A Dimension-Independent Commutator Bound
Abstract
We prove that every trace-zero matrix A∈ Mn(ℂ) admits a representation A=BC−CB with B,C∈ Mn(ℂ) and lVert BrVertlVert CrVert≤ KlVert ArVert, where K is an absolute constant independent of n, and lVert·rVert denotes the operator norm. For a fixed t>0, the proof splits according to whether lVertRe(eiθA)rVert1≥ tnlVert ArVert holds for all θ∈ℝ, where lVert·rVert1 denotes the trace norm. When this lower bound holds, we construct a commutator representation directly. Otherwise, the vector-selection theorem of Marcus, Spielman, and Srivastava yields smaller trace-zero compressions whose norms are small enough for the induction to close. We also construct an explicit family of zero-diagonal Hermitian unitaries that forces a lower bound of order √(log n) for lVert BrVertlVert CrVert when either factor is required to be diagonal in the prescribed basis. The same family admits ε-pavings with fewer than 2ε−2 blocks and representations by two normal factors with optimal norm product 1/2. This establishes a distinction between unrestricted commutator bounds and bounds under a prescribed diagonal restriction. The main results and their essential inputs are formalized in Lean 4 using Mathlib. The development also includes a formal derivation of the Kadison-Singer state-extension theorem from the same vector-selection theorem.
Related
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BibTeX
@misc{shen2026dimensionindependent,
title = {{A Dimension-Independent Commutator Bound}},
author = {Hao Shen and Jiaqi Wang and Lihong Zhi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09938},
eprint = {2609.09938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09938},
}
RIS
TY - UNPB
TI - A Dimension-Independent Commutator Bound
AU - Hao Shen
AU - Jiaqi Wang
AU - Lihong Zhi
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09938
UR - https://arxiv.org/abs/2609.09938
AB - We prove that every trace-zero matrix A∈ M_n(ℂ) admits a representation A=BC−CB with B,C∈ M_n(ℂ) and lVert BrVertlVert CrVert≤ KlVert ArVert, where K is an absolute constant independent of n, and lVert·rVert denotes the operator norm. For a fixed t>0, the proof splits according to whether lVertRe(e^iθA)rVert_1≥ tnlVert ArVert holds for all θ∈ℝ, where lVert·rVert_1 denotes the trace norm. When this lower bound holds, we construct a commutator representation directly. Otherwise, the vector-selection theorem of Marcus, Spielman, and Srivastava yields smaller trace-zero compressions whose norms are small enough for the induction to close. We also construct an explicit family of zero-diagonal Hermitian unitaries that forces a lower bound of order √(log n) for lVert BrVertlVert CrVert when either factor is required to be diagonal in the prescribed basis. The same family admits ε-pavings with fewer than 2ε^−2 blocks and representations by two normal factors with optimal norm product 1/2. This establishes a distinction between unrestricted commutator bounds and bounds under a prescribed diagonal restriction. The main results and their essential inputs are formalized in Lean 4 using Mathlib. The development also includes a formal derivation of the Kadison-Singer state-extension theorem from the same vector-selection theorem.
ER -