman and Ozawa inequalities.
The uncertainty principle constitutes a cornerstone of quantum mechanics. However, its conventional formulations do not adequately describe uncertainty in scenarios involving conditional measurement. In this work, we propose uncertainty relations based on conditional variance and conditional expectation. We define the conditional expectation operator E ( A ̂ | B ̂ ) , its dual operator E ( B ̂ | A ̂ ) , and a corresponding conditional covariance term R corr . We demonstrate that even for noncommuting observables A ̂ and B ̂ , their conditional expectation operators can possess a nonvanishing commutator. This leads to a strengthened uncertainty relation applicable to both pure and mixed states. In addition to the standard term arising from the noncommutativity of the observables themselves, our relation incorporates three distinct contributions: one from the noncommutativity of the conditional expectations, one from quantum coherence, and one from the conditional covariance. These contributions collectively yield a significantly tighter lower bound. Numerical simulations verify the generality and enhanced tightness of the proposed relation. Our results provide a refined theoretical tool for the joint estimation of multiple observables in composite quantum systems, with potential applications in quantum precision metrology and secure quantum key distribution.
The uncertainty principle is fundamentally rooted in the algebraic asymmetry between observables. We introduce a new class of uncertainty relations grounded in the resource theory of asymmetry, where incompatibility is quantified by an observable's intrinsic, state-independent capacity to break the symmetry associated with another. This ``operator asymmetry,'' formalized as the incompatibility norm, leads to a variance-based uncertainty relation for pure states that can be tighter than the standard Robertson bound. Most significantly, this framework resolves a long-standing open problem in quantum information theory: the formulation of a universally valid, product-form uncertainty relation for the Wigner-Yanase skew information. We demonstrate the practical power of our framework by deriving tighter quantum speed limits for the dynamics of nearly conserved quantities, which are crucial for understanding non-equilibrium phenomena such as prethermalization and many-body localization. This work provides both a new conceptual lens for understanding quantum uncertainty and a powerful, versatile toolkit for its application.
The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.
Uncertainty relations occupy a fundamental position in quantum mechanics. We propose stronger variance-based uncertainty relations for the product and sum of variances of two incompatible observables in a finite dimensional Hilbert space. It is shown that the new uncertainty relations provide near-optimal state-dependent bounds, which can be useful for quantum metrology, entanglement detection etc. in quantum information theory. It is further shown that the uncertainty relations are related to the spreads of the distribution of measurement outcomes caused by incompatible observables. Intuitively, this means that the ability of learning the distribution has both the upper and lower bounds. Combination of these bounds provides naturally an uncertainty interval which captures the essence of uncertainty in quantum theory. Finally, we explain how to employ entropic uncertainty relations to derive lower bounds for the product of variances of incompatible observables.