In this work our aim is to study several aspects related to quantum coherence as understood to correspond with the non-classical behaviour that can be observed for certain particular states of a physical system. In particular we are interested in the possible mechanisms that result in dynamically induced transitions between quantum and classical regimes. The thesis is organized as follows: The first chapter dubs as an introduction and serves to set out the basic philosophy underlying the questions addressed in this thesis. It also presents some elementary properties of states and state spaces in Quantum Theory including what we have chosen to define as classical and quantum behaviour. In chapter 2 we study some of the aspects related to observing quantum behaviour and of the properties of our main definition of classicality (and quantumness). Here we also study some of the restrictions imposed on measurements by the existence of globally conserved quantities (Wigner-Araki-Yanase theorem) and their relationship to weak measurements coupled to postselection. In the following chapter we review some of the basic tools used in the description of open quantum system dynamics that will be applied in other chapters. In chapter 4 we review the basics of decoherence and analyse the importance of the choice of initial conditions when trying to study the dynamical emergence of classical behaviour within Quantum Theory. Next we study the other direction of the transition and focus on how to obtain pure quantum states from states that originally were classically mixed. Along the same lines, in chapter 6 we cover some topics related to the production of pure quantum states from measurements. We pay special attention to a model of the non-selective continuous monitoring of a system coupled to another unmonitored system. Lastly we explore some of the possible similarities between the theory of phase transitions and the quantum-classical transition. We must emphasize that all the work done in this thesis assumes that Quantum Theory is generally valid (at least within a broad enough range of energies). Thus, when we say that a state is “classical” we will mainly be referring to one of all the possible states contained in Quantum Theory, but that is susceptible to being interpreted as corresponding to “classical” behaviour. Similarly when we speak of creating a “quantum”, or “quantum coherent”, state we mean that the system has evolved to this state from one of the “classical” ones, but all of these still correspond to valid states within Quantum Theory. In the opinion of the author the main original contributions that can be found in this thesis are the following: - The recognition of the relationship between the Wigner-Araki-Yanase theorem and weak measurements coupled to postselection (sections 2.2 and 2.4); - A mathematical proof of the possible ambiguities arising when two observers try to decide if a state corresponds to quantum or classical behaviour (section 2.6); - The implications of initial correlations in decoherence models. In particular how the choice of certain (correlated) initial conditions can result in residual coherence and the production of pure quantum states in a model that otherwise results in ideal decoherence when (locally equivalent) uncorrelated initial conditions are used (section 4.2); - Various results related to the production of quantum states from initially classical states (sections 5.2 to 5.4); - The analysis of the inverse of a generalized depolarizing channel (section 5.7); - The study of a model of the non-selective continuous monitoring, in the quantum Zeno limit, of a subsystem A interacting with an unmonitored subsystem B. In particular the absence of the purification of B, which has been previously predicted in the selective case, but the possibility of coherent dynamics for B (section 6.4); - The identification of the loose equivalent of a broken symmetry and order parameter in the quantum-classical transition (section 7.2).
In this work our aim is to study several aspects related to quantum coherence as understood to correspond with the non-classical behaviour that can be observed for certain particular states of a physical system. In particular we are interested in the possible mechanisms that result in dynamically induced transitions between quantum and classical regimes. The thesis is organized as follows: The first chapter dubs as an introduction and serves to set out the basic philosophy underlying the questions addressed in this thesis. It also presents some elementary properties of states and state spaces in Quantum Theory including what we have chosen to define as classical and quantum behaviour. In chapter 2 we study some of the aspects related to observing quantum behaviour and of the properties of our main definition of classicality (and quantumness). Here we also study some of the restrictions imposed on measurements by the existence of globally conserved quantities (Wigner-Araki-Yanase theorem) and their relationship to weak measurements coupled to postselection. In the following chapter we review some of the basic tools used in the description of open quantum system dynamics that will be applied in other chapters. In chapter 4 we review the basics of decoherence and analyse the importance of the choice of initial conditions when trying to study the dynamical emergence of classical behaviour within Quantum Theory. Next we study the other direction of the transition and focus on how to obtain pure quantum states from states that originally were classically mixed. Along the same lines, in chapter 6 we cover some topics related to the production of pure quantum states from measurements. We pay special attention to a model of the non-selective continuous monitoring of a system coupled to another unmonitored system. Lastly we explore some of the possible similarities between the theory of phase transitions and the quantum-classical transition. We must emphasize that all the work done in this thesis assumes that Quantum Theory is generally valid (at least within a broad enough range of energies). Thus, when we say that a state is “classical” we will mainly be referring to one of all the possible states contained in Quantum Theory, but that is susceptible to being interpreted as corresponding to “classical” behaviour. Similarly when we speak of creating a “quantum”, or “quantum coherent”, state we mean that the system has evolved to this state from one of the “classical” ones, but all of these still correspond to valid states within Quantum Theory. In the opinion of the author the main original contributions that can be found in this thesis are the following: - The recognition of the relationship between the Wigner-Araki-Yanase theorem and weak measurements coupled to postselection (sections 2.2 and 2.4); - A mathematical proof of the possible ambiguities arising when two observers try to decide if a state corresponds to quantum or classical behaviour (section 2.6); - The implications of initial correlations in decoherence models. In particular how the choice of certain (correlated) initial conditions can result in residual coherence and the production of pure quantum states in a model that otherwise results in ideal decoherence when (locally equivalent) uncorrelated initial conditions are used (section 4.2); - Various results related to the production of quantum states from initially classical states (sections 5.2 to 5.4); - The analysis of the inverse of a generalized depolarizing channel (section 5.7); - The study of a model of the non-selective continuous monitoring, in the quantum Zeno limit, of a subsystem A interacting with an unmonitored subsystem B. In particular the absence of the purification of B, which has been previously predicted in the selective case, but the possibility of coherent dynamics for B (section 6.4); - The identification of the loose equivalent of a broken symmetry and order parameter in the quantum-classical transition (section 7.2).
In the last decade decoherence has become a very popular topic mainly due to the progress in experimental techniques which allow monitoring of the process of decoherence for single microscopic or mesoscopic systems. The other motivation is the rapid development of quantum information and quantum computation theory where decoherence is the main obstacle in the implementation of bold theoretical ideas. All that makes the second improved and extended edition of this book very timely. Despite the enormous efforts of many authors decoherence with its consequences still remains a rather controversial subject. It touches on, namely, the notoriously confusing issues of quantum measurement theory and interpretation of quantum mechanics. The existence of different points of view is reflected by the structure and content of the book. The first three authors (Joos, Zeh and Kiefer) accept the standard formalism of quantum mechanics but seem to reject orthodox Copenhagen interpretation, Giulini and Kupsch stick to both while Stamatescu discusses models which go beyond the standard quantum theory. Fortunately, most of the presented results are independent of the interpretation and the mathematical formalism is common for the (meta)physically different approaches. After a short introduction by Joos followed by a more detailed review of the basic concepts by Zeh, chapter 3 (the longest chapter) by Joos is devoted to the environmental decoherence. Here the author considers mostly rather `down to earth' and well-motivated mechanisms of decoherence through collisions with atoms or molecules and the processes of emission, absorption and scattering of photons. The issues of decoherence induced superselection rules and localization of objects including the possible explanation of the molecular structure are discussed in details. Many other topics are also reviewed in this chapter, e.g., the so-called Zeno effect, relationships between quantum chaos and decoherence, the role of decoherence in quantum information processing and even decoherence in the brain. The next chapter, written by Kiefer, is devoted to decoherence in quantum field theory and quantum gravity which is a much more speculative and less explored topic. Two complementary aspects are studied in this approach: decoherence of particle states by the quantum fields and decoherence of field states by the particles. Cosmological issues related to decoherence are discussed, not only within the standard Friedmann cosmology, but also using the elements of the theory of black holes, wormholes and strings. The relations between the formalism of consistent histories defined in terms of decoherence functionals and the environmental decoherence are discussed in chapter 5, also written by Kiefer. The Feynman--Vernon influence functional for the quantum open system is presented in detail as the first example of decoherence functional. Then the general theory is outlined together with possible interpretations including cosmological aspects. The next chapter by Giulini presents an overview of the superselection rules arising from physical symmetries and gauge transformations both for nonrelativistic quantum mechanics and quantum field theory. Critical discussion of kinematical superselection rules versus dynamical ones is illustrated by numerous examples like Galilei invariant quantum mechanics, quantum electrodynamics and quantum gravity. The introduction to the theory of quantum open systems and its applications to decoherence models is given in chapter 7 by Kupsch. Generalized master equations, Markovian approximation and a few Hamiltonian models relevant for decoherence are discussed. Some mathematical tools, e.g., complete positivity and entropy inequalities are also presented. The last chapter by Stamatescu is devoted to stochastic collapse models which can be interpreted either as certain representations of the dynamics of open quantum systems or as fundamental modifications of the Schr\"odinger equation. The final part of the book consists of remarks by Zeh on related concepts and methods and seven appendices. The broad spectrum, mathematically-friendly presentation, inclusion of the very recent developments and the extensive bibliography (about 550 references) make this book a valuable reference for all researchers, graduate and PhD students interested in the foundations of quantum mechanics, quantum open systems and quantum information. The relative independence of the chapters and numerous redundancies allow for selective reading, which is very helpful for newcomers to this field.
In the thesis I study various quantum coherence phenomena and create some of the foundations for a systematic coherence theory. So far, the approach to quantum coherence in science has been purely phenomenological. In my thesis I try to answer the question what quantum coherence is and how it should be approached within the framework of physics, the metatheory of physics and the terminology related to them. It is worth noticing that quantum coherence is a conserved quantity that can be exactly defined. I propose a way to define quantum coherence mathematically from the density matrix of the system. Degenerate quantum gases, i.e., Bose condensates and ultracold Fermi systems, form a good laboratory to study coherence, since their entropy is small and coherence is large, and thus they possess strong coherence phenomena. Concerning coherence phenomena in degenerate quantum gases, I concentrate in my thesis mainly on collective association from atoms to molecules, Rabi oscillations and decoherence. It appears that collective association and oscillations do not depend on the spin-statistics of particles. Moreover, I study the logical features of decoherence in closed systems via a simple spin-model. I argue that decoherence is a valid concept also in systems with a possibility to experience recoherence, i.e., Poincaré recurrences. Metatheoretically this is a remarkable result, since it justifies quantum cosmology: to study the whole universe (i.e., physical reality) purely quantum physically is meaningful and valid science, in which decoherence explains why the quantum physical universe appears to cosmologists and other scientists very classical-like. The study of the logical structure of closed systems also reveals that complex enough closed (physical) systems obey a principle that is similar to Gödel's incompleteness theorem of logic. According to the theorem it is impossible to describe completely a closed system within the system, and the inside and outside descriptions of the system can be remarkably different. Via understanding this feature it may be possible to comprehend coarse-graining better and to define uniquely the mutual entanglement of quantum systems.
Introduction In this chapter I attempt to review a few somewhat loosely connected themes, which figure in recent work by physicists on decoherence and irreversibility in quantum mechanics. The looseness is forced upon us by the subject – any proper discussion inevitably drags in questions about thermodynamic reversibility and the various arrows of time, about the foundations of quantum mechanics and quantum measurement, about the nature of quantum coherence, and about the relationship between quantum and classical mechanics. There are inevitably many gaps in our understanding of these questions, both for the physicist and for the philosopher. Some philosophers may feel that the subject is too immature to be interesting to them – that it is not yet ripe for philosophical analysis. Indeed I have recently heard the opinion expressed that quantum mechanics in its entirety was still not amenable to such analysis – that if the physicists could not yet understand it, there was no point in philosophers trying! I hope this feeling is not too widespread amongst philosophers of science – I suspect that it may arise in part because the philosophical community (along with a good part of the scientific community) is unaware of some important advances that have been made in the last decade in understanding the physics of quantum processes at the macroscopic level, as well as the relation between classical and quantum mechanics. Much of the chapter is therefore unashamedly tutorial in nature; however, given the wide range of topics covered, I imagine that even physicists will find at least some of the material to be novel.
In the works leading to this thesis the aim was to give a clear operational interpretation of different aspects of non-classicality. Doing so, enables to see the different properties as different aspects of the same underlying question, namely whether there is a classical explanation of the measurements at hand. If one incorporates the additional assumptions in the setting (such as locality by the separation of the two parties in entanglement theory), the question of whether some system is classical, boils down to ask whether one can simulate everything one can possibly observe by classical variables governed by a stochastic process. This new perspective allows to characterise dynamic properties of non-classicality, and also to connect them to more static types of non-classicality such as entanglement. Indeed we unify entanglement, discord and coherence from one single perspective. We do this by restricting our setting such that the operations have a clear classical interpretation and hence incorporating natural restrictions of classical systems, if the initial state of the system is not entangled does not have discord or does not have coherence. If the initial state of the system is different, we see the magic of quantum mechanics happening again. For instance we show for one specific quantum algorithm, which is thought to be better than any classical algorithm for its task, that the precision of the algorithm scales with the coherence present at its start. In other words, it is these resources (coherence, discord and entanglement) that make the magic possible and with our perspective we can interpret the role of the different resources and analyse how they are linked to each other. This does not give an explanation of the magic of quantum mechanics, but it does make the mechanisms more transparent allowing for new explanations and there are some examples already in this thesis. Finally we also look at the dual of when states are non-classical, asking, and answering, in a specific setting, what makes measurements non-classical.
A useful and unifying concept in quantum information theory is the idea of a consumable "resource". The basic idea behind quantum resource theories is that whenever we face restrictions on the set of operations we can perform, some states (resource states) enable us to perform particular tasks that otherwise would have been impossible. In this thesis we first review a resource theory that recently has gained a lot of attention, namely the resource theory of asymmetry in which one is restricted to symmetric operations and therefore asymmetric states are the resource states. The asymmetry properties of a state relative to some symmetry group specify how and to what extent the given symmetry is broken by the state. In the remainder of the thesis we explain how the ideas developed in this resource theory are related to three different areas in quantum theory: theory of quantum reference frames, the Wigner-Araki-Yanase(WAY) theorem and the conditional probability interpretation of time in quantum mechanics. Theory of quantum reference frames treats reference frames like any other physical system within the formalism of quantum theory, which causes the measurements to be an approximation of the measurements against their classical counterparts. We consider the dynamics of a quantum directional reference frame undergoing repeated interactions. These interactions induce a back-action on the reference which is the central focus of our study. The effect of a precise sequence of measurement outcomes on the reference frame is studied by looking at both the case that the measurement record is averaged over and the case wherein it is retained. We find, in particular, that there is interesting dynamics in the latter situation, which cannot be revealed by considering the average case. We then consider in detail how a sequence of rotationally invariant unitary interactions affects the reference frame, a situation, which leads to quite different dynamics than the case of repeated measurements. Different strategies for correcting reference frame drift is considered given that we have access to a set of particles with polarization opposite to the direction of drift. In particular, we find that, by implementing a suitably chosen unitary interaction after every two measurements, we can eliminate the rotational drift of the reference frame. The WAY theorem establishes an important constraint that conservation laws impose on quantum mechanical measurements. We formulate the WAY theorem in the broader context of resource theories, where one is constrained to a subset of quantum mechanical operations described by a symmetry group. Establishing connections with the theory of quantum state discrimination we obtain optimal unitaries describing the measurement of arbitrary observables, explain how prior information can permit perfect measurements that circumvent the WAY constraint, and provide a framework that establishes a natural ordering on measurement apparatuses through a decomposition into asymmetry and charge subsystems. Finally we review two di erent schemes in the conditional probability interpretation (CPI) of time in quantum mechanics. In these schemes parameter time "t" as it appears in Schrodinder equation is considered to be unobservable, yet one can use an extension of conditional probabilities in order to study dynamics of a system relative to another system, namely the quantum clock. We phrase CPI as a quantum communication protocol which enables us to use the machinery developed in the resource theory of asymmetry. This provides us with a deeper understanding of the decoherence effect caused due to the inaccessibility of the parameter time "t" and that how this effect can be suppressed by choosing an optimal initial state for the quantum clock.
In this chapter we shall deal with the quantum dynamics of an open system. By open quantum system we mean here a system which interacts with an environment (see Sec. 9.4): since we are not interested in the dynamics of the environment, we shall have to describe the evolution of the system in some “effective” way. In particular, if we consider only the evolution of the system, it will be non-unitary, and this will represent the subject of this chapter. As we know, Hamiltonian quantum dynamics is unitary and changes pure states into pure states. On the other hand, non-unitary dynamics changes an initially pure state into a mixture, which must be described by a density matrix (see Ch. 5). In the case of macroscopic systems, the coupling with the environment may be arbitrarily reduced and therefore its influence can be made correspondingly small (see Sec. 1.1). Microscopic systems, however, always couple to the environment and this coupling cannot be considered negligible. This is the reason why the quantum theory of open systems is one of the most important and fundamental chapters of quantum mechanics that, though born in quantum optics, has many implications in almost all fields of physics. The present chapter can be seen as a further development of the measurement theory (see Ch. 9), as open systems manifest a decoherent dynamics (see in particular Sec. 9.4).