Alberto Barchielli

• Professor
• Full professor (retired) at Politecnico di Milano

Quantum trajectory techniques have been used in the theory of open systems as a starting point for numerical computations and to describe the monitoring of a quantum system in continuous time. We extend this technique to develop a general approach to the dynamics of quantum/classical hybrid systems. By using two coupled stochastic differential equa...

A. Barchielli, Markovian dynamics for a quantum/classical system and quantum trajectories, arXiv:2403.16065 [quant-ph] (2024). A. Barchielli, R.F. Werner, Hybrid quantum-classical systems: Quasi-free Markovian dynamics, Int. J. Quantum Inf. (2024); arXiv:2307.02611 [quant-ph] (2023) A. Barchielli, Markovian master equations for quantum-classical hy...

In the case of a quantum-classical hybrid system with a finite number of degrees of freedom, the problem of characterizing the most general dynamical semigroup is solved, under the restriction of being quasi-free. This is a generalization of a Gaussian dynamics, and it is defined by the property of sending (hybrid) Weyl operators into Weyl operator...

In the case of a quantum-classical hybrid system with a finite number of degrees of freedom, the problem of characterizing the most general dynamical semigroup is solved, under the restriction of being "quasi-free". This is a generalization of a Gaussian dynamics, and it is defined by the property of sending (hybrid) Weyl operators into Weyl operat...

The eight-port homodyne detector is an optical circuit designed to perform the monitoring of two quadratures of an optical field: the signal. By using quantum Bose fields and quantum stochastic calculus, we give a complete quantum description of this apparatus, when used as quadrature detector in continuous time. We can treat either the traveling w...

Abstract. The eight-port homodyne detector is an optical circuit designed to perform the monitoring of two quadratures of an optical field, the signal. By using quantum Bose fields and quantum stochastic calculus, we give a complete quantum description of this apparatus, when used as quadrature detector in continuous time. We can treat either the...

The eight-port homodyne detector is an optical circuit designed to perform the monitoring of two quadratures of an optical field, the signal. By using quantum Bose fields and quantum stochastic calculus, we give a complete quantum description of this apparatus, when used as quadrature detector in continuous time. The analysis includes imperfections...

We consider an oscillating micromirror replacing one of the two fixed mirrors of a Mach-Zehnder interferometer. In this ideal optical set-up the quantum oscillator is subjected to the radiation pressure interaction of traveling light waves, no cavity is involved. The aim of this configuration is to show that squeezed light can be generated by pure...

We consider an oscillating micromirror replacing one of the two fixed mirrors of a Mach-Zehnder interferometer. In this ideal optical set-up the quantum oscillator is subjected to the radiation pressure interaction of travelling light waves, no cavity is involved. The aim of this configuration is to show that squeezed light can be generated by pure...

Presentation and slides: http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=28892

The information-theoretic formulation of quantum measurement uncertainty relations (MURs), based on the notion of relative entropy between measurement probabilities, is extended to the set of all the spin components for a generic spin s. For an approximate measurement of a spin vector, which gives approximate joint measurements of the spin componen...

The information-theoretic formulation of quantum measurement uncertainty relations (MURs), based on the notion of relative entropy between measurement probabilities, is extended to the set of all the spin components for a generic spin $s$. For a physical class of approximate joint measurements of the spin components, we define the device informatio...

Optomechanical systems; optical circuits

Talk given at the "International conference on Quantum Information, Statistics, Probability" Moscow, Steklov Mathematical Institute, September 12th 2018.

We formulate entropic measurements uncertainty relations (MURs) for a spin-1/2 system. When incompatible observables are approximatively jointly measured, we use relative entropy to quantify the information lost in approximation and we prove positive lower bounds for such a loss: there is an unavoidable information loss. Firstly we allow only for c...

We introduce a new information-theoretic formulation of quantum measurement uncertainty relations, based on the notion of relative entropy between measurement probabilities. In the case of a finite-dimensional system and for any approximate joint measurement of two target discrete observables, we define the entropic divergence as the maximal total...

Talk given at Meeting on Functional Analysis and Quantum Information Theory 27 November 2017, Université Bourgogne/Franche Compté, Besançon, France

Heisenberg’s uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss wh...

Heisenberg's uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss wh...

The stochastic Schrödinger equation approach to quantum feedback control

Talk given at Institut für Theoretische Physik, ETH, Zurich

We introduce a new information-theoretic formulation of quantum measurement uncertainty relations, based on the notion of relative entropy between measurement probabilities. In the case of a finite-dimensional system and for any approximate joint measurement of two target discrete observables, we define the entropic divergence as the maximal total...

The quantum stochastic Schroedinger equation or Hudson-Parthasarathy (HP) equation is a powerful tool to construct unitary dilations of quantum dynamical semigroups and to develop the theory of measurements in continuous time via the construction of output fields. An important feature of such an equation is that it allows to treat not only absorpti...

Corrections for the book: A. Barchielli, M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time: The Diffusive Case, Lect. Notes Phys. 782 (Springer, Berlin & Heidelberg, 2009). DOI 10.1007/978-3-642-01298-3.

We provide a fully quantum description of a mechanical oscillator in the presence of thermal environmental noise by means of a quantum Langevin formulation based on quantum stochastic calculus. The system dynamics is determined by symmetry requirements and equipartition at equilibrium, while the environment is described by quantum Bose fields in a...

Firstly, the Markovian stochastic Schr\"odinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples - the damped harmonic oscillator and a two-level atom with...

The stochastic Schroedinger equation, of classical or quantum type, allows to describe open quantum systems under measurement in continuous time. In this paper we review the link between these two descriptions and we study the properties of the output of the measurement. For simplicity we deal only with the diffusive case. Firstly, we discuss the q...

Entanglement between two quantum systems is a resource in quantum information, but dissipation usually destroys it. In this article we consider two qubits without direct interaction. We show that, even in cases where the entanglement is destroyed by the open system dynamics, the entanglement can be preserved or created by the mere monitoring of the...

Starting from a generalization of the quantum trajectory theory (based on the stochastic Schr\"odinger equation - SSE), non-Markovian models of quantum dynamics are derived. In order to describe non-Markovian effects, the approach used in this article is based on the introduction of random coefficients in the usual linear SSE. A major interest is t...

In this article, we reconsider a version of quantum trajectory theory based on the stochastic Schrödinger equation with stochastic coefficients, which was mathematically introduced in the 1990s, and we develop it in order to describe the non-Markovian evolution of a quantum system continuously measured and controlled, thanks to a measurement-based...

In this article we reconsider a version of quantum trajectory theory based on the stochastic Schr\"odinger equation with stochastic coefficients, which was mathematically introduced in the '90s, and we develop it in order to describe the non Markovian evolution of a quantum system continuously measured and controlled thanks to a measurement based f...

A natural formulation of the theory of quantum measurements in continuous time is based on quantum stochastic differential equations (Hudson-Parthasarathy equations). However, such a theory was developed only in the case of Hudson-Parthasarathy equations with bounded coefficients. By using some results on Hudson-Parthasarathy equations with unbound...

By starting from the stochastic Schroedinger equation and quantum trajectory theory, we introduce memory effects by considering stochastic adapted coefficients. As an example of a natural non-Markovian extension of the theory of white noise quantum trajectories we use an Ornstein-Uhlenbeck coloured noise as the output driving process. Under certain...

Slides

w The "correlated-projection technique" has been successfully applied to derive a large class of highly non-Markovian dynamics, the so called non-Markovian generalized Lindblad-type equations or Lindblad rate equations. In this article, general unravelings are presented for these equations, described in terms of jump-diffusion stochastic differenti...

We consider a trapped two-level atom stimulated by a coherent monochromatic laser and we study how to enhance the squeezing of the fluorescence light in the presence of a Wiseman-Milburn feedback mechanism, based on the homodyne detection of a fraction of the emitted light. We analyze the effect of the control parameters on the properties of the sp...

A natural non-Markovian extension of the theory of white noise quantum trajectories is presented. In order to introduce memory effects in the formalism an Ornstein-Uhlenbeck coloured noise is considered as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, non-Markovian stochastic Schrodinger...

A natural non-Markovian extension of the theory of white noise quantum trajectories is presented. In order to introduce memory effects in the formalism an Ornstein-Uhlenbeck coloured noise is considered as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, non-Markovian stochastic Schrödinger...

A satisfactory theory of continuous measurements has to be developed according to the axioms of quantum mechanics, that is by introducing, more or less explicitly, the associated instruments (Sect. B.4). This approach requires the statistical formulation of quantum mechanics (see Sect. B.3). This chapter generalises to this framework the theory dev...

Among the formulations of the theory of quantum measurements in continuous time, quantum trajectory theory is very suitable for the introduction of measurement based feedback and closed loop control of quantum systems. In this paper we present such a construction in the concrete case of a two-level atom stimulated by a coherent, monochromatic laser...

In this chapter, we introduce the theory of measurements in continuous time (diffusive case) starting from the particular but important case of complete observation. This allows to present the Hilbert space formulation of the theory, where the state of the observed quantum system is described by a vector in the Hilbert space H of the system. Even i...

In this chapter we study the atomic spectra, a concept which depends on the detection type, heterodyning or homodyning. These spectra give information on the atom, the atom/field interaction and the properties of the emitted light. In particular we shall see the important phenomena of dynamical Stark effect and of squeezing of the fluorescence ligh...

We already saw that there exist peculiar cases (Sects. 2.4.4, 2.5.2.1) in which no information on the quantum system is extracted by the continuous measurement. Obviously, in other cases we get some information on the system, but the question arises of how to quantify the gain in information. The answer coming out from the whole development of clas...

In Chap. 3, starting from the linear formulation of the quantum trajectory theory, it has been shown that the a posteriori states satisfy the nonlinear SDE (3.69). Now we show that this equation can be taken as starting point of the whole theory and we study some of its properties.

In this chapter we complete the task of showing that the SDE approach can be reduced to the usual formulation of quantum mechanics. The last notion which we need is the one of “instrument”; its definition, meaning and properties are presented in Sect. B.4. We give also the important concept of characteristic operator, a kind of Fourier transform of...

A two-level atom is perhaps the simplest quantum system, but in spite of this it has a very rich behaviour. So, it represents an ideal physical system to illustrate the theory presented so far [1]. The fluorescence spectrum of a two-level atom stimulated by an intense monochromatic laser is highly non-trivial; it presents a typical three-peaked str...

This course-based monograph introduces the reader to the theory of continuous measurements in quantum mechanics and provides some benchmark applications. The approach chosen, quantum trajectory theory, is based on the stochastic Schrödinger and master equations, which determine the evolution of the a-posteriori state of a continuously observed quan...

Quantum mechanics started as a theory of closed systems: the state of the system is a vector of norm one in a Hilbert space and it evolves in time according to the Schrödinger equation (B.11). In order to describe also a possible uncertainty on the initial state, a “statistical” formulation of quantum mechanics has been developed: the states are re...

Inspired by works on information transmission through quantum channels, we propose the use of a couple of mutual entropies to quantify the efficiency of continual measurement schemes in extracting information on the measured quantum system. Properties of these measures of information are studied and bounds on them are derived.

When a quantum system is monitored in continuous time, the result of the measurement is a stochastic process. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. A typical continuous measurement for quantum optical systems is the so called homodyne detection. In...

Quantum trajectory theory is the best mathematical set up to model continual observations of a quantum system and feedback based on the observed output. Inside this framework, we study how to enhance the squeezing of the fluorescence light emitted by a two-level atom, stimulated by a coherent monochromatic laser. In the presence of a Wiseman-Milbur...

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 th...

Quantum trajectory theory is the best mathematical set up to model continual observations of a quantum system and feedback based on the observed output. Inside this framework, we study how to enhance the squeezing of the fluorescence light emitted by a two-level atom, stimulated by a coherent monochromatic laser. In the presence of a Wiseman-Milbur...

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 monotonicity theorem for relative entropies many bounds on the classical information extracted in a quantum measuremen...

General quantum measurements are represented by instruments. In this paper the mathematical formalization is given of the idea that an instrument is a channel which accepts a quantum state as input and produces a probability and an a posteriori state as output. Then, by using mutual entropies on von Neumann algebras and the identification of instru...

Some bounds on the entropic informational quantities related to a quantum continual measurement are obtained and the time dependencies of these quantities are studied. Comment: 10 pages

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 extrac...

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 monotonicity theorem for relative entropies many bounds on the classical information extracted in a quantum measuremen...

General quantum measurements are represented by instruments. In this paper the mathematical formalization is given of the idea that an instrument is a channel which accepts a quantum state as input and produces a probability and an a posteriori state as output. Then, by using mutual entropies on von Neumann algebras and the identification of instru...

In this paper we will give a short presentation of the quantum Levy-Khinchin formula and of the formulation of quantum continual measurements based on stochastic differential equations, matters which we had the pleasure to work on in collaboration with Prof. Holevo. Then we will begin the study of various entropies and relative entropies, which see...

In this paper we will give a short presentation of the quantum Lévy-Khinchin formula andof the formulation of quantum continual measurements based on stochastic differentialequations, matters which we had the pleasure to work on in collaboration with Prof.Holevo. Then we will begin the study of various entropies and relative entropies, whichseem to...

In this article we study the long time behaviour of a class of stochastic dierential equations introduced in the theory of measurements continuous in time for quantum open systems. Such equations give the time evolution of the a posteriori states for a system underlying a continual measurement. First of all we give conditions for the equation to pr...

By means of quantum stochastic calculus we construct a model for an atom with two degenerate levels and stimulated by a laser and we compute its fluorescence spectrum; let us stress that, once the model for the unitary atom-field dynamics has been given, then the spectrum is computed without further approximations. If only the absorption/emission t...

The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure, functional integrals, quantum stochastic differential equations and classical stochastic differential equations (SDE's). Various types of SDE's are involved,...

The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument, positive operator valued measure, etc., by using quantum stochastic differential equations and by using classical stochastic differential equations (SDE's) for vectors in Hilbert spaces or for t...

Quantum stochastic differential equations have been used to describe the dynamics of an atom interacting with the electromagnetic field via absorption/emission processes. Here, by using the full quantum stochastic Schroedinger equation proposed by Hudson and Parthasarathy fifteen years ago, we show that such models can be generalized to include oth...

Some “classical” stochastic differential equations have been used in the theory of measurements continuous in time in quantum mechanics and, more generally, in quantum open system theory. In this paper, we introduce and study a class of such equations which allow us to achieve the same level of generality as the one obtained by the approach to cont...

Stochastic differential equations for processes with values in Hilbert spaces are now largely used in the quantum theory of open systems. In this work we present a class of such equations and discuss their main properties; moreover, we explain how they are derived from purely quantum mechanical models, where the dynamics is represented by a unitary...

Just at the beginning of quantum stochastic calculus Hudson and Parthasarathy proposed a quantum stochastic Schrodinger equation linked to dilations of quantum dynamical semigroups. Such an equation has found applications in physics, mainly in quantum optics, but not in its full generality. It has been used to give, at least approximately, the dyna...

By using the theory of measurements continuous in time in quantum mechanics [1][8], a photon detection theory has been formulated [9]– [12]; see Refs. [10]– [12] and [8] for detailed references. A quantum source as an atom, an ion or a more complicated system, eventually placed inside an optical cavity, is stimulated by lasers or by a thermal bath....

In the past few years there has been an increasing interest in a certain class of stochastic differential equations (SDE’s) in Hilbert spaces for applications in quantum mechanics (measurements continuous in time [1-5]) and in quantum optics (photon-detection theory and numerical simulations of master equations [6-10]). Part of the mathematical the...

Within the quantum theory of measurements continuous in time, a photon detection theory was formulated by using quantum stochastic calculus; this is a purely quantum formulation, where the usual notions of quantum mechanics appear: the dynamics is given by unitary operators and the observables are represented by commuting self-adjoint operators. In...

A class of linear stochastic differential equations in Hilbert spaces is studied, which allows to construct probability densities and to generate changes in the probability measure one started with. Related linear equations for trace-class operators are discussed. Moreover, some analogue of filtering theory gives rise to related non-linear stochast...

Stochastic differential equations of jump type are used in the theory of measurements continuous in time in quantum mechanics and have a concrete application in describing direct detection in quantum optics (counting of photons). In the paper the connections are explained among various types of stochastic equations: linear for Hilbert-space unnorma...

In recent years a consistent theory describing measurements continuous in time in quantum mechanics has been developed. The result of such a measurement is atrajectoryfor one or more quantities observed with continuity in time. Applications are connected especially with detection theory in quantum optics. In such a theory of continuous measurements...

The idea of measurements continuous in time in quantum mechanics [7,8] led to the introduction of the notion of convolution semigroup of instruments [9–12] and to the study of dilations of such semigroups obtained by the use of quantum stochastic calculus [13–16]. In [17] we discussed the connections between these notions and that of stochastic pro...

A quantum theory of measurements continuous in time has been developed at various levels of generality and by means of various mathematical tools. Here, only some points of this theory are presented. By using the notion of convolution semigroup of instruments one can give a prescription for extracting from quantum mechanics a consistent set of fini...

In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of, instruments on groups and the associated semigroups of probability operators, which now are defin...

Quantum stochastic calculus (QSC) is a noncommutative analogue of Ito's stochastic calculus. Its usefulness in quantum optics is mainly due to the fact that it is based on the use of certain Bose fields which can be taken as an approximlation of the electromagnetic field...

Nowadays quantum stochastic calculus (QSC) begins to be applied to problems in quantum optics. The explicit introduction of QSC in quantum optics was made by C. W. Gardiner and M. J. Collet [Phys. Rev. A. 31, 3761-6774 (1985)], but the use of the related δ-correlated noise is older [see M. Lax, Phys. Rev. 145, 110-129 (1966)]. It is possible to dis...

In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of instruments on groups and the associated semigroups of probability operators. In this paper the ca...

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 p...

Quantum stochastic calculus is an operator analogue of classical Ito's stochastic calculus which was originally developed for treating quantum noise. However, it turned out to be also useful in other cases of open systems, as in the case of measurement theory in quantum mechanics and in the treatment of quantum input and output channels. In the pre...

The theoretical derivation from QCD of the q[`(q)]q\bar q potential with velocity-dependent corrections is reconsidered. In particular the problem of ordering between momenta and functions of position in the Hamiltonian is discussed and exact relations among the various potentials are derived. Explicit expressions for the long-range behaviour of th...