On the General Form of Quantum Stochastic Evolution Equation

A characterisation of the quantum stochastic bounded generators of irreversible quantum state evolutions is given. This suggests the general form of quantum stochastic evolution equation with respect to the Poisson (jumps), Wiener (diffusion) or general Quantum Noise. The corresponding irreversible Heisenberg evolution in terms of stochastic completely positive (CP) cocycles is also characterized and the general form of the stochastic completely dissipative (CD) operator equation is discovered. 1. Quantum Stochastic Filtering Equations The quantum filtering theory, which was outlined in [1, 2] and developed then since [3], provides the derivations for new types of irreversible stochastic equations for quantum states, giving the dynamical solution for the well-known quantum measurement problem. Some particular types of such equations have been considered recently in the phenomenological theories of quantum permanent reduction [4, 5], continuous measurement collapse [6, 7], spontaneous jumps [8, 9], diffusions and localizations [10, 11]. The main feature of such dynamics is that the reduced irreversible evolution can be described in terms of a linear dissipative stochastic wave equation, the solution to which is normalized only in the mean square sense. The simplest dynamics of this kind is described by the continuous filtering wave propagators Vt (ω), defined on the space Ω of all Brownian trajectories as an adapted operator-valued stochastic process in the system Hilbert space H, satisfying the stochastic diffusion equation (1.1) dVt +KVtdt = LVtdQ, V0 = I in the Itô sense, which was derived from a unitary evolution in [13]. Here Q (t, ω) is the standard Wiener process, which is described by the independent increments dQ (t) = Q (t+ dt) − Q(t), having the zero mean values 〈dQ〉 = 0 and the multiplication property (dQ) = dt, K is an accretive operator, K +K ≥ LL, and L is a linear operator D → H. Using the Itô formula (1.2) d ( V † t Vt ) = dV † t Vt + V † t dVt + dV † t dVt, Date: July 20, 1995. 1991 Mathematics Subject Classification. Quantum Stochastics.