Discrete Quantum Processes

Abstract

A discrete quantum process is defined as a sequence of local states $\rho_t$, $t=0,1,2,...$, satisfying certain conditions on an $L_2$ Hilbert space $H$. If $\rho =\lim\rho_t$ exists, then $\rho$ 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 $\mu$ on the algebra of cylinder sets $\cscript$. We consider the problem of extending $\mu$ to other physically relevant sets in a systematic way. To this end we show that $\mu$ can be properly extended to a quantum measure $\mutilde$ on a "quadratic algebra" containing $\cscript$. We also show that a random variable $f$ can be "quantized" to form a self-adjoint operator $\fhat$ on $H$. We then employ $\fhat$ to define a quantum integral $\int fd\mutilde$. Various examples are given