arXiv subjects

Aina Mayumi

Publications and source records attributed to Aina Mayumi.

5 recordsLinked to original sources

arXiv2026

Optimal Uncertainty Relations for a Single Observable

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.

quant-ph↗

arXiv2026

Beyond Robertson-Schrödinger: A General Uncertainty Relation Unveiling Hidden Noncommutative Trade-offs

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 \emph{exact equality} for \emph{any} state and \emph{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.

quant-ph↗

arXiv2025

Tight Generalization of Robertson-Type Uncertainty Relations

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 a state-dependent coefficient $\frac{(λ_{\max} + λ_{\min})^2}{4(λ_{\max} - λ_{\min})^2}$, where $ λ_{\max} $ and $ λ_{\min}$ denote the largest and smallest eigenvalues of the density operator $ρ$, respectively. This coefficient is optimal among all Robertson-type generalizations and does not admit further improvement.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 Schröedinger'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.

quant-ph↗

arXiv2024

Uncertainty relations based on state-dependent norm of commutator

We introduce two uncertainty relations based on the state-dependent norm of commutators, utilizing generalizations of the Böttcher-Wenzel inequality. The first relation is mathematically proven, while the second, tighter relation is strongly supported by numerical evidence. Both relations surpass the conventional Robertson and Schrödinger bounds, particularly as the quantum state becomes increasingly mixed. This reveals a previously undetected complementarity of quantum uncertainty, stemming from the non-commutativity of observables. We also compare our results with the Luo-Park uncertainty relation, demonstrating that our bounds can outperform especially for mutually unbiased observables.

quant-ph↗

arXiv2024

Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics

By employing a weighted Frobenius norm with a positive matrix $ω$, we introduce natural generalizations of the famous Böttcher-Wenzel (BW) inequality. Based on the combination of the weighted Frobenius norm $\|A\|_ω:= \sqrt{{\rm tr}(A^\ast A ω)}$ and the standard Frobenius norm $\|A\| := \sqrt{{\rm tr}(A^\ast A)}$, there are exactly five possible generalizations, labeled (i) through (v), for the bounds on the norms of the commutator $[A,B]:= AB - BA$. In this paper, we establish the tight bounds for cases (iii) and (v), and propose conjectures regarding the tight bounds for cases (i) and (ii). Additionally, the tight bound for case (iv) is derived as a corollary of case (i). All these bounds (i)-(v) serve as generalizations of the BW inequality. The conjectured bounds for cases (i) and (ii) (and thus also (iv)) are numerically supported for matrices up to size $n=15$. Proofs are provided for $n=2$ and certain special cases. Interestingly, we find applications of these bounds in quantum physics, particularly in the contexts of the uncertainty relation and open quantum dynamics.

math-ph↗