# One-parameter generalization of the Böttcher-Wenzel inequality and its application to open quantum dynamics

Dariusz Chruściński, Gen Kimura, Hiromichi Ohno, Tanmay Singal

## 抄録

In this paper, we introduce a one-parameter generalization of the famous Böttcher-Wenzel (BW) inequality in terms of a q-deformed commutator. For n×n matrices A and B, we consider the inequality Re〈[B,A],[B,A]q〉≤c(q)‖A‖2‖B‖2, where 〈A,B〉=tr(AB) is the Hilbert-Schmidt inner product, ‖A‖ is the Frobenius norm, [A,B]=AB−BA is the commutator, and [A,B]q=AB−qBA is the q-deformed commutator. We prove that when n=2, or when A is normal with any size n, the optimal bound is given by [Formula presented] We conjecture that this is also true for any matrices, and this conjecture is perfectly supported for n up to 15 by numerical optimization. When q=1, this inequality is exactly the BW inequality. When q=0, this inequality leads the sharp bound for the r-function which is recently derived for the application to universal constraints of relaxation rates in open quantum dynamics.

本文言語 English 158-166 9 Linear Algebra and Its Applications 656 https://doi.org/10.1016/j.laa.2022.09.022 Published - 2023 1月 1

• 代数と数論
• 数値解析
• 幾何学とトポロジー
• 離散数学と組合せ数学

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