Unbounded length minimal synchronizing words for quantum channels over qutrits

2026-02-28

AI Summary

This study challenges the applicability of Černý’s conjecture in the quantum realm by investigating whether arbitrarily long minimal synchronizing words can exist in quantum channels. By constructing a quantum channel tailored for qutrit systems and leveraging tools from quantum information theory alongside state-space analysis, the authors rigorously establish—for the first time—the existence of minimal synchronizing words of unbounded length. This result not only overturns the classical automata-theoretic assumption that synchronizing word lengths are inherently bounded but also substantially extends the prior work of Grudka et al., whose construction achieved only length three. Furthermore, the work introduces a novel paradigm for quantum control and synchronization, offering a profound contrast between quantum and classical synchronization theories.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Applications

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• quantum channels
• synchronizing words
• qutrits
• unbounded length
• minimal synchronizing words

Innovation

• quantum channels
• synchronizing words
• qutrits
• unbounded length
• Černý conjecture