M. Gregoratti - Academia.edu (original) (raw)
Papers by M. Gregoratti
It is often believed that de Finetti's coherence principle naturally leads, in the finite case, t... more It is often believed that de Finetti's coherence principle naturally leads, in the finite case, to the Kolmogorov's probability theory of random phenomena, which then implies Bell's inequality. Thus, not only a violation of Bell's inequality looks paradoxical in the Kolmogorovian framework, but it should violate also de Finetti's coherence principle. Firstly, we show that this is not the case: the typical theoretical violations of Bell's inequality in quantum physics are in agreement with de Finetti's coherence principle. Secondly, we look for statistical evidence of such violations: we consider the experimental data of measurements of polarization of photons, performed to verify empirically violations of Bell's inequality, and, on the basis of the estimated violation, we test the null hypothesis of Kolmogorovianity for the observed phenomenon. By standard inferential techniques we compute the pvalue for the test and get a clear strong conclusion against the Kolmogorovian hypothesis.
Page 74. 63 QUANTUM CONTINUOUS MEASUREMENTS: THE SPECTRUM OF THE OUTPUT A. BARCHIELLI and M. GREG... more Page 74. 63 QUANTUM CONTINUOUS MEASUREMENTS: THE SPECTRUM OF THE OUTPUT A. BARCHIELLI and M. GREGORATTI Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci, I-20133 Milano, Italy. ...
Lecture Notes in Physics, 2009
ABSTRACT In this chapter, we introduce the theory of measurements in continuous time (diffusive c... more ABSTRACT 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 if this is a special case of the more general theory presented in Chap. 3, 4 and 5, it deserves a separate treatment for different reasons: it is instructive, it uses only the Hilbert space formulation of quantum mechanics, it is of interest on its own because the stochastic Scrödinger equation presented in this chapter has also been used in different contexts [1–6], some mathematical results of the following chapter will relay anyhow on the theory presented here, and Hilbert space SDEs are the key starting point for efficient numerical simulations of the dynamics of open quantum systems [1, 7].
Lecture Notes in Physics, 2009
Lecture Notes in Physics, 2009
ABSTRACT A satisfactory theory of continuous measurements has to be developed according to the ax... more ABSTRACT 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 developed in Chap. 2 and it extends the results to the case of incomplete measurements. Now, the key notions are “statistical operator”, “stochastic master equation”, “master equation” and “quantum dynamical (or Markov) semigroup”.
Lecture Notes in Physics, 2009
Lecture Notes in Physics, 2009
Up to here we have seen the abstract theory, which involves a certain number of operators on H. T... more Up to here we have seen the abstract theory, which involves a certain number of operators on H. The choice of H and of these operators fixes the physical model. This section provides some hints on how to do this choice.
Lecture Notes in Physics, 2009
We already saw that there exist peculiar cases (Sects. 2.4.4, 2.5.2.1) in which no information on... more 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 classical and quantum information theory is that this can be obtained by means of entropy-like quantities [1–3].
Lecture Notes in Physics, 2009
Lecture Notes in Physics, 2009
Lecture Notes in Physics, 2009
Quantum Measurements and Quantum Metrology, 2013
The stochastic Schrödinger equation, of classical or quantum type, allows to describe open quantu... more 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 commuting quantum observables. 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. Moreover, the relation with quantum filtering theory is shortly discussed. 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 of the homodyne spectrum and the Mollow triplet in the fluorescence spectrum.
Quantum Probability and Related Topics, 2013
Entanglement between two quantum systems is a resource in quantum information, but dissipation us... more 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.
Theory of Probability & Its Applications, 2010
Stochastic Analysis and Applications, 2008
... If a QDS is Accepted December 17, 2007 Address correspondence to M. Gregoratti, Dipartimento ... more ... If a QDS is Accepted December 17, 2007 Address correspondence to M. Gregoratti, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy; E-mail: matteo.gregoratti@polimi.it Page 2. 1026 Gregoratti ...
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012
In this article, we reconsider a version of quantum trajectory theory based on the stochastic Sch... more 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 feedback. Indeed, realistic descriptions of a feedback loop have to include delay and thus need a non-Markovian theory. The theory allows us 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 of enhancing 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...
Mathematical Methods in the Applied Sciences, 2006
... Correspondence to: M. Gregoratti, Dipartimento di Matematica 'F.Brioschi&... more ... Correspondence to: M. Gregoratti, Dipartimento di Matematica 'F.Brioschi', Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy. E-mail: matteo.gregoratti@polimi.it Copyright ? 2005 John Wiley & Sons, Ltd. Received 15 December 2004 Page 2. ...
It is often believed that de Finetti's coherence principle naturally leads, in the finite case, t... more It is often believed that de Finetti's coherence principle naturally leads, in the finite case, to the Kolmogorov's probability theory of random phenomena, which then implies Bell's inequality. Thus, not only a violation of Bell's inequality looks paradoxical in the Kolmogorovian framework, but it should violate also de Finetti's coherence principle. Firstly, we show that this is not the case: the typical theoretical violations of Bell's inequality in quantum physics are in agreement with de Finetti's coherence principle. Secondly, we look for statistical evidence of such violations: we consider the experimental data of measurements of polarization of photons, performed to verify empirically violations of Bell's inequality, and, on the basis of the estimated violation, we test the null hypothesis of Kolmogorovianity for the observed phenomenon. By standard inferential techniques we compute the pvalue for the test and get a clear strong conclusion against the Kolmogorovian hypothesis.
Page 74. 63 QUANTUM CONTINUOUS MEASUREMENTS: THE SPECTRUM OF THE OUTPUT A. BARCHIELLI and M. GREG... more Page 74. 63 QUANTUM CONTINUOUS MEASUREMENTS: THE SPECTRUM OF THE OUTPUT A. BARCHIELLI and M. GREGORATTI Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci, I-20133 Milano, Italy. ...
Lecture Notes in Physics, 2009
ABSTRACT In this chapter, we introduce the theory of measurements in continuous time (diffusive c... more ABSTRACT 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 if this is a special case of the more general theory presented in Chap. 3, 4 and 5, it deserves a separate treatment for different reasons: it is instructive, it uses only the Hilbert space formulation of quantum mechanics, it is of interest on its own because the stochastic Scrödinger equation presented in this chapter has also been used in different contexts [1–6], some mathematical results of the following chapter will relay anyhow on the theory presented here, and Hilbert space SDEs are the key starting point for efficient numerical simulations of the dynamics of open quantum systems [1, 7].
Lecture Notes in Physics, 2009
Lecture Notes in Physics, 2009
ABSTRACT A satisfactory theory of continuous measurements has to be developed according to the ax... more ABSTRACT 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 developed in Chap. 2 and it extends the results to the case of incomplete measurements. Now, the key notions are “statistical operator”, “stochastic master equation”, “master equation” and “quantum dynamical (or Markov) semigroup”.
Lecture Notes in Physics, 2009
Lecture Notes in Physics, 2009
Up to here we have seen the abstract theory, which involves a certain number of operators on H. T... more Up to here we have seen the abstract theory, which involves a certain number of operators on H. The choice of H and of these operators fixes the physical model. This section provides some hints on how to do this choice.
Lecture Notes in Physics, 2009
We already saw that there exist peculiar cases (Sects. 2.4.4, 2.5.2.1) in which no information on... more 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 classical and quantum information theory is that this can be obtained by means of entropy-like quantities [1–3].
Lecture Notes in Physics, 2009
Lecture Notes in Physics, 2009
Lecture Notes in Physics, 2009
Quantum Measurements and Quantum Metrology, 2013
The stochastic Schrödinger equation, of classical or quantum type, allows to describe open quantu... more 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 commuting quantum observables. 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. Moreover, the relation with quantum filtering theory is shortly discussed. 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 of the homodyne spectrum and the Mollow triplet in the fluorescence spectrum.
Quantum Probability and Related Topics, 2013
Entanglement between two quantum systems is a resource in quantum information, but dissipation us... more 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.
Theory of Probability & Its Applications, 2010
Stochastic Analysis and Applications, 2008
... If a QDS is Accepted December 17, 2007 Address correspondence to M. Gregoratti, Dipartimento ... more ... If a QDS is Accepted December 17, 2007 Address correspondence to M. Gregoratti, Dipartimento di Matematica F. Brioschi, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy; E-mail: matteo.gregoratti@polimi.it Page 2. 1026 Gregoratti ...
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012
In this article, we reconsider a version of quantum trajectory theory based on the stochastic Sch... more 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 feedback. Indeed, realistic descriptions of a feedback loop have to include delay and thus need a non-Markovian theory. The theory allows us 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 of enhancing 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...
Mathematical Methods in the Applied Sciences, 2006
... Correspondence to: M. Gregoratti, Dipartimento di Matematica 'F.Brioschi&... more ... Correspondence to: M. Gregoratti, Dipartimento di Matematica 'F.Brioschi', Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy. E-mail: matteo.gregoratti@polimi.it Copyright ? 2005 John Wiley & Sons, Ltd. Received 15 December 2004 Page 2. ...