## Abstract

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.

Original language | English |
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Pages (from-to) | 293-305 |

Number of pages | 13 |

Journal | Linear Algebra and Its Applications |

Volume | 630 |

DOIs | |

Publication status | Published - 2021 Dec 1 |

## Keywords

- 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