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arXiv:2504.20404 · npj Quantum Information

Beyond Robertson–Schrödinger: a general uncertainty relation unveiling hidden noncommutative trade-offs

Gen Kimura, Aina Mayumi, Hiromichi Ohno, Jaeha Lee, Dariusz Chruściński

Published Sep 21, 2026• 1 citation
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Abstract

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.

DOI
10.1038/s41534-026-01374-0

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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.
Uncertainty relations are usually formulated as trade-offs between two or more observables. Yet the uncertainty of a single observable can already contain an intrinsically quantum contribution arising from its noncommutativity with the state. In the sense of Luo, such a contribution can be quantified by a quantum uncertainty Qρ(A)Q_ρ(A)Qρ​(A), with prominent examples including the Wigner--Yanase and Wigner--Yanase--Dyson skew informations and the quantum Fisher information. Here we ask how strong such single-observable uncertainty relations can be when the quantum state is fixed. We solve this fixed-state optimization problem by determining the largest coefficient copt(ρ)c_{\rm opt}(ρ)copt​(ρ) for which Vρ(A)≥copt(ρ)Qρ(A)V_ρ(A)\ge c_{\rm opt}(ρ)Q_ρ(A)Vρ​(A)≥copt​(ρ)Qρ​(A) holds for every observable AAA, and show that it depends only on the smallest and largest eigenvalues of ρρρ. The same optimization principle extends to a new power-commutator family 12∣[ρs,A]∣2\frac{1}{2}|[ρ^s,A]|^221​∣[ρs,A]∣2 (s≥1/2)(s\ge 1/2)(s≥1/2), providing a direct measure of state--observable noncommutativity beyond the standard quantum-uncertainty framework. More importantly, the optimal noncommutative contribution can be combined with the maximal classical contribution compatible with the state, yielding a sharpened variance decomposition without weakening the optimal bound. For states with exactly two distinct eigenvalues, this relation becomes an identity for every observable; in particular, qubit variances decompose exactly into classical and noncommutative parts. Finally, the corresponding variance-product bounds are no weaker than the Robertson relation for qubit spin observables and can be strictly stronger, showing that a substantial part of the uncertainty usually expressed as a trade-off between distinct observables is already encoded in their individual noncommutativity with the state.
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.
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.
In this paper, we propose two uncertainty relations that reveal a previously unnoticed complementarity in quantum uncertainty not captured by the conventional Robertson and Schr\"odinger relations, particularly in the case of mixed quantum states. These relations are formulated based on the state-dependent norm of the commutator, generalizing the B\"ottcher-Wenzel inequality. Furthermore, we compare our results with the Luo-Park uncertainty relation and demonstrate that our bound outperforms their bound for mutually unbiased observables.
Abstract The uncertainty principle sets a limit to our capacity to predict the outcomes of two incompatible measurements. The Heisenberg uncertainty relation for two arbitrary observables A and B is usually discussed in textbooks. However, little or no attention is paid to the fact that Schrödinger generalised the Heisenberg relation taking into account the covariance between the observables A and B . This extended inequality is known as the Robertson–Schrödinger uncertainty relation. Here, we demonstrate the less known fact that two-level quantum states, i.e., qubits, satisfy the equality of the Robertson–Schrödinger uncertainty relation for two arbitrary observables A and B . Taking advantage of the homomorphism between SU(2) and SO(3) groups, it is possible to map the distributions of the expectation values and variances of the observables, and the Heisenberg and covariance terms on the Bloch sphere. The graphical visualisation of the relevant quantities involved in the uncertainty relations allows us to distinguish specific properties and symmetries that are not so evident in the algebraic formalism.