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Mesures continues en mécanique quantique : quelques résultats et applications

Antoine Tilloy

Published Jun 24, 2016• 1 citation
OpenAlex

Abstract

This thesis is devoted to the study of the quantum trajectories obtained from the theory of continuous measurement in non relativistic quantum mechanics. New theoretical results as well as examples of applications are presented. On the theoretical front, we study mostly the limit of "strong" measurement where we put forward the emergence of quantum jumps and quantum spikes, two phenomena we characterize in detail. Out of the strong measurement limit, we investigate a method to extract information from a register of qubits optimally. On the application front, we introduce an original method to control quantum systems exploiting only the freedom of changing the measurement intensity and we explain the transition between a ballistic and a diffusive behavior in open quantum random walks; two byproduct of the theoretical study of the strong measurement regime. We further study the problem of semi-classical gravity and show that continuous measurement theory allows to construct a consistent model in the Newtonian regime. We eventually suggest possible extensions of the formalism to a posteriori estimation and hint at generalizations of the results for the strong measurement limit in the wider context of discrete repeated measurements. In the course of our presentation, we emphasize the link with other approaches to the theory of continuous measurement (parallels with stochastic filtering and collapse models in foundations) rather than aim for mathematical rigor.

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This thesis is devoted to the study of the quantum trajectories obtained from the theory of continuous measurement in non relativistic quantum mechanics. New theoretical results as well as examples of applications are presented. On the theoretical front, we study mostly the limit of "strong" measurement where we put forward the emergence of quantum jumps and quantum spikes, two phenomena we characterize in detail. Out of the strong measurement limit, we investigate a method to extract information from a register of qubits optimally. On the application front, we introduce an original method to control quantum systems exploiting only the freedom of changing the measurement intensity and we explain the transition between a ballistic and a diffusive behavior in open quantum random walks; two byproduct of the theoretical study of the strong measurement regime. We further study the problem of semi-classical gravity and show that continuous measurement theory allows to construct a consistent model in the Newtonian regime. We eventually suggest possible extensions of the formalism to a posteriori estimation and hint at generalizations of the results for the strong measurement limit in the wider context of discrete repeated measurements. In the course of our presentation, we emphasize the link with other approaches to the theory of continuous measurement (parallels with stochastic filtering and collapse models in foundations) rather than aim for mathematical rigor.
Cette thèse est consacrée à l’étude des trajectoires quantiques issues de la théorie desmesures continues en mécanique quantique non relativiste. On y présente de nouveaux résultatsthéoriques ainsi que des exemples d’applications. Sur le front théorique, on étudie principalementla limite de mesure «forte» dans laquelle on met en évidence l’émergence de sauts quantiques etd’échardes quantiques, deux phénomènes dont on précise la statistique. Hors de la limite forte, onpropose une méthode d’extraction optimale d’information pour un registre de qubits. Sur le frontdes applications, on introduit une méthode originale de contrôle utilisant l’intensité de la mesurecomme unique variable et on explique la transition balistique-diffusif dans les marches aléatoiresquantiques ouvertes; deux sous produits de l’étude théorique préalable des situations de mesureforte. On s’intéresse aussi au problème de la gravité semi-classique et montre que la théorie desmesures continues peut permettre d’en construire un modèle cohérent à la limite newtonienne. Onsuggère enfin quelques extensions possibles de la théorie à l’estimation a posteriori et d’éventuellesgénéralisations des résultats théoriques à des situations de mesures répétées discrètes. Dans laprésentation des résultats, l’accent est mis davantage sur l’explicitation des liens entre les multiplespoints de vue possibles sur les trajectoires quantiques (parallèles avec la théorie classique du filtrageet les modèles de collapse objectif utilisés dans les fondements) que sur la rigueur mathématique.
Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement which excludes in principle the singular direct observability continual case. Quantum theory of time continuous measurements and quantum prediction theory, developed by the author on the basis of an independent-increment model for quantum noise and nondemolition causality principle in the 80's, solves this problem allowing continual quantum predictions and reducing many quantum information problems like problems of quantum feedback control to the classical stochastic ones. Using explicit indirect observation models for diffusive and counting measurements we derive quantum filtering (prediction) equations to describe the stochastic evolution of the open quantum system under the continuous partial observation. Working in parallel with classical indeterministic control theory, we show the Markov Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these two kinds of measurements. The resulting filtering and Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities and differences with the corresponding classical nonlinear filtering and control problems and demonstrates microduality between quantum filtering and classical control.
Measurements continuous in time have been 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 a 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 for 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 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.
"Quantum trajectories" are solutions of stochastic differential equations of non-usual type. Such equations are called ``Belavkin'' or ``Stochastic Schrödinger Equations'' and describe random phenomena in continuous measurement theory of Open Quantum System. Many recent investigations deal with the control theory in such model. In this article, stochastic models are mathematically and physically justified as limit of concrete discrete procedures called ``Quantum Repeated Measurements''. In particular, this gives a rigorous justification of the Poisson and diffusion approximation in quantum measurement theory with control. Furthermore we investigate some examples using control in quantum mechanics.
In this thesis we study the problem of continuous measurement on an open quantum system.The starting point of our analysis is the time-evolution of an open quantum system where the evolution of the environment is not explicitly described, i.e. traced out.More explicitly we restrict to time-evolutions having the structure of a semigroup or evolution system.That is, for time-points 0 ≤ r ≤ s ≤ t there exists a two-parameter family of completely-positive maps E * (t, s), such that E * (t, s)E * (s, r) = E * (t, r) and the systems state evolved according to ρ(tPhysically this means that in our Ansatz the interaction between system and environment is already fixed and can only be weak.The main goal of this thesis is now to describe, mathematically rigorous, all continuous measurements compatible with the given evolution.A measurement is compatible with the given time-evolution, if the evolution does not change when measurement results are ignored.In other words the measurement is indirect and does not further disturb the systems evolution.Our results can be interpreted as a rigorous construction of continuous matrix product states, and are related to the Hudson-Parthasarathy quantum stochastic calculus.The analysis in this thesis separates into two parts.In the first we study the class of quantum systems having an evolution system structure as described above.Precisely we study the theory of minimal solutions to a Cauchy equation with a generator of Gorini-Kossakowski-Sudarshan-Lindblad type.We do this for the case of time-dependent and unbounded generators.This analysis culminates in the description of a counterexample due to Holevo of a semigroup which is not the minimal solution to such a standard Lindblad equation.In the second part we construct a description of all measurements compatible with a given evolution and thus achieve our main goal.The analysis starts with the discrete-time version of the problem, i.e.where the time-points r, s, t in the above description of the evolution are chosen from a finite set.This discrete-time problem is completely solved by Stinespring dilation theory.We then construct our solution of the continuous-time case as a refinement limit over arbitrary discretizations of the evolution.Hence, our construction gives a continuous-time analogue of the Stinespring dilation for semigroups and evolution systems.It can be interpreted as a complete quantum description of the information the system radiates into the environment, i.e. it yields a quantum state of the environment.We complement the description of the state with a description of the observables for continuous measurements and the calculation expectation values.
Small quantum systems can now be continuously monitored experimentally which allows for the reconstruction of quantum trajectories. A peculiar feature of these trajectories is the emergence of jumps between the eigenstates of the observable which is measured. Using the stochastic master equation (SME) formalism for continuous quantum measurements, we show that the density matrix of a system indeed shows a jumpy behaviour when it is subjected to a tight measurement (even if the noise in the SME is Gaussian ). We are able to compute the jump rates analytically for any system evolution, i.e. any Lindbladian, and we illustrate how our general recipe can be applied to two simple examples. We then discuss the mathematical, foundational and practical applications of our results. The analysis we present is based on a study of the strong noise limit of a class of stochastic differential equations (the SME) and as such the method may be applicable to other physical situations in which a strong noise limit plays a role.