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Conjectured tight weighted Frobenius norm bounds for matrix commutators

From papers

Let ω∈Mn(C)\omega \in M_n(\mathbb{C}) be a positive matrix, and let λM\lambda_{\rm M}, λm\lambda_{\rm m}, and λsm\lambda_{\rm sm} denote respectively its largest, smallest, and second smallest eigenvalues. For matrices A,B∈Mn(C)A,B \in M_n(\mathbb{C}), write [A,B]=AB−BA[A,B]=AB-BA, and let ∥⋅∥ω\|\cdot\|_\omega and ∥⋅∥\|\cdot\| denote the weighted and ordinary Frobenius norms used in the paper.

Conjectured tight bounds. The tight bounds for cases (i), (ii), and (iv) are respectively

∥[A,B]∥ω≤λm+λsmλmλsm ∥A∥ω∥B∥ω,\|[A,B]\|_\omega \leq \sqrt{\frac{\lambda_{\rm m}+\lambda_{\rm sm}}{\lambda_{\rm m}\lambda_{\rm sm}}}\,\|A\|_\omega\|B\|_\omega, ∥[A,B]∥ω≤λm+λMλm ∥A∥ω∥B∥,\|[A,B]\|_\omega \leq \sqrt{\frac{\lambda_{\rm m}+\lambda_{\rm M}}{\lambda_{\rm m}}}\,\|A\|_\omega\|B\|,

and

∥[A,B]∥≤λm+λsmλm2λsm ∥A∥ω∥B∥ω.\|[A,B]\|\leq \sqrt{\frac{\lambda_{\rm m}+\lambda_{\rm sm}}{\lambda_{\rm m}^2\lambda_{\rm sm}}}\,\|A\|_\omega\|B\|_\omega.

In each case, the bound is tight: there are nonzero matrices AA and BB attaining equality. These claims are motivated by numerical optimizations; the paper presents them as conjectures, so their general validity remains open.

Progress summary

Open

The three proposed sharp bounds remain conjectures: numerical tests and low-dimensional special cases support them, but no general proof or counterexample has been reported.

A 2024 paper asks whether three eigenvalue-dependent commutator inequalities for weighted Frobenius norms are universally valid and sharp. The proposed constants depend on the smallest, second-smallest, and largest eigenvalues of the positive weight matrix.

Known results

  • The paper proves the corresponding tight bounds for cases (iii)(iii) and (v)(v).
  • For the conjectured cases (i)(i), (ii)(ii), and (iv)(iv), proofs are given when n=2n=2 and in certain special cases.
  • Numerical optimization agrees with the conjectured constants for randomly generated weights through n=15n=15.

2024 conjecture and subsequent status

The paper presents cases (i)(i) and (ii)(ii) as conjectures, with case (iv)(iv) derived from case (i)(i), and supplies equality examples. The scan found no later proof, counterexample, withdrawal, retraction, or independent verification.

Current status (as of August 2026): the bounds for cases (iii)(iii) and (v)(v) are settled, while the proposed tight bounds for cases (i)(i), (ii)(ii), and (iv)(iv) remain open.

Sources
Sources & referencesView supporting material

Primary source

Aina Mayumi, Gen Kimura, Hiromichi Ohno and Dariusz Chruściński, “Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics”, arXiv:2403.04199 (2024).

matrix-inequalitiesquantum-informationweighted-frobenius-norms

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