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Alberto Barchielli

Publications and source records attributed to Alberto Barchielli.

At least 19 recordsLinked to original sources
arXiv2026

Quantum jump trajectories, hybrid systems, non-Hermitian evolutions, quantum/classical walks

Quantum stochastic master equations of jump type are formulated in a general way and connections with quantum/classical hybrid systems and quantum filtering theory are discussed. By introducing the notion of ``typical trajectory", we show how to recursively construct the solution of the non-linear stochastic master equation (the conditional state). Moreover, by the notion of ``exclusive probability densities" we can describe all the probabilities related to the jumps, in particular, the waiting times of the jumps and their probability distributions. This general formulation and the idea of hybrid system allow to unify and generalize different fields: evolutions under non-Hermitian Hamiltonians, unitary dynamics interspersed by quantum channels at random times, quantum renewal processes, continuous time open quantum walks, Lindblad rate equation, ...

quant-ph↗
arXiv2024

Markovian dynamics for a quantum/classical system and quantum trajectories

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. Here we extend this technique and use it to develop a general approach to the dynamics of quantum/classical hybrid systems. By using two coupled stochastic differential equations, we can describe a classical component and a quantum one which have their own intrinsic dynamics and which interact with each other. A mathematically rigorous construction is given, under the restriction of having a Markovian joint dynamics and of involving only bounded operators on the Hilbert space of the quantum component. An important feature is that, if the interaction allows for a flow of information from the quantum component to the classical one, necessarily the dynamics is dissipative. We show also how this theory is connected to a suitable hybrid dynamical semigroup, which reduces to a quantum dynamical semigroup in the purely quantum case and includes Liouville and Kolmogorov-Fokker-Plank equations in the purely classical case. Moreover, this semigroup allows to compare the proposed stochastic dynamics with various other proposals based on hybrid master equations. Some simple examples are constructed in order to show the variety of physical behaviours which can be described; in particular, a model presenting hidden entanglement is introduced.

quant-ph↗
arXiv2023

Markovian master equations for quantum-classical hybrid systems

The problem of constructing a consistent quantum-classical hybrid dynamics is afforded in the case of a quantum component in a separable Hilbert space and a continuous, finite-dimensional classical component. In the Markovian case, the problem is formalized by the notion of hybrid dynamical semigroup. A classical component can be observed without perturbing the system and information on the quantum component can be extracted, thanks to the quantum-classical interaction. This point is formalized by showing how to introduce positive operator valued measures and operations compatible with the hybrid dynamical semigroup; in this way the notion of hybrid dynamics is connected to quantum measurements in continuous time. Then, the case of the most general quasi-free generator is presented and the various quantum-classical interaction terms are discussed. To bee quasi-free means to send, in the Heisenberg description, hybrid Weyl operators into multiples of Weyl operators; the results on the structure of quasi-free semigroups were proved in the article arXiv:2307.02611. Even in the pure quantum case, a quasi-free semigroup is not restricted to have only a Gaussian structure, but also jump-type terms are allowed. An important result is that, to have interactions producing a flow of information from the quantum component to the classical one, suitable dissipative terms must be present in the generator. Finally, some possibilities are discussed to go beyond the quasi-free case.

quant-ph↗
arXiv2023

Hybrid quantum-classical systems: Quasi-free Markovian dynamics

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 operators in the Heisenberg description. The result is a quantum generalization of the L\'evy-Khintchine formula; Gaussian and jump contributions are included. As a byproduct, the most general quasi-free quantum-dynamical semigroup is obtained; on the classical side the Liouville equation and the Kolmogorov-Fokker-Planck equation are included. As a classical subsystem can be observed, in principle, without perturbing it, information can be extracted from the quantum system, even in continuous time; indeed, the whole construction is related to the theory of quantum measurements in continuous time. While the dynamics is formulated to give the hybrid state at a generic time $t$, we show how to extract multi-time probabilities and how to connect them to the quantum notions of positive operator valued measure and instrument. The structure of the generator of the dynamical semigroup is analized, in order to understand how to go on to non quasi-free cases and to understand the possible classical-quantum interactions; in particular, all the interaction terms which allow to extract information from the quantum system necessarily vanish if no dissipation is present in the dynamics of the quantum component. A concrete example is given, showing how a classical component can input noise into a quantum one and how the classical system can extract information on the behaviour of the quantum one.

quant-ph↗
arXiv2022

The eight-port homodyne detector: the effect of imperfections on quantum random number generation and on detection of quadratures

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 travelling waves in the optical circuit, either the observables involved in the detection part: two couples of photodiodes, postprocessing of the output currents. The analysis includes imperfections, such as not perfectly balanced beam splitters, detector efficiency, electronic noise, phase and intensity noise in the laser acting as local oscillator; this last noise is modelled by using mixtures of field coherent states as statistical operator of the laser component. Due to the monitoring in continuous time, the output is a stochastic process and its full probability distribution is obtained. When the output process is sampled at discrete times, the quantum description can be reduced to discrete mode operators, but at the price of having random operators, which contain also the noise of the local oscillator. Consequently, the local oscillator noise has a very different effect on the detection results with respect to an additive noise, such as the noise in the electronic components. As an application, the problem of secure random number generation is considered, based on the local oscillator shot noise. The rate of random bits that can be generated is quantified by the min-entropy; the possibility of classical and quantum side information is taken into account by suitable conditional min-entropies. The final rate depends on which parts of the apparatus are considered to be secure and on which ones are considered to be exposed to the intervention of an intruder. In some experimentally realistic situations, the entropy losses are computed.

quant-ph↗
arXiv2021

Quantum optomechanical system in a Mach-Zehnder interferometer

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. This configuration shows that squeezed light can be generated by pure scattering on a quantum system, without involving a cavity. The squeezing can be detected at the output ports of the interferometer either by direct detection or by measuring the spectrum of the difference current. We use the Hudson-Parthasarathy equation to model the global evolution. It can describe the scattering of photons and the resulting radiation pressure interaction on the quantum oscillator. It allows to consider also the interaction with a thermal bath. In this way we have a unitary dynamics giving the evolution of oscillator and fields. The Bose fields of quantum stochastic calculus and the related generalized Weyl operators allow to describe the whole optical circuit. By working in the Heisenberg picture, the quantum Langevin equations for position and momentum and the output fields arise, which are used to describe the monitoring in continuous time of the light at the output ports. In the case of strong laser and weak radiation pressure interaction highly non-classical light is produced, and this can be revealed either by direct detection (a negative Mandel Q-parameter is found), either by the intensity spectrum of the difference current of two photodetector; in the second case a nearly complete cancellation of the shot noise can be reached. In this last case it appears that the Mach-Zehnder configuration together with the detection of the difference current corresponds to an homodyne detection scheme, so that we can say that the apparatus is measuring the spectrum of squeezing.

quant-ph↗
arXiv2020

Entropic measurement uncertainty relations for all the infinite components of a spin vector

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 components, we define the device information loss as the maximum loss of information per observable occurring in approximating the ideal incompatible components with the joint measurement at hand. By optimizing on the measuring device, we define the notion of minimum information loss. By using these notions, we show how to give a significant formulation of state independent MURs in the case of infinitely many target observables. The same construction works as well for finitely many observables, and we study the related MURs for two and three orthogonal spin components. The minimum information loss plays also the role of measure of incompatibility and in this respect it allows us to compare quantitatively the incompatibility of various sets of spin observables, with different number of involved components and different values of s.

quant-ph↗
arXiv2018

Uncertainty relations and information loss for spin-1/2 measurements

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 covariant approximate joint measurements and we find state-dependent MURs for two or three orthogonal spin-1/2 components. Secondly we consider any possible approximate joint measurement and we find state-independent MURs for two or three spin-1/2 components. In particular we study how MURs depend on the angle between two spin directions. Finally, we extend our approach to infinitely many incompatible observables, namely to the spin components in all possible directions. In every scenario, we always consider also the characterization of the optimal approximate joint measurements.

quant-ph↗
arXiv2018

Measurement uncertainty relations for discrete observables: Relative entropy formulation

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 loss of information occurring in the approximation at hand. For fixed target observables, we study the joint measurements minimizing the entropic divergence, and we prove the general properties of its minimum value. Such a minimum is our uncertainty lower bound: the total information lost by replacing the target observables with their optimal approximations, evaluated at the worst possible state. The bound turns out to be also an entropic incompatibility degree, that is, a good information-theoretic measure of incompatibility: indeed, it vanishes if and only if the target observables are compatible, it is state-independent, and it enjoys all the invariance properties which are desirable for such a measure. In this context, we point out the difference between general approximate joint measurements and sequential approximate joint measurements; to do this, we introduce a separate index for the tradeoff between the error of the first measurement and the disturbance of the second one. By exploiting the symmetry properties of the target observables, exact values, lower bounds and optimal approximations are evaluated in two different concrete examples: (1) a couple of spin-1/2 components (not necessarily orthogonal); (2) two Fourier conjugate mutually unbiased bases in prime power dimension. Finally, the entropic incompatibility degree straightforwardly generalizes to the case of many observables, still maintaining all its relevant properties; we explicitly compute it for three orthogonal spin-1/2 components.

math-ph↗
arXiv2017

Measurement uncertainty relations for position and momentum: Relative entropy formulation

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 when a `true' probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) $n$-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension $n$; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements.

quant-ph↗
arXiv2015

Quantum stochastic equations for an opto-mechanical oscillator with radiation pressure interaction and non-Markovian effects

The quantum stochastic Schroedinger equation or Hudson-Parthasareathy (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 absorption and emission of quanta, but also scattering processes, which however had very few applications in physical modelling. Moreover, recent developments have shown that also some non-Markovian dynamics can be generated by suitable choices of the state of the quantum noises involved in the HP-equation. This paper is devoted to an application involving these two features, non-Markovianity and scattering process. We consider a micro-mirror mounted on a vibrating structure and reflecting a laser beam, a process giving rise to a radiation-pressure force on the mirror. We show that this process needs the scattering part of the HP-equation to be described. On the other side, non-Markovianity is introduced by the dissipation due to the interaction with some thermal environment which we represent by a phonon field, with a nearly arbitrary excitation spectrum, and by the introduction of phase noise in the laser beam. Finally, we study the full power spectrum of the reflected light and we show how the laser beam can be used as a temperature probe.

quant-ph↗
arXiv2015

Quantum Langevin equations for optomechanical systems

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 suitable non-Fock representation which allows for the introduction of temperature. A generic spectral density of the environment can be described by introducing its state trough a suitable P-representation. Including interaction of the mechanical oscillator with a cavity mode via radiation pressure we obtain a description of a simple optomechanical system in which, besides the Langevin equations for the system, one has the exact input-output relations for the quantum noises. The whole theory is valid at arbitrarily low temperature. This allows the exact calculation of the stationary value of the mean energy of the mechanical oscillator, as well as both homodyne and heterodyne spectra. The present analysis allows in particular to study possible cooling scenarios and to obtain the exact connection between observed spectra and fluctuation spectra of the position of the mechanical oscillator.

quant-ph↗
arXiv2013

Quantum continuous measurements: The stochastic Schroedinger equations and the spectrum of the output

The stochastic Schrödinger 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 quantum stochastic Schrödinger equation, which is based on quantum stochastic calculus, and we show how to transform it into the classical stochastic Schrödinger equation by diagonalization of suitable quantum observables, based on the isomorphism between Fock space and Wiener space. Then, we give the a posteriori state, the conditional system state at time $t$ given the output up to that time and we link its evolution to the classical stochastic Schrödinger equation. Finally, we study the output of the continuous measurement, which is a stochastic process with probability distribution given by the rules of quantum mechanics. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. In particular we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible spectra and we discuss how this is related to the typical quantum phenomenon of squeezing. We use a simple quantum system, a two-level atom stimulated by a laser, to discuss the differences between homodyne and heterodyne detection and to explicitly show squeezing and anti-squeezing and the Mollow triplet in the fluorescence spectrum.

quant-ph↗
arXiv2012

Quantum measurements in continuous time, non Markovian evolutions and feedback

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 '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 feedback. Indeed, realistic descriptions of a feedback loop have to include delay and thus need a non Markovian theory. The theory allows to put together non Markovian evolutions and measurements in continuous time in agreement with the modern axiomatic formulation of quantum mechanics. To illustrate the possibilities of such a theory, we apply it to a two-level atom stimulated by a laser. We introduce closed loop control too, via the stimulating laser, with the aim to enhance the "squeezing" of the emitted light, or other typical quantum properties. Note that here we change the point of view with respect to the usual applications of control theory. In our model the "system" is the two-level atom, but we do not want to control its state, to bring the atom to a final target state. Our aim is to control the "Mandel $Q$-parameter" and the spectrum of the emitted light; in particular the spectrum is not a property at a single time, but involves a long interval of times (a Fourier transform of the autocorrelation function of the observed output is needed).

quant-ph↗
arXiv2012

Entanglement protection and generation under continuous monitoring

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 and we show that, even in cases where the open system dynamics destroys any initial entanglement, the mere monitoring of the environment can preserve or create the entanglement, by filtering the state of the qubits. While the systems we study are very simple, we can show examples with entanglement protection or entanglement birth, death, rebirth due to monitoring.

quant-ph↗
arXiv2010

Quantum stochastic differential equations and continuous measurements: unbounded coefficients

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 unbounded coefficients, we are able to extend the theory of quantum continuous measurements to cases in which unbounded operators on the system space are involved. A significant example of a quantum optical system (the degenerate parametric oscillator) is shown to fulfill the hypotheses introduced in the general theory.

math.PR↗
arXiv2009

Feedback control of the squeezing of the fluorescence light

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 particular, we show how fast feedback à la Wiseman and Milburn can be introduced in the formulation of the theory. Then, the spectrum of the free fluorescence light is studied and typical quantum phenomena, squeezing and sub-natural line-narrowing, are presented.

quant-ph↗
arXiv2008

Quantum continual measurements: the spectrum of the output

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 continual measurement for quantum optical systems is the so called homodyne detection. In this paper we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible homodyne spectra and we discuss how this is related to the typical quantum phenomenon of squeezing.

quant-ph↗