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arXiv:2506.16850

Refined uncertainty relation for q-commutator

Yanagi, Kenjiro

Published Jun 20, 2025• 0 citations
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Abstract

We show that Robertson uncertainty relation can be refined for q-commutator which is defined by $[A,B]_q = AB-qBA$, where $A, B$ are self-adjoint operators and real number $q\in \mathbb{R}$. The coefficient is represented by the eigenvalues of state $ρ$.

DOI
10.48550/arxiv.2506.16850

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We show that Robertson uncertainty relation can be refined for q-commutator which is defined by [A,B]q=AB−qBA[A,B]_q = AB-qBA[A,B]q​=AB−qBA, where A,BA, BA,B are self-adjoint operators and real number q∈Rq\in \mathbb{R}q∈R. The coefficient is represented by the eigenvalues of state ρρρ.
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.
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.
The Ehrenfest theorem and the Robertson uncertainty relation are well-known basic equations in quantum mechanics. However, there exist problematic cases, where the Ehrenfest theorem and the Robertson uncertainty relation are not correct. These cases occur when the azimuthal angle in polar or spherical coordinates is used within these equations. The purpose of this paper is to present and discuss a generalized Ehrenfest theorem and a generalized uncertainty relation which are still valid for these cases. Hereby, we define and use a mathematical operation called expectation commutator.
We report a refinement of Robertson-Schroedinger uncertainty relation via Wigner-Yanase skew information. Besides the well known quantum uncertainty arising from the noncommutativity of observables, there is classical uncertainty arising from the mixedness of the states that is quantified by the difference between the variance and the skew information. Our refined uncertainty relation for canonical observables is saturated by all the Gaussian states, pure or mixed, and thus provides an alternative measure for the non-Gaussianity of quantum states. Generalizations to the case of metric adjusted skew information are presented, unifying and refining most of previous results.
Abstract Real Hilbert space is a Euclidean space of infinite dimensionality. Complex Hilbert space is less amenable to elementary geometrical intuitions, but these do serve to give structure to the present argument. The familiar process of finding, relative to a vector x, the expectation Ā of a self-adjoint linear operator A, and then using the deviation A — Ā to obtain a dispersion ΔA, may be interpreted as a projection leading to a unit vector e orthogonal to x. If we have n self-adjoint operators Ar, we thus form n unit vectors er orthogonal to x. From a basic property of Hilbert space, the vector c1e1 + c2e2 + ... + cnen has a non-negative norm for arbitrary complex coefficients. This demands the positive-indefinite character of a real quadratic form in 2n variables; equivalently, all its eigenvalues must be non-negative. This leads to inequalities on the dispersions ΔAr, the expectations of the commutators i[Ar, As] and other expectations. For n = 2, this process leads to Schrödinger’s improvement of the Heisenberg inequality. The case n = 3 is here worked out explicitly.