Decomposition of Quantum Markov Chains and Its Applications

TL;DR

This paper introduces a new periodic decomposition method for quantum Markov chains, providing a limit theorem and an algorithm to identify maximum dimensional noiseless subsystems in quantum systems.

Contribution

It presents a novel periodic decomposition technique for quantum Markov chains and establishes a limit theorem, advancing analysis and control of quantum systems.

Findings

Introduces periodic decomposition for quantum Markov chains

Establishes a limit theorem for the new decomposition

Provides an algorithm to find maximum dimensional noiseless subsystems

Abstract

Markov chains have been widely employed as a fundamental model in the studies of probabilistic and stochastic communicating and concurrent systems. It is well-understood that decomposition techniques play a key role in reachability analysis and model-checking of Markov chains. (Discrete-time) quantum Markov chains have been introduced as a model of quantum communicating systems [1] and also a semantic model of quantum programs [2]. The BSCC (Bottom Strongly Connected Component) and stationary coherence decompositions of quantum Markov chains were introduced in [3, 4, 5]. This paper presents a new decomposition technique, namely periodic decomposition, for quantum Markov chains. We further establish a limit theorem for them. As an application, an algorithm to find a maximum dimensional noiseless subsystem of a quantum communicating system is given using decomposition techniques of quantum Markov chains.