You are presented with multiple paper abstracts, alongside the associated literature review and chapter titles. Given the provided information, you are going to do a literature review. Please generate a coherent literature review that synthesizes the content, draws connections between different works, highlights major findings, and identifies any gaps in the existing research.

Literature Review Title: Model Checking Quantum Systems --- A Survey 
Chapter Title:  Definition 4.1 (Quantum Markov chains [47]): 
1. Abstract of Cited Paper (BIB001):  This paper studies three kinds of long-term behaviour, namely reachability, repeated reachability and persistence, of quantum Markov chains (qMCs). As a stepping-stone, we introduce the notion of bottom strongly connected component (BSCC) of a qMC and develop an algorithm for finding BSCC decompositions of the state space of a qMC. As the major contribution, several (classical) algorithms for computing the reachability, repeated reachability and persistence probabilities of a qMC are presented, and their complexities are analysed. 
2. Abstract of Cited Paper (BIB002):  We present quantum observable Markov decision processes (QOMDPs), the quantum analogs of partially observable Markov decision processes (POMDPs). In a QOMDP, an agent is acting in a world where the state is represented as a quantum state and the agent can choose a superoperator to apply. This is similar to the POMDP belief state, which is a probability distribution over world states and evolves via a stochastic matrix. We show that the existence of a policy of at least a certain value has the same complexity for QOMDPs and POMDPs in the polynomial and infinite horizon cases. However, we also prove that the existence of a policy that can reach a goal state is decidable for goal POMDPs and undecidable for goal QOMDPs. 
3. Abstract of Cited Paper (BIB003):  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.

Literature Review:
A quantum Markov chain is a triple (H, E, H 0 ), where H and H 0 are the same as in Definition 3.1, and E is a super-operator on H. A quantum Markov chain starts in an initial state in H 0 , and at each step it performs (the same) quantum operation modelled by the super-operator E. Note that the (discrete-time) dynamics of closed quantum systems are usually depicted by unitary operators, and the behaviours of open quantum systems are described by super-operators (see BIB001 , Section 8.2). Obviously, the notion of quantum automata can be generalised by replacing unitary operators U α (α ∈ Act) in Definition 3.1 by super-operators E α (α ∈ Act). Furthermore, quantum Markov decision processes can be defined by introducing decision strategies into such generalised quantum automata BIB003 ,  . Several algorithms for checking reachability of quantum Markov chains and quantum Markov decision processes were developed in BIB002 ,  . As in checking classical Markov chains and Markov decision processes, graph reachability is a key to these algorithms. However, classical graph theory is not suited to our purpose; instead a new theory of quantum graphs (i.e. graphs in a Hilbert space with adjacency relation induced by a super-operator) was developed, and in particular, an algorithm for the BSCC (bottom strongly connected components) decomposition of the state Hilbert spaces was found in BIB002 . Another decomposition technique, namely periodic decomposition, for quantum Markov chains was recently proposed in BIB004 .