5 Reconstruction Theorem for Quantum Stochastic Processes

Statistically interpretable axioms are formulated that define a quantum stochastic process (QSP) as a causally ordered field in an arbitrary space–time region T of an open quantum system under a sequential observation at a discrete space-time localization. It is shown that to every QSP described in the weak sense by a self-consistent system of causally ordered correlation kernels there corresponds a unique, up to unitary equivalence, minimal QSP in the strong sense. It is shown that the proposed QSP construction, which reduces in the case of the linearly ordered discrete T = Z to the construction of the inductive limit of Lindblad's canonical representations [8], corresponds to Kolmogorov's classical reconstruction [12] if the order on T is ignored and leads to Lewis construction [14] if one uses the system of all (not only causal) correlation kernels, regarding this system as lexicographically preordered on Z × T. The approach presented encompasses both nonrelativistic and relativis-tic irreversible dynamics of open quantum systems and fields satisfying the conditions of local commutativity semigroup covariance. Also given are necessary and sufficient conditions of dynamicity (or conditional Markovianity) and regularity, these leading to the properties of complete mixing (relaxation) and ergodicity of the QSP. The problem of statistical foundation for the irreversible processes in open systems of quantum thermodynamics and measurement theory encountered in coherent optics, quantum communications and microelectronics [1]–[3] requires the development of a general theory of quantum stochastic processes (QSP) that contains the classical theory and the known QSP models as special cases. This theory must be operational and should admit a microscopically consistent statistical interpretation for the irreversible successive physical maps like quantum dynamical transformations and quantum measurement operations. On the phenomenological level such quantum operations were introduced already by von Neumann [4] and considered in a more general framework by Haag and Kastler [5] and Davies and Lewis [6]. A microscopically consistent operational approach to the general QSP theory which will be followed here, containing the classical and quantum statistical theories as special cases, was outlined in [7]. In physical applications, QSPs are usually described by special chronologically ordered correlation functions, the only functions which can be dynamically defined and tested on the basis of the statistics of successive measurements in real time, called causal. An axiomatic definition of QSPs based on causal correlation operators corresponding to a discrete time T = Z was given by Lindblad [8] for the case …