Constraints for the spectra of generators of quantum dynamical semigroups

Dariusz Chruściński, Ryohei Fujii, Gen Kimura, Hiromichi Ohno

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


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(AB) 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.

Original languageEnglish
Pages (from-to)293-305
Number of pages13
JournalLinear Algebra and Its Applications
Publication statusPublished - 2021 Dec 1


  • Commutator
  • Complete positivity
  • Frobenius norm
  • Hilbert-Schmidt inner product
  • Quantum dynamical semigroup

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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