This chapter introduces quantum spin chains for two different purposes: prototypes of infinite quantum systems discretized in space, and general models of stochastic processes in discrete time on a discrete quantum state space. The algebra of local observables is constructed. The probability functionals, i.e. the states, are defined in terms of reduced density matrices with their compatibility relations. Several examples of shift-invariant states are discussed: product states, classical states, and limiting Gibbs states. The chapter concludes with the construction of nearest-neighbour dynamics and the need for an appropriate algebra of quasi-local observables.
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