Measurements continuous in time and a posteriori states in quantum mechanics

Abstract

Measurements continuous in time were consistently introduced in quantum mechanics and applications worked out, mainly in quantum optics. In this context a quantum filtering theory has been developed giving the reduced state after the measurement when a certain trajectory of the measured observables is registered (the a posteriori states). In this paper a new derivation of filtering equations is presented, in the cases of counting processes and of measurement processes of diffusive type. It is also shown that the equation for the a posteriori dynamics in the diffusive case can be obtained, by a suitable limit, from that one in the counting case. Moreover, the paper is intended to clarify the meaning of the various concepts involved and to discuss the connections among them. As an illustration of the theory, simple models are worked out.

Quantum computing in the presence of spontaneous emission by a combined dynamical decoupling and quantum-error-correction strategy

“…It is then well known that the Markovian quantum master equation can be unravelled into a set of quantum trajectories, consisting of a conditional evolution (governed by a non-Hermitian conditional Hamiltonian H c , defined below), randomly interrupted by quantum jumps (wavefunction collapse) into different observed decay channels [11,12,13,14]. The time evolution conditional to a given set of time-ordered observations is called "a posteriori dynamics" [15], and is not Markovian. The continuous observation can lead to a Zeno-effect type suppression of decoherence, a fact that was exploited in [9], in conjunction with an encoding into a decoherence-free subspace (DFS) [16,17], in order to resist SE.…”

Principles and applications of quantum control engineering

“…This was extended at the end of the 1980s by Belavkin to the quantum conditionally Markov setting in a series of papers [28][29][30][31][32]. For a general discussion on continual measurement of quantum systems, see Barchielli & Gregoratti [33] and Barchielli & Belavkin [34], as well as Barchielli & Gregoratti [35].…”

“…There is an analogue [34] of the Kallianpur-Striebel representation p t (X ) = 6 t (X )/6 t (1), where the corresponding quantum Zakai equation [34] in the diffusive case is…”

Quantum stochastic semigroups and their generators

“…Some particular types of such equations have been considered also in the phenomenological theories of quantum permanent reduction [16,17], continuous measurement collapse [18,19], spontaneous jumps [26,20], diffusions and localizations [21,22]. 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.…”