sequence of quantum measures, Hilbert space
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 = lim 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 C. 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 µe on a “quadratic algebra” containing C. We also show that a random variable f can be “quantized” to form a self-adjoint operator fb on H. We then employ fb to define a quantum integral R f dµe. Various examples are given.
Gudder, S., "Discrete Quantum Processes" (2011). Mathematics Preprint Series. Paper 38.