The Hamiltonian Operator Associated with Some Quantum Stochastic Evolutions

Matteo Gregoratti

Manuel Gadella, Luis Miguel Nieto, J. M. Noriega +1 more

The definition and some general properties of the generalized Weyl correspondence between stochastic processes and operator-valued real functions on a Hilbert space plus a trace class operator are given. The relation between the derivatives of an operator-valued function and the derivative of the corresponding stochastic process are studied. When the operator-valued function is the position (or momentum) in the Heisenberg picture, a condition for the positivity of the joint distribution functions of the corresponding process is given, provided that the evolution Hamiltonian be quadratic in the position and momentum. Finally, the case of an arbitrary Hamiltonian evolution for the position operator is studied and the two-dimensional density functions of the process is related to the Wigner function associated to some state ρ̂, and a necessary condition for the positivity of the densities is given.

G. O. S. Ekhaguere

Alain Sarlette, Pierre Rouchon

This paper studies the stochastic differential equation (SDE) associated with a two-level quantum system (qubit) subject to Hamiltonian evolution as well as unmonitored and monitored decoherence channels. The latter imply a stochastic evolution of the quantum state (density operator), whose associated probability distribution we characterize. We first show that for two sets of typical experimental settings, corresponding either to weak quantum non-demolition measurements or to weak fluorescence measurements, the three Bloch coordinates of the qubit remain confined to a deterministically evolving surface or curve inside the Bloch sphere. We explicitly solve the deterministic evolution, and we provide a closed-form expression for the probability distribution on this surface or curve. Then we relate the existence in general of such deterministically evolving submanifolds to an accessibility question of control theory, which can be answered with an explicit algebraic criterion on the SDE. This allows us to show that, for a qubit, the above two sets of weak measurements are essentially the only ones featuring deterministic surfaces or curves.

Andrzej Łuczak

Christof Wetterich

Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.