Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unbounded length minimal synchronizing words for quantum channels over qutrits

Bjørn Kjos-Hanssen, Swarnalakshmi Lakshmanan

TL;DR

This work extends Grudka, Karczewski, Kurzynski, Stempin, W\'ojcik and Wojcik's result to arbitrarily long minimal synchronizing words, providing a contrast to Kern's conjecture for finite automata.

Abstract

Grudka, Karczewski, Kurzynski, Stempin, Wójcik and Wojcik (2025) constructed quantum channels with synchronizing words of length 3 for qutrits. We extend their result to arbitrarily long minimal synchronizing words, providing a contrast to Černý's conjecture for finite automata.

Unbounded length minimal synchronizing words for quantum channels over qutrits

TL;DR

This work extends Grudka, Karczewski, Kurzynski, Stempin, W\'ojcik and Wojcik's result to arbitrarily long minimal synchronizing words, providing a contrast to Kern's conjecture for finite automata.

Abstract

Grudka, Karczewski, Kurzynski, Stempin, Wójcik and Wojcik (2025) constructed quantum channels with synchronizing words of length 3 for qutrits. We extend their result to arbitrarily long minimal synchronizing words, providing a contrast to Černý's conjecture for finite automata.
Paper Structure (4 sections, 8 theorems, 31 equations, 1 figure)

This paper contains 4 sections, 8 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Construction
  3. Main results
  4. Future work

Key Result

Lemma 5

For any quantum channel $\Phi$,

Figures (1)

  • Figure 1: A finite part of $\mathscr{M}_2'$. Self-loops are not shown and each density matrix should be normalized to have trace 1.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • Lemma 6: Hölder inequality for Schatten norms MR2978290 (5.6)
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 8 more