Measurements continuous in time and a posteriori states in quantum

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.Comment: 31 page. See also related papers at http://www.maths.nott.ac.uk/personal/vpb/research/mes_fou.html and http://www.maths.nott.ac.uk/personal/vpb/research/fil_con.htm

Feedback control of the fluorescence light squeezing

We consider a two-level atom stimulated by a coherent monochromatic laser and we study how to enhance the squeezing of the fluorescence light and of the atom itself in the presence of a Wiseman-Milburn feedback mechanism, based on the homodyne detection of a fraction of the emitted light. Besides analyzing the effect of the control parameters on the squeezing properties of the light and of the atom, we also discuss the relations among these. The problem is tackled inside the framework of quantum trajectory theory.Comment: RevTeX4, 4 pages, 2 figure

Description of isolated macroscopic systems inside quantum mechanics

For an isolated macrosystem classical state parameters $\zeta(t)$ are introduced inside a quantum mechanical treatment. By a suitable mathematical representation of the actual preparation procedure in the time interval $[T,t_0]$ a statistical operator is constructed as a solution of the Liouville von Neumann equation, exhibiting at time $t$ the state parameters $\zeta(t')$, $t_0\leq t' \leq t$, and {\it preparation parameters} related to times $T \leq t'\leq t_0$. Relation with Zubarev's non-equilibrium statistical operator is discussed. A mechanism for memory loss is investigated and time evolution by a semigroup is obtained for a restricted set of relevant observables, slowly varying on a suitable time scale.Comment: 13 pages, latex, romp31 style, no figures, to appear in the Proceedings of the XXXI Symposium on Mathematical Physics (Torun, Poland), to be published in Rep. Math. Phy

Instruments and channels in quantum information theory

While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the typical inequalities for the quantum and classical relative entropies, many bounds on the classical information extracted in a quantum measurement, of the type of Holevo's bound, are obtained in a unified manner.Comment: 12 pages, revtex

Translation-covariant Markovian master equation for a test particle in a quantum fluid

A recently proposed master equation in the Lindblad form is studied with respect to covariance properties and existence of a stationary solution. The master equation describes the interaction of a test particle with a quantum fluid, the so-called Rayleigh gas, and is characterized by the appearance of a two-point correlation function known as dynamic structure factor, which reflects symmetry and statistical mechanics properties of the fluid. In the case of a free gas all relevant physical parameters, such as fugacity, ratio between the masses, momentum transfer and energy transfer are put into evidence, giving an exact expansion of the dynamic structure factor. The limit in which these quantities are small is then considered. In particular in the Brownian limit a Fokker-Planck equation is obtained in which the corrections due to quantum statistics can be explicitly evaluated and are given in terms of the Bose function $g_0 (z)$ and the Fermi function $f_0 (z)$.Comment: 18 pages, revtex, no figures, to appear in J. Math. Phy

Physics of a microsystem starting from non-equilibrium quantum statistical mechanics

In this paper we address the problem to give a concrete support to the idea, originally stemming from Niels Bohr, that quantum mechanics must be rooted inside the physics of macroscopic systems. It is shown that, starting from the formalism of the non-equilibrium statistical operator, which is now a consolidated part of quantum statistical mechanics, particular correlations between two isolated systems can be singled out and interpreted as microsystems. In this way also a new framework is established in which questions of decoherence can be naturally addressed.Comment: 14 pages, latex, no figures, contribution to the Proceedings of the XXXIII Symposium on Mathematical Physics (Torun, Poland

Linear stochastic wave-equations for continuously measured quantum systems

While the linearity of the Schr\"odinger equation and the superposition principle are fundamental to quantum mechanics, so are the backaction of measurements and the resulting nonlinearity. It is remarkable, therefore, that the wave-equation of systems in continuous interaction with some reservoir, which may be a measuring device, can be cast into a linear form, even after the degrees of freedom of the reservoir have been eliminated. The superposition principle still holds for the stochastic wave-function of the observed system, and exact analytical solutions are possible in sufficiently simple cases. We discuss here the coupling to Markovian reservoirs appropriate for homodyne, heterodyne, and photon counting measurements. For these we present a derivation of the linear stochastic wave-equation from first principles and analyze its physical content.Comment: 34 pages, Revte

Maximal Commutative Subalgebras Invariant for CP-Maps: (Counter-)Examples

We solve, mainly by counterexamples, many natural questions regarding maximal commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a von Neumann algebra. In particular, we discuss the structure of the generators of norm continuous semigroups on B(G) leaving a maximal commutative subalgebra invariant and show that there exists Markov CP-semigroups on M_d without invariant maximal commutative subalgebras for any d>2.Comment: After the elemenitation in Version 2 of a false class of examples in Version 1, we now provide also correct examples for unital CP-maps and Markov semigroups on M_d for d>2 without invariant masa

Unraveling quantum dissipation in the frequency domain

We present a quantum Monte Carlo method for solving the evolution of an open quantum system. In our approach, the density operator evolution is unraveled in the frequency domain. Significant advantages of this approach arise when the frequency of each dissipative event conveys information about the state of the system.Comment: 4 pages, 4 Postscript figures, uses RevTe

Quantum optical versus quantum Brownian motion master-equation in terms of covariance and equilibrium properties

Structures of quantum Fokker-Planck equations are characterized with respect to the properties of complete positivity, covariance under symmetry transformations and satisfaction of equipartition, referring to recent mathematical work on structures of unbounded generators of covariant quantum dynamical semigroups. In particular the quantum optical master-equation and the quantum Brownian motion master-equation are shown to be associated to $\mathrm{U(1)}$ and $\mathrm{R}$ symmetry respectively. Considering the motion of a Brownian particle, where the expression of the quantum Fokker-Planck equation is not completely fixed by the aforementioned requirements, a recently introduced microphysical kinetic model is briefly recalled, where a quantum generalization of the linear Boltzmann equation in the small energy and momentum transfer limit straightforwardly leads to quantum Brownian motion.Comment: 11 pages, latex, no figures, slight changes and a few references added, to appear in J. Math. Phy