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Asymptotic equivalence of a subclass of WCLT non degenerate GKSL generator

Jorge R. Bolaños-Servín

Abstract

We prove the assymptotic equivalence of a sequence of block diagonal matrices with Toeplitz blocks. The blocks are the principal submatrices of an originating Toeplitz sequence with generating symbol of the Wiener class. As an application, using the invariance of certain \textit{diagonal} and \textit{cyclic-diagonal} operator subspaces of the GKSL generators of circulant and WCLT quantum Markov semigroups, the asymptotic equivalence of the families is proved under suitable hypothesis.

Asymptotic equivalence of a subclass of WCLT non degenerate GKSL generator

Abstract

We prove the assymptotic equivalence of a sequence of block diagonal matrices with Toeplitz blocks. The blocks are the principal submatrices of an originating Toeplitz sequence with generating symbol of the Wiener class. As an application, using the invariance of certain \textit{diagonal} and \textit{cyclic-diagonal} operator subspaces of the GKSL generators of circulant and WCLT quantum Markov semigroups, the asymptotic equivalence of the families is proved under suitable hypothesis.
Paper Structure (11 sections, 14 theorems, 66 equations)

This paper contains 11 sections, 14 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic behaviour of matrices
  4. Circulant and Toeplitz asymptotic equivalence
  5. Toeplitz and circulant CP maps and GKSL generators
  6. Toeplitz and circulant induced vectorizations
  7. Circulant C.P. maps
  8. CP Toeplitz maps
  9. Asymptotic equivalence of CP Toeplitz and Circulant maps
  10. Toeplitz class of non degenerate WCLT QMS
  11. Assymptotic equivalence of WCLT and circulant generators

Key Result

Theorem 2

Let $\{A_n\}_n$, $\{B_n\}_n$ be asymptotically equivalent sequences of matrices with eigenvalues $\alpha_{n,k}$ and $\beta_{n,k}$, respectively. Assume that the eigenvalues of either matrix converge, i.e., exists and is finite for any positive integer $s$. Then

Theorems & Definitions (26)

  • Definition 1
  • Theorem 2
  • Proposition 3
  • Definition 4
  • Definition 5
  • Remark 6
  • Proposition 7
  • Lemma 8
  • proof
  • Theorem 9
  • ...and 16 more