TY - JOUR

T1 - One-parameter generalization of the Böttcher-Wenzel inequality and its application to open quantum dynamics

AU - Chruściński, Dariusz

AU - Kimura, Gen

AU - Ohno, Hiromichi

AU - Singal, Tanmay

N1 - Funding Information:
D.C. was supported by the Polish National Science Centre Project No. 2018/30/A/ST2/00837 . G.K. was supported in part by JSPS KAKENHI Grant No. 17K18107 .
Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - 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(A⁎B) 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.

AB - 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(A⁎B) 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.

KW - Böttcher-Wenzel inequality

KW - Commutator

KW - Frobenius norm

KW - Hilbert-Schmidt inner product

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U2 - 10.1016/j.laa.2022.09.022

DO - 10.1016/j.laa.2022.09.022

M3 - Article

AN - SCOPUS:85139250175

SN - 0024-3795

VL - 656

SP - 158

EP - 166

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -