Discrete Quantum Processes

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A discrete quantum process is defined as a sequence of local states ρₜ, t=0,1,2,..., satisfying certain conditions on an L₂ Hilbert space H. If ρ=ρₜ exists, then ρ is called a global state for the system. In important cases, the global state does not exist and we must then work with the local states. In a natural way, the local states generate a sequence of quantum measures which in turn define a single quantum measure μ on the algebra of cylinder sets. We consider the problem of extending μ to other physically relevant sets in a systematic way. To this end we show that μ can be properly extended to a quantum measure on a "quadratic algebra" containing. We also show that a random variable f can be "quantized" to form a self-adjoint operator on H. We then employ to define a quantum integral ∫ fd. Various examples are given