Quantum stochastic calculus and quantum nonlinear filtering

Abstract

A ⋆-algebraic indefinite structure of quantum stochastic (QS) calculus is introduced and a continuity property of generalized nonadapted QS integrals is proved under the natural integrability conditions in an infinitely dimensional nuclear space. The class of nondemolition output QS processes in quantum open systems is characterized in terms of the QS calculus, and the problem of QS nonlinear filtering with respect to nondemolition continuous measurments is investigated. The stochastic calculus of a posteriori conditional expectations in quantum observed systems is developed and a general quantum filtering stochastic equation for a QS process is derived. An application to the description of the spontaneous collapse of the quantum spin under continuous observation is given.

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].…”

“…which satisfies the nonlinear quantum filtering equation first derived by Belavkin for diffusive and counting observations [28][29][30][31][32].…”

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

“…One motivation for extending the study of (0.1) from quantum diffusions to CP flows was to develop an approach to Belavkin's quantum filtering theory ([Be1]) in which processes of the form (0.4) arise, by viewing these as "inner" CP flows. As stated in [LP1] the broader framework of completely positive flows is required in order to formulate a quantum theory of measure-valued diffusions -quantum diffusions corresponding to stochastic flows of diffeomorphisms.…”

Quantum stochastic semigroups and their generators

“…The quantum filtering theory, which was outlined in [13,14] and developed then since [15], 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 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].…”

“…For simplicity we shall assume that the pre-Hilbert Fréchet space E is separable, E ⊆ ℓ 2 . Then the index • can take any value in {1, 2, ...} and Λ ν µ (t) are indexed with µ ∈ {−, 1, 2, ...}, ν ∈ {+, 1, 2, ...} as the standard time Λ + − (t) = tI, annihilation Λ m − (t), creation Λ + n (t) and exchange-number Λ m n (t) operator integrators with m, n ∈ N. The infinitesimal increments dΛ µ ν (t) = Λ tµ ν (dt) are formally defined by the HP multiplication table [6] and the ⋆ -property [15],…”