TY - JOUR

T1 - Constraints for the spectra of generators of quantum dynamical semigroups

AU - Chruściński, Dariusz

AU - Fujii, Ryohei

AU - Kimura, Gen

AU - Ohno, Hiromichi

N1 - Funding Information:
We would like to thank Prof. Andrzej Kossakowski for his fruitful comments and advices. During the preparation of this manuscript, he passed away on 1st February 2021. We would like to dedicate this paper to his memory. D.C. is supported by the National Science Center project 2018/30/A/ST2/00837 . G.K. is supported in part by JSPS KAKENHI Grants No. 17K18107 .
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/12/1

Y1 - 2021/12/1

N2 - Motivated by a spectral analysis of the generator of a completely positive trace-preserving semigroup, we analyze the real functional [Formula presented] where 〈A,B〉:=tr(A⁎B) is the Hilbert-Schmidt inner product, and [A,B]:=AB−BA is the commutator. In particular we discuss upper and lower bounds of the form c−‖A‖2‖B‖2≤r(A,B)≤c+‖A‖2‖B‖2 where ‖A‖ is the Frobenius norm. We prove that the optimal upper and lower bounds are given by [Formula presented]. If A is restricted to be traceless, the bounds are further improved to be [Formula presented]. Interestingly, these upper bounds, especially the latter one, provide new constraints on relaxation rates for the quantum dynamical semigroup tighter than previously known constraints in the literature. A relation with the Böttcher-Wenzel inequality is also discussed.

AB - Motivated by a spectral analysis of the generator of a completely positive trace-preserving semigroup, we analyze the real functional [Formula presented] where 〈A,B〉:=tr(A⁎B) is the Hilbert-Schmidt inner product, and [A,B]:=AB−BA is the commutator. In particular we discuss upper and lower bounds of the form c−‖A‖2‖B‖2≤r(A,B)≤c+‖A‖2‖B‖2 where ‖A‖ is the Frobenius norm. We prove that the optimal upper and lower bounds are given by [Formula presented]. If A is restricted to be traceless, the bounds are further improved to be [Formula presented]. Interestingly, these upper bounds, especially the latter one, provide new constraints on relaxation rates for the quantum dynamical semigroup tighter than previously known constraints in the literature. A relation with the Böttcher-Wenzel inequality is also discussed.

KW - Commutator

KW - Complete positivity

KW - Frobenius norm

KW - Hilbert-Schmidt inner product

KW - Quantum dynamical semigroup

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U2 - 10.1016/j.laa.2021.08.012

DO - 10.1016/j.laa.2021.08.012

M3 - Article

AN - SCOPUS:85113379341

SN - 0024-3795

VL - 630

SP - 293

EP - 305

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -