Quantum continual measurements and a posteriori collapse on CCR

Abstract

Abstract. A quantum theory for the Markovian dynamics of an open system under the unsharp observation which is continuous in time, is developed within the CCR stochastic approach. A stochastic classical equation for the posterior evolution of quantum continuously observed system is derived and the spontaneous collapse (stochastically continuous reduction of the wave packet) is described. The quantum Langevin evolution equation is solved for the general linear case of a quasi-free Hamiltonian in the initial CCR algebra with a fixed output observable field, and the posterior Kalman dynamics coresponding to an initial Gaussian state is found. It is shown for an example of the posterior dynamics of quantum unstable open system that any mixed state under a complete nondemolition measurement collapses exponentially to a pure Gaussian one.

Principles and applications of quantum control engineering

“…The general theory of nonlinear filtering was subsequently developed in a time continuous setting by Stratonovich, Kallianpur, Striebel, Zakai and others. 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].…”

Stochastic Differential Equations for Trace-Class Operators and Quantum Continual Measurements

“…Let M be a d-dimensional C ∞ real compact manifold and let B 0 , B 1 be C ∞ vector fields; we consider the diffusion process defined by a Stratonovich SDE of type (19) with m = 1. If there exists a smooth hypersurface Γ in M such that the field A is transversal to Γ and the equationẊ t = B 1 (X t ) is a contraction outside Γ (i.e.…”

General-dyne unravelling of a thermal master equation

“…In the context of the coherent control of quantum continuous variables, consistently pursued, over the last thirty years, since early works by Belavkin [7][8][9], the class of general-dyne measurements stands out as it is associated to all diffusive unravellings of the dynamics, i.e. to all the unravellings that can be treated as multivariate quantum Wiener processes [10][11][12].…”