Introduction

DISSERTATION
Titel der Dissertation
Quantum violation of macroscopic realism
and the transition to classical physics
Verfasser
Johannes Kofler
angestrebter akademischer Grad
Doktor der Naturwissenschaften

Ich weiß ehrlich nicht, was die Leute meinen, wenn sie von der Freiheit des menschlichen Willens sprechen. Ich habe zum Beispiel das Gefühl, daß ich irgend etwas will; aber was das mit Freiheit zu tun hat, kann ich überhaupt nicht verstehen. Ich spüre, daß ich meine Pfeife anzünden will und tue das auch; aber wie kann ich das mit der Idee der Freiheit verbinden? Was liegt hinter dem Willensakt, daß ich meine Pfeife anzünden will? Ein anderer Willensakt? Schopenhauer hat einmal gesagt: “Der Mensch kann tun was er will; er kann aber nicht wollen was er will.” [Honestly I cannot understand what people mean when they talk about the freedom of the human will. I have a feeling, for instance, that I will something or other; but what relation this has with freedom I cannot understand at all. I feel that I will to light my pipe and I do it; but how can I connect that up with the idea of freedom? What is behind the act of willing to light the pipe? Another act of willing? Schopenhauer once said: “Man can do what he wills but he cannot will what he wills.”] Albert Einstein

Abstract

The descriptions of the quantum realm and the macroscopic classical world differ significantly not only in their mathematical formulations but also in their foundational concepts and philosophical consequences. The assumptions of a genuine classical world—local realism and macroscopic realism (macrorealism)—are at variance with quantum mechanical predictions as characterized by the violation of the Bell and the Leggett-Garg inequality, respectively. When and how physical systems stop to behave quantumly and begin to behave classically is still heavily debated in the physics community and subject to theoretical and experimental research.

The first chapter of this dissertation puts forward a novel approach to the quantum-to-classical transition fully within quantum theory and conceptually different from already existing models. It neither needs to refer to the uncontrollable environment of a system (decoherence) nor to change the quantum laws itself (collapse models), but puts the stress on the limits of observability of quantum phenomena due to imprecisions of our measurement apparatuses. Naively, one would say that the predictions of quantum mechanics reduce to those of classical physics merely by going to large quantum numbers. Using a quantum spin as a model object, we first demonstrate that for unrestricted measurement accuracy the system’s time evolution cannot be described classically and is conflict with macrorealism through violation of the Leggett-Garg inequality, no matter how large the spin is. How then does the classical world arise? Under realistic conditions in every-day life, we are only able to perform coarse-grained measurements and do not resolve individual quantum levels of the macroscopic system. We show that for some “classical” Hamiltonians it is this mere restriction to fuzzy measurements which is sufficient to see the natural emergence of macrorealism and the classical Newtonian laws out of the full quantum formalism. This resolves the apparent impossibility of how classical realism and deterministic laws can emerge out of fundamentally random quantum events.

In the second chapter, we find that the restriction of coarse-grained measurements usually allows to describe the time evolution of any quantum spin state by a time evolution of a statistical mixture. However, we demonstrate that there exist “non-classical” Hamiltonians for which the time evolution of this mixture cannot be understood classically, leading to a violation of macrorealism. We derive the necessary condition for these non-classical time evolutions and illustrate it with the example of an oscillating Schrödinger cat-like state. Constant interaction of the system with an environment establishes macrorealism but cannot account for a continuous spatiotemporal description of the system’s non-classical time evolution in terms of classical laws of motion. We argue that non-classical Hamiltonians are unlikely to appear in nature because they require interactions between a large number of particles or are of high computational complexity.

The third chapter investigates entanglement between collective operators in two specific physical systems, namely in a linear chain of harmonic oscillators and in ensembles of spin- particles. We show that under certain conditions entanglement between macroscopic observables can persist for large system sizes. However, since this analysis uses sharp measurements, it is not in disagreement with our quantum-to-classical approach.

The last chapter addresses the question of the origin of quantum randomness and proposes a link with mathematical undecidability. We demonstrate that the states of elementary quantum systems are capable of encoding a set of mathematical axioms. Quantum measurements reveal whether a given proposition is decidable or undecidable within this set. We theoretically find and experimentally confirm that whenever a mathematical proposition is undecidable within the axiomatic set encoded in the state, the measurement associated to the proposition has random outcomes. This supports the view that quantum randomness is irreducible and a manifestation of mathematical undecidability.

Zusammenfassung

Die Beschreibungen der quantenmechanischen und der klassischen Welt unterscheiden sich nicht nur signifikant in ihren mathematischen Formulierungen sondern auch in ihren grundsätzlichen Konzepten und philosophischen Implikationen. Die Annahmen einer klassischen Welt – lokaler und makroskopischer Realismus (Makrorealismus) – widersprechen den Vorhersagen der Quantenphysik, was durch die Verletzung der Bell- und der Leggett-Garg-Ungleichung charakterisiert wird. Die Frage, wann und wie physikalische Systeme aufhören sich quantenmechanisch und anfangen sich klassisch zu verhalten, wird in der wissenschaftlichen Gemeinschaft noch immer heftig diskutiert und ist Gegenstand experimenteller und theoretischer Forschung.

Das erste Kapitel der vorliegenden Dissertation entwickelt einen neuen Zugang zum Übergang der Quanten- zur klassischen Physik, und zwar vollkommen innerhalb der Quantentheorie und konzeptionell verschieden von bereits bestehenden Modellen. Dieser Zugang muss sich weder auf die unkontrollierbare Umgebung von Systemen beziehen (Dekohärenz) noch die Gesetze der Quantenmechanik selbst abändern (Kollaps-Modelle). Er fokussiert sich vielmehr auf die Limitierung der Beobachtbarkeit von Quantenphänomen aufgrund der Ungenauigkeit unserer Messapparate. Naiverweise würde man annehmen, dass sich die Vorhersagen der Quantenmechanik auf jene der klassischen Physik allein dadurch reduzieren, indem man zu großen Quantenzahlen geht. Wir verwenden einen Quantenspin als Modellobjekt und zeigen zunächst, dass bei uneingeschränkter Messgenauigkeit die Zeitevolution des Systems nicht klassisch verstanden werden kann und aufgrund der Verletzung der Leggett-Garg-Ungleichung im Widerspruch zu Makrorealismus steht, selbst wenn der Spin beliebig groß ist. Wie entsteht dann die klassische Welt? Unter realistischen alltäglichen Bedingungen sind wir nur in der Lage, grobkörnige Messungen durchzuführen, die die einzelnen Quantenniveaus des makroskopischen Systems nicht auflösen können. Wir zeigen, dass für bestimmte “klassische” Hamilton-Operatoren diese bloße Einschränkung zu unscharfen Messungen ausreicht, um die natürliche Emergenz von Makrorealismus und klassischer Newtonscher Gesetze aus dem vollen quantenmechanischen Formalismus zu sehen. Dies löst die scheinbare Unmöglichkeit auf, wie klassischer Realismus und deterministische Gesetze aus fundamental zufälligen Quantenereignissen entstehen können.

Im zweiten Kapitel zeigen wir, dass die Einschränkung grobkörniger Messungen es üblicherweise erlaubt, die zeitliche Evolution jedes beliebigen quantenmechanischen Spinzustands durch die zeitliche Evolution einer statistischen Mischung zu beschreiben. Ungeachtet dessen demonstrieren wir, dass es “nicht-klassische” Hamilton-Operatoren gibt, für die die Zeitentwicklung dieser Mischung nicht klassisch verstanden werden kann. Wir leiten die allgemeine Bedingung für solche nicht-klassischen Zeitentwicklungen her und veranschaulichen sie anhand des Beispiels einer oszillierenden Schrödinger-Katze. Andauernde Interaktion des Systems mit einer Umgebung etabliert Makrorealismus, kann aber keine kontinuierliche raumzeitliche Beschreibung der nicht-klassischen Zeitevolution des Systems durch klassische Bewegungsgleichungen liefern. Wir argumentieren, dass nicht-klassische Hamilton-Operatoren in der Natur wahrscheinlich nicht vorkommen, weil sie Vielteilchen-Wechselwirkungen benötigen oder von hoher Komplexität sind.

Das dritte Kapitel untersucht Verschränkung zwischen kollektiven Operatoren in zwei spezifischen physikalischen Systemen, nämlich in einer linearen Kette von harmonischen Oszillatoren und in Ensembles von Spin--Teilchen. Wir zeigen, dass Verschränkung zwischen makroskopischen Observablen für große Systeme bestehen bleiben kann. Zumal diese Analyse scharfe Messungen verwendet, steht sie nicht im Widerspruch zu unserem Zugang zum Übergang von der Quanten- zur klassischen Physik.

Das letzte Kapitel beschäftigt sich mit der Frage nach dem Ursprung von quantenmechanischem Zufall und schlägt eine Verbindung mit mathematischer Unentscheidbarkeit vor. Wir demonstrieren, dass Zustände elementarer Quantensysteme einen Satz von mathematischen Axiomen kodieren können. Quantenmechanische Messungen bringen zum Vorschein, ob eine gegebene Proposition innerhalb dieses Satzes entscheidbar oder unentscheidbar ist. Wir finden theoretisch und bestätigen experimentell, dass Messungen, die mit innerhalb des vom Quantenzustand kodierten Axiomensatzes unentscheidbaren mathematischen Propositionen assoziiert sind, zu zufälligen Resultaten führen. Dies unterstützt die Sichtweise, dass quantenmechanischer Zufall irreduzibel und eine Manifestation von mathematischer Unentscheidbarkeit ist.

Acknowledgement

When I was writing my diploma thesis at the University of Linz in summer 2004, I applied for a PhD position in Vienna because of a strong interest in quantum physics and foundational questions. In retrospect, this was likely one of the best decisions that I could have made—both from a professional and a private perspective.

A few days after my arrival in Vienna in January 2005, I was the first person to move into the building of the Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences. I did not yet know how great the pleasure would be to learn and work in this extremely inspiring and challenging environment during the following years. Now is the time to thank:

First and foremost Časlav Brukner for his perfect scientific and personal support. I am sincerely grateful for all the time and effort he has put into our joint research, and without exaggeration, I could not have wished for a better supervision. His vast knowledge and insight into quantum theory as well as the true joy and interest with which he is doing science have been an enduring inspiration through my whole doctoral studies.

Nikita Arnold and Urbaan M. Titulaer for their objective advice on my interest to go to Vienna after my graduate studies.

Anton Zeilinger for advocating my application for a theoretical PhD position, for his open-minded way of leading the institute, and for integrating me into experimental activities.

Markus Aspelmeyer, Tomasz Paterek, and Rupert Ursin for many fascinating discussions, not only on quantum mechanics.

Simon Gröblacher and Robert Prevedel for forcing me every now and then to have a glass of wine. Katharina Gugler for sharing with me the disbelieve in free will.

The whole Vienna quantum group—Academy and University—for creating a very pleasant atmosphere.

The Austrian Academy of Sciences for a doctoral fellowship that allowed me to pursue the second half of my research in a fully independent way.

And last but not least my family for their unconditional support and interest in my work throughout the years, and my friends for all the hiking trips, cinema evenings, and discussions on Life, the Universe, and Everything.

Johannes Kofler

Vienna, June 2008

List of publications

The titles of publications that are directly relevant for this dissertation are written in bold face.

Articles in refereed journals

  • J. Kofler and Č. Brukner
    The conditions for quantum violation of macroscopic realism
    Phys. Rev. Lett. (accepted); arXiv:0706.0668 [quant-ph].

  • J. Kofler and Č. Brukner
    Classical world arising out of quantum physics under the restriction of coarse-grained measurements
    Phys. Rev. Lett. 99, 180403 (2007).

  • J. Kofler and Č. Brukner
    Entanglement distribution revealed by macroscopic observations
    Phys. Rev. A 74, 050304(R) (2006).

  • M. Lindenthal and J. Kofler
    Measuring the absolute photo detection efficiency using photon number correlations
    Appl. Opt. 45, 6059 (2006).

  • J. Kofler and N. Arnold
    Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie
    Phys. Rev. B 73, 235401 (2006).

  • J. Kofler, V. Vedral, M. S. Kim, and Č. Brukner
    Entanglement between collective operators in a linear harmonic chain
    Phys. Rev. A 73, 052107 (2006).

  • J. Kofler, T. Paterek, and Č. Brukner
    Experimenter’s freedom in Bell’s theorem and quantum cryptography
    Phys. Rev. A 73, 022104 (2006).

Submitted or in preparation

  • J. Kofler and Č. Brukner
    The conditions for quantum violation of macroscopic realism
    ArXiv:0706.0668 [quant-ph] (submitted).

  • X. Ma, A. Qarry, J. Kofler, T. Jennewein, and A. Zeilinger
    Experimental violation of a Bell inequality with two different degrees of freedom
    In preparation (2008).

  • X. Ma, A. Qarry, N. Tetik, J. Kofler, T. Jennewein, and A. Zeilinger
    Entanglement-assisted delayed-choice experiment
    In preparation (2008).

  • J. Kofler and Č. Brukner
    Fundamental limits on observing quantum phenomena from within quantum theory
    In preparation (2008).

Contributions in books

  • J. Kofler and Č. Brukner
    A coarse-grained Schrödinger cat
    In: Quantum Communication and Security, ed. M. Żukowski, S. Kilin, and J. Kowalik (IOS Press 2007).

  • J. Kofler and N. Arnold
    Axially symmetric focusing of light in dry laser cleaning and nanopatterning
    In: Laser Cleaning II, ed. D. M. Kane (World Scientific Publishing, 2006).

  • D. Bäuerle, T. Gumpenberger, D. Brodoceanu, G. Langer, J. Kofler, J. Heitz, and K. Piglmayer
    Laser cleaning and surface modifications: Applications in nano- and biotechnology
    In: Laser Cleaning II, ed. D. M. Kane (World Scientific Publishing, 2006).

Contributions in proceedings

  • R. Ursin, T. Jennewein, J. Kofler, J. Perdigues, L. Cacciapuoti, C. J. de Matos, M. Aspelmeyer, A. Valencia, T. Scheidl, A. Fedrizzi, A. Acin, C. Barbieri, G. Bianco, Č. Brukner, J. Capmany, S. Cova, D. Giggenbach, W. Leeb, R. H. Hadfield, R. Laflamme, N. Lütkenhaus, G. Milburn, M. Peev, T. Ralph, J. Rarity, R. Renner, E. Samain, N. Solomos, W. Tittel, J. P. Torres, M. Toyoshima, A. Ortigosa-Blanch, V. Pruneri, P. Villoresi, I. Walmsley, G. Weihs, H. Weinfurter, M. Żukowski, and A. Zeilinger
    Space-QUEST: Experiments with quantum entanglement in space
    Accepted for the 59th International Astronautical Congress (2008); arXiv:0806.0945v1 [quant-ph].

  • D. Bäuerle, L. Landström, J. Kofler, N. Arnold, and K. Piglmayer
    Laser-processing with colloid monolayers
    Proc. SPIE 5339, 20 (2004).

Articles in popular journals

  • A. Zeilinger and J. Kofler
    La dissolution du paradoxe
    Sciences et Avenir Hors-Série 148, 54 (2006).

Theses

  • J. Kofler
    Focusing of light in axially symmetric systems within the wave optics approximation
    Diploma Thesis, Johannes Kepler University Linz, Austria (2004).

Chapter 0 Introduction

Since the birth of quantum theory in the 1920s, quantum entanglement and quantum superposition have been used to highlight a number of counter-intuitive phenomena. They lie at the heart of the Einstein-Podolsky-Rosen [31] and the Schrödinger cat paradox [84] (Figure 0.1). The corresponding conflicts between quantum mechanics on one side and a classical world—local realism and macroscopic realism (macrorealism)—on the other, are quantitatively expressed by the violation of Bell’s [10] and the Leggett-Garg inequality [61], respectively. The quantum violation of local realism shows that the view is untenable that space-like separated events do not influence each other and that objects have their properties prior to and independent of measurement. The quantum violation of macrorealism means that it is wrong to believe that a macroscopic object has definite properties at any time and that it can be measured without effecting them or their subsequent dynamics.

Figure 0.1: In 1935, Schrödinger put forward the “burlesque” Gedankenexperiment of a “hell machine” to illustrate that, according to quantum physics, it is possible to prepare a cat in a superposition of ‘dead’ and ‘alive’ [84]. [Picture taken from Sciences et Avenir Hors-Série 148, 54 (2006).]

The importance of this incongruousness today exceeds the realm of the foundations of quantum physics and has become an important conceptual tool for developing new quantum information technology. Entanglement and superposition allow to perform certain computation and communication tasks such as quantum cryptography [11, 39], teleportation [12, 15] or quantum computation [29, 68], which are not possible classically. Experiments in the near future will be realized with increasingly complex objects, either by entangling more and more systems with each other, or by entangling systems with a very large number of degrees of freedom. Eventually, all these developments will push the realm of quantum physics well into the macroscopic world. Moreover, implications on society in a cultural sense may manifest themselves, for the characteristics and peculiarities of the quantum world—in particular quantum entanglement and quantum superposition—could eventually become part of the every-day experience.

However, the macroscopic classical world that we perceive around us does not show any characteristics of the quantum realm. The question “Why do classical systems stop to show quantum features?” is still answered in radically different ways within the physics community. Since classical apparatuses are needed for performing measurements on quantum systems, this question is also related to the so called “measurement problem” and the various interpretations of quantum mechanics, ranging from the Copenhagen over the Bohmian to the many-worlds interpretation [47].

On the one hand, there exist a number of so-called collapse models [37, 74] which try to explain the discrepancy between the quantum and the classical world by introducing a fundamental breakdown of quantum superpositions at some quantum-classical border. On the other, the decoherence program [104, 105] demonstrates that the states of complex systems interacting with an environment, which cannot be accessed and controlled in detail, rapidly evolve into statistical mixtures and lose their quantum character.

While neither of these approaches can give a definite or already experimentally settled answer, the understanding of the quantum-to-classical transition is not only of prior importance for the future development towards macroscopic superpositions and entanglement but also necessary for a consistent description of the physical world. Collapse models make assumptions about inherently non-quantum mechanical background fields or gravitational mechanisms which are still to be tested experimentally. The decoherence program is inherently quantum mechanical and can give good explanations for many observations though it has to rely on the assumption of a preferred pointer basis. However, the effects of decoherence can in principle be always reduced and the experimental progress of the last years has already demonstrated quantum interference of (Schrödinger-cat like) macroscopic superpositions, e.g., interference fringes with large molecules of atomic mass units [5], entanglement between clouds of atoms [48], or superpositions of macroscopically distinct flux states in superconducting rings corresponding to amperes of current flowing clock- or anticlockwise [33]. These experiments circumvent the problem of decoherence but did not yet come into the region where they could exclude collapse models.

Until today there exists no definite answer to the problem of the quantum-to-classical transition. Hence, certainly one of the most fundamental and interesting questions in modern physics still remains unanswered:

How does the classical physical world emerge out of the quantum realm?

Chapter 1 of this dissertation addresses this question from a novel perspective and develops an approach to the quantum-to-classical transition fully within quantum theory and conceptually different from already existing models. It neither needs to refer to the environment of a system (decoherence) nor to change the quantum laws itself (collapse models) but puts the stress on the limits of observability of quantum phenomena due to our measurement apparatuses. Using a quantum spin as a model object, we first demonstrate that for unrestricted measurement accuracy the system’s time evolution cannot be described classically and is in conflict with macrorealism through violation of the Leggett-Garg inequality. This conflict remains even if the spin is arbitrarily large and macroscopic.

Under realistic conditions in every-day life, however, we are only able to perform coarse-grained measurements and do not resolve individual quantum levels of the macroscopic system. As we show, it is this mere restriction to fuzzy measurements which is sufficient to see the natural emergence of macrorealism and the classical Newtonian laws out of the full quantum formalism: the system’s time evolution governed by the Schrödinger equation and the state projection induced by measurements. This resolves the apparent impossibility of how classical realism and deterministic laws can emerge out of fundamentally random quantum events. Figure 0.2 presents an illustration of this approach.

Figure 0.2: Under the magnifying glass of sharp measurements Albert Einstein sees a strange and colorful quantum picture of the face next to him. Its abstractness is symbolized by Pablo Picasso’s “Head of a Reading Woman”. Under an every-day coarse-grained view the classical appearance of Charlie Chaplin emerges.

Chapter 2 first shows that a violation of the Leggett-Garg inequality itself is possible for arbitrary Hamiltonians given the ability to perform sharp quantum measurements. Apparatus decoherence or the restriction of coarse-grained measurements usually allow to describe the time evolution of any quantum spin state by a time evolution of a statistical mixture. However, we demonstrate that there are “non-classical” Hamiltonians for which the time evolution of this mixture cannot be understood classically, leading to a violation of macrorealism. We derive the necessary condition for these non-classical time evolutions and illustrate it with the example of an oscillating Schrödinger cat-like state. System decoherence, i.e. the continuous monitoring of the system by an environment, leads to macrorealism but a dynamical description of non-classical time evolutions in terms of classical laws of motion remains impossible.

In the last part we argue that non-classical Hamiltonians either require interactions between a large number of particles or are of high computational complexity. This might be understood as the reason why they are unlikely to appear in nature.

Chapter 3 investigates entanglement between collective operators in two specific physical systems, namely in a linear chain of harmonic oscillators and in ensembles of spin- particles. We demonstrate that under certain conditions entanglement between macroscopic observables can indeed persist for large system sizes. However, since this analysis uses sharp measurements, it is not in disagreement with our quantum-to-classical approach.

Chapter 4 addresses the question of the origin of quantum randomness. In our view, classical physics emerges out of the quantum world but the randomness in the classical mixture is still irreducible and of quantum nature. We propose to link quantum randomness with mathematical undecidability in the sense of Chaitin’s version of Gödel’s theorem. It states that given a set of axioms that contains a certain amount of information, it is impossible to deduce the truth value of a proposition which, together with the axioms, contains more information than the set of axioms itself.

First, we demonstrate that the states of elementary quantum systems are capable of encoding mathematical axioms. Quantum mechanics imposes an upper limit on how much information can be carried by a quantum state, thus limiting the information content of the set of axioms. Then, we show that quantum measurements are capable of revealing whether a given proposition is decidable or undecidable within this set. This allows for an experimental test of mathematical undecidability by realizing in the laboratory the actual quantum states and operations required. We theoretically find and experimentally confirm that whenever a mathematical proposition is undecidable within the system of axioms encoded in the state, the measurement associated to the proposition gives random outcomes. Our results support the view that quantum randomness is irreducible and a manifestation of mathematical undecidability.

Chapter 1 Classical world emerging from quantum physics

Summary:

Inspired by the thoughts of Peres on the classical limit [75]—we present a novel theoretical approach to macroscopic realism and classical physics within quantum theory. While our approach is not at variance with the decoherence program [104, 105], it differs conceptually from it. It is not dynamical and puts the stress on the limits of observability of quantum effects of macroscopic objects, i.e. on the required precision of our measurement apparatuses such that quantum phenomena can still be observed. The term “macroscopic” is used here to denote a system with a high dimensionality rather than a low-dimensional system with a large parameter such as mass or size. Furthermore, there is no need to change the quantum laws itself like in collapse models [37, 74].

Using a quantum spin as a model system, we first show that, if consecutive eigenvalues of a spin component can be experimentally resolved in sharp quantum measurements, the Leggett-Garg inequality is violated for arbitrary spin lengths and the violation persists even in the limit of infinitely large spins. This contradicts the naive assumption that the predictions of quantum mechanics reduce to those of classical physics merely due to the fact that a system becomes “large”. For local realism this persistence of quantum features was demonstrated by Garg and Mermin [34], and the violation even increases with the systems’ dimensionality [49, 26].

In every-day life, however, one not only encounters very high-dimensional systems but is experimentally restricted to coarse-grained measurements. They only distinguish between eigenvalues which are separated by much more than the intrinsic quantum uncertainty. We show for arbitrary spin states that, given a certain time evolution, the macroscopically distinct outcomes obey the classical Newtonian laws which emerge out of the Schrödinger equation and the projection postulate.

This suggests that classical physics can be seen as implied by quantum mechanics under the restriction of fuzzy measurements and resolves the apparent impossibility of how classical realism and deterministic laws can emerge out of fundamentally random quantum events.

This chapter mainly bases on and also uses parts of Reference [57]:

  • J. Kofler and Č. Brukner
    Classical world arising out of quantum physics under the restriction of coarse-grained measurements
    Phys. Rev. Lett. 99, 180403 (2007).

1.1 The Leggett-Garg inequality

In this section we introduce the concept of macroscopic realism (macrorealism) and show how to derive a Leggett-Garg inequality, which can be used as a tool to indicate whether or not a system’s time evolution can be understood in classical terms. In agreement with this, we then briefly demonstrate explicitly that the inequality can always be violated for genuine quantum systems but always satisfied for classical objects.

Macrorealism is defined by the conjunction of the following three postulates [63]:

  1. Macrorealism per se. A macroscopic object which has available to it two or more macroscopically distinct states is at any given time in a definite one of those states.

  2. Non-invasive measurability. It is possible in principle to determine which of these states the system is in without any effect on the state itself or on the subsequent system dynamics.

  3. Induction. The properties of ensembles are determined exclusively by initial conditions (and in particular not by final conditions).

The last two postulates can be phrased into the single assumption that the object’s state is independent of past and future measurements [62]. Classical (Newtonian) physics belongs to the class of macrorealistic theories.

Now consider a macroscopic physical system and a dichotomic quantity , which whenever measured is found to take one of the values only. Further consider a series of runs starting from identical initial conditions at time such that on the first set of runs is measured only at times and , only at and on the second, at and on the third, and at and on the fourth . Let denote the value of at time (see Figure 1.1). Consider the algebraic combination of the Clauser-Horne-Shimony-Holt (CHSH) type [24]:

(1.1)
Figure 1.1: In a macrorealistic theory macrorealism per se implies that a time-dependent quantity of a macroscopic object has a well defined unambiguous value at every time . Non-invasive measurability (together with induction) is reflected by the fact that the ’s do not depend on whether the system was or was not measured at earlier times.

It can only have the values or for one of the two brackets has to vanish and the other is or and then multiplied with or . Macrorealism per se is reflected by the objective existence of unambiguous values of at all times, and non-invasive measurability together with induction is reflected by the fact that the ’s are the same, independent in which combination they appear. E.g., is independent of previous measurements, i.e. whether it appears in a run together with or . Repeating the experimental runs many times, we introduce the temporal correlation functions

(1.2)

By averaging (1.1) it follows that any macrorealistic theory has to satisfy the Leggett-Garg inequality [61]

(1.3)

Its violation implies that the object’s time evolution cannot be understood classically.

Let us briefly analyze the quantum evolution of a microscopic quantum object, say the precession of a spin- particle with the Hamiltonian , where is the angular precession frequency and is the Pauli -matrix.1 If we measure the spin along the -direction, then we obtain the temporal correlations . Choosing the four possible measurement times as equidistant, with time distance , the Leggett-Garg inequality becomes

(1.4)

This is maximally violated for the time distance for which (see red line in Figure 1.2). The violation is not surprising as a spin- particle is a genuine quantum system and cannot have the objective properties tentatively attributed to macroscopic objects prior to and independent of measurements. In contrast, we consider an arbitrarily sized uniformly rotating classical spin vector, again precessing around and pointing along at time . As dichotomic observable quantity we use sgn such that () if the spin is pointing upwards (downwards) along . As expected, the inequality (1.3) is always satisfied (see blue line in Figure 1.2).

Figure 1.2: Violation of the Leggett-Garg inequality (1.4) for a rotating spin- particle with precession frequency . The left-hand side of the inequality, , is shown by a red line, while the classical limit, , is indicated by a dashed line. If the distance between the four possible equidistant measurement times is chosen as , then the inequality is maximally violated with . The blue line shows the left-hand side of the Leggett-Garg inequality for a classical rotating spin vector. Its time evolution can be understood classically and does not violate the inequality.

1.2 Violation of the Leggett-Garg inequality for arbitrarily large spins

In this section, we demonstrate that the Leggett-Garg inequality (1.3) is violated for arbitrarily large (macroscopic) spin lengths as long as accurate measurements can be performed. In any run, the first of the two measurements acts as a preparation of the state for the subsequent measurement. Therefore, the initial state of the spin is not decisive and it is sufficient for us to consider as initial state the maximally mixed one:

(1.5)

Here, 11 is the identity operator and are the (spin -component) eigenstates with the possible eigenvalues . We consider the Hamiltonian

(1.6)

where is the rotor’s total spin vector operator, its -component, the moment of inertia and the angular precession frequency. The constant of motion can be ignored since commutes with the individual spin components and does not contribute to their time evolution. The solution of the Schrödinger equation produces a rotation about the -axis, represented by the time evolution operator

(1.7)

We assume that individual eigenstates can be experimentally resolved and use the parity measurement

(1.8)

with the possible dichotomic outcomes (identifying ). The temporal correlation function between results of the parity measurement at different (arbitrary) times and () is

(1.9)

where () is the probability for measuring () at and is the probability for measuring at given that was measured at (). Furthermore,

(1.10)
(1.11)

Here is the expectation value of at and is the expectation value of at given that was the outcome at . The totally mixed state is not changed until the first measurement: and we find

(1.12)

The approximate sign is accurate for half integer as well as in the macroscopic limit , which is assumed from now on. Hence, we have , which is self-evident for a totally mixed state. Depending on the measurement result at , the state is reduced to

(1.13)

with the projection operator onto positive (negative) parity states. Denoting , the remaining expectation value Tr becomes

(1.14)

Here we used the geometrical meaning of the rotations in the first line. From it follows . Using this and from above, the temporal correlation function becomes

(1.15)

Having four possible equidistant measurements with time distance , and using the abbreviation

(1.16)

the Leggett–Garg inequality (1.3) now reads

(1.17)

We approximated the sine function in the denominator, assuming . Inequality (1.17) is violated for all positive and maximally violated for where (compare with Reference [75] for the violation of local realism) as can be seen in Figure 1.3. For every spin size , and given a precession frequency , it is always possible to choose the time distance such that .

Figure 1.3: Violation of the Leggett-Garg inequality (1.17) for arbitrarily large spin size . The left hand side of the inequality, , is indicated by a red line and the classical limit is (dashed line). The initial quantum state is totally mixed, eq. (1.5), and the Hamiltonian (1.6) produces a precession around the -axis with frequency . The dichotomic observable is the parity in a measurement (1.8) of the spin’s -component. For all spin sizes and precession frequencies , one can always find a time distance between the four possible equidistant measurement times such that the quantity is around the value and the classical limit is violated. The maximal violation is achieved for .

We can conclude that a violation of the Leggett-Garg inequality is possible for arbitrarily high-dimensional systems and even for initially totally mixed states.

Note, however, that the temporal precision of our measurement apparatuses, which is required for seeing the violation, increases with , as has to scale with in order to keep . Moreover, due to the nature of the parity measurement, consecutive values of have to be resolved.

1.3 The quantum-to-classical transition for a spin-coherent state

In this section we will show that coarse-grained measurements not only lead to the validity of macrorealism but even to the emergence of classical physics for a certain class of quantum states. The generalization to arbitrary states will be done in the next section.

Let us start with a preliminary remark about distinguishability of states in quantum theory. Any two different eigenvalues and in a measurement of a spin’s -component correspond to orthogonal states without any concept of closeness or distance. In Hilbert space the vectors and are as orthogonal as and :

(1.18)
(1.19)

The terms “close” or “distant” only make sense in a classical context, where those eigenvalues are treated as close which correspond to neighboring outcomes in the real configuration space.

For example, the “eigenvalue labels”  and of a spin component observable correspond to neighboring outcomes in a Stern-Gerlach experiment. (Such observables are sometimes called classical or reasonable [102, 50, 75].) If our measurement accuracy is limited, it is those neighboring eigenvalues which we conflate to coarse-grained observables. It seems thus unavoidable that certain features of classicality have to be assumed beforehand to give the Hilbert space some structure which it does not have a priori.

In what follows, we will first consider the special case of a single spin coherent state and then generalize the transition to classicality for arbitrary states. Spin- coherent states  [79, 7] are the eigenstates with maximal eigenvalue of a spin operator pointing into the direction , where and are the polar and azimuthal angle in spherical coordinates, respectively:

(1.20)

Let us consider the initial spin coherent state at time pointing into the direction (). In the basis of eigenstates it reads

(1.21)

Under time evolution e, eq. (1.7), the probability that a measurement at some later time has the particular outcome is given by the binomial distribution

(1.22)

with , , where and are the polar and azimuthal angle of the (rotated) spin coherent state at time . In the macroscopic limit, , the binomial distribution (1.22) can be very well approximated by a Gaussian distribution

(1.23)

with the width (standard deviation) and the mean value.

Under the “magnifying glass” of sharp measurements individual eigenvalues can be distinguished and the Gaussian probability distribution can be resolved, as shown in Figure 1.4(a), allowing a violation of the Leggett-Garg inequality. Let us now assume that, as in every-day life, the resolution of the measurement apparatus, , is restricted and that it subdivides the possible different outcomes into a much smaller number of coarse-grained “slots” . If the slot size is much larger than the standard deviation , i.e.

(1.24)

the sharply peaked Gaussian cannot be distinguished anymore from the discrete Kronecker delta,

(1.25)

Here, is numbering the slots (from to in steps ) and is the number of the slot in which the center of the Gaussian lies. This is indicated in Figure 1.4(b).

Figure 1.4: An initial spin- coherent state precesses into the coherent state at time under a quantum time evolution. (a) The probability for the outcome in a measurement of the spin’s -component is given by a Gaussian distribution with width and mean , which can be seen under the magnifying glass of sharp measurements. (b) The measurement resolution is finite and subdivides the possible outcomes into a smaller number of coarse-grained “slots”. If the measurement accuracy is much poorer than the width , i.e., , the sharply peaked Gaussian cannot be distinguished anymore from the discrete Kronecker delta where is numbering the slots and is the slot in which the center of the Gaussian lies. (c) In the limit the slots seem to become infinitely narrow and becomes the Dirac delta function .

In the classical limit, , one can distinguish two cases: (1) If the inaccuracy scales linearly with , i.e. , the discreteness remains. (2) If scales slower than , i.e.  but still , then the slots seem to become infinitely narrow. Pictorially, the real space length of the eigenvalue axis, representing the possible outcomes , is limited in any laboratory, e.g., by the size of the observation screen after a Stern-Gerlach magnet, whereas the number of slots grows as . Then, in the limit , the discrete Kronecker delta becomes the Dirac delta function,

(1.26)

which is shown in Figure 1.4(c).

Now we have to focus on the question in which sense coarse-grained von Neumann measurements disturb the spin coherent state. Let

(1.27)

denote the projector onto the slot with the set of all belonging to . Then is almost (the zero vector ) for all coherent states lying inside (outside) the slot, respectively:

(1.28)

This means that the reduced (projected) state is essentially the state before the measurement or projected away. If is centered well inside the slot, the above relation holds with merely exponentially small deviation. Only in the cases where is close to the border between two slots, the measurement is invasive and disturbs the state. Presuming that the measurement times and/or slot positions chosen by the observer are statistically independent of the (initial) position of the coherent state, a significant disturbance happens merely in the fraction of all measurements. This is equivalent to the already assumed condition . Therefore, fuzzy measurements of a spin coherent state are almost always non-invasive such as in any macrorealistic theory, in particular classical Newtonian physics. Small errors may accumulate over many measurements and eventually there might appear deviations from the classical time evolution. This, however, is unavoidable in any explanation of classicality gradually emerging out of quantum theory.2

Hence, at the coarse-grained level the physics of the (quantum) spin system can completely be described by a “new” formalism, utilizing an initial (classical) spin vector at time , pointing in the ()-direction with length , where , and a (Hamilton) function

(1.29)

At any time the probability that the spin vector’s -component is in slot is given by , eq. (1.25), as if the time evolution of the spin components () is given by the Poisson brackets,

(1.30)

and measurements are non-invasive. Only the term in eq. (1.29) governs the time evolution and the solutions correspond to a rotation around the -axis. The spin vector at time points in the ()-direction where and are the same as for the spin coherent state and the probability of measurement outcomes is given by , eq. (1.26).

This is classical (Newtonian) mechanics of a single spin emerging from quantum physics.

1.4 The quantum-to-classical transition for an arbitrary spin state

Now we demonstrate that the time evolution of any spin- quantum state becomes classical under the restriction of coarse-grained measurements. At all times any (pure or mixed) spin- density matrix can be written in the quasi-diagonal form [4]

(1.31)

with ddd the infinitesimal solid angle element and a not necessarily positive real function with the normalization d.

The probability for an outcome in a measurement in the state (1.31) is given by

(1.32)

where is written in eq. (1.22). At the coarse-grained level of classical physics only the probability for a slot outcome can be measured, i.e.

(1.33)

with from eq. (1.27). Inserting eq. (1.31), we get Trd. Using eq. (1.28), , and Tr, this can be well approximated by

(1.34)

where is the region between two circles of latitude at polar angles and corresponding to the slot (Figure 1.5). We will show that can be obtained from a positive probability distribution of classical spin vectors. Consider the well known -function [2, 3]

(1.35)
Figure 1.5: Given a coarse-grained measurement of the spin’s -component (), one can compute the probability that the outcome will be a certain slot via integration of the quantum state’s -function over the region . This is the region between two circles of latitude corresponding to the slot of eigenstates. The coarse-graining condition ensures that alternatively the integration can be carried out over the state’s -function, representing the probability distribution of an ensemble of classical spins.

with ddd and the angle between the directions and . In the case of large spins the factor in the integrand is sharply peaked around vanishing relative angle and significant contributions arise only from regions where . The normalization factor in eq. (1.35) is the inverse size of the solid angle element for which the integrand contributes significantly and makes normalized: d.

The distribution is positive because it is, up to a normalization factor, the expectation value of the state :

(1.36)

For fuzzy measurements with (angular) inaccuracy , which is equivalent to , the probability for having an outcome can now be expressed only in terms of the positive distribution :

(1.37)

Figure 1.5 shows the integration region over which and have to be integrated. The approximate equivalence of eqs. (1.34) and (1.37) is verified by substituting eq. (1.35) into (1.37) and is not accurately fulfilled for quantum states directly at a slot border.3

Note that is a mere mathematical tool and not experimentally accessible in coarse-grained measurements. Operationally, because of an averaged version of , denoted as , is used by the experimenter to describe the system. Mathematically, this function is obtained by integrating over solid angle elements corresponding to the actual measurement inaccuracy. In the classical limit, without the “magnifying glass”, the regions given by the experimenter’s resolution become “points” on the sphere where is defined.

Thus, under coarse-grained measurements, a full description of an arbitrary quantum spin state is provided by an ensemble of classical spins with a positive probability distribution.

In other words, there exists a hidden variable description. The time evolution of the general state (1.31) is determined by (1.6). In the classical limit it can be described by an ensemble of classical spins characterized by the initial distribution (), where each spin is rotating according to the Hamilton function (1.29). From eq. (1.37) one can see that for the non-invasiveness at the classical level it is the change of the distribution () which is important and not the change of the quantum state or equivalently itself. In fact, upon a fuzzy measurement the state is reduced to one particular state depending on the outcome ,

(1.38)

with the corresponding (normalized) functions , and . The reduction to happens with probability , which is given by eq. (1.34) or (1.37). Whereas the -function can change dramatically upon reduction, is (up to normalization) approximately the same as the original in the region . Thus,

(1.39)

Therefore, at the coarse-grained level the distribution () of the reduced state after the measurement can always be understood approximately as a subensemble of the (classical) distribution before the measurement.

Effectively, the measurement only reveals already existing properties in the mixture and does not alter the subsequent rotation of the individual classical spins.

The disturbance at the slot borders at that level is quantified by how much differs from a function which is (up to normalization) within and zero outside. One may think of dividing all quantum states and their -distributions into two extreme classes: The ones which show narrow pronounced regions of size comparable to individual coherent states and the ones which change smoothly over regions larger or comparable to the slot size. The former can be highly disturbed but in an extremely rare fraction of all measurements. The latter is disturbed in general in a single measurement but to very small extent, as the weight—in terms of the -distribution—on the slot borders () is small compared to the weight well inside the slot (). (In the intermediate cases one has a trade-off between these two scenarios.) The typical fraction of these weights is . In any case, classicality arises with overwhelming statistical weight. In the next chapter we will introduce measurements with smooth borders (POVM) and therefore reduce the disturbance dramatically, even for states near a slot border.

Finally, we want to point out explicitly: The angular resolution which is necessary to see the quantumness, i.e. the superposition character, of a given quantum state is of the order of . In other words, it is necessary to be able to distinguish at least of the order of different measurement outcomes. For a macroscopic object, , it would be necessary to resolve different measurement outcomes. If this precision cannot be met, macrorealism emerges out of quantum physics for the rotation Hamiltonian.

1.5 An alternative derivation

For the sake of completeness we now present an alternative way to derive that classicality emerges under coarse-grained measurements. Again, we consider the totally mixed state (1.5) and the time evolution (1.7). We remind that this allows to violate the Leggett-Garg inequality if sharp measurements can be performed. Now we are interested in the probability for obtaining the results at time and at in measurements of the spin operator’s -component —in analogy to [75], where a generalized singlet state and correlations in space are considered. This probability can be written as

(1.40)

i.e. as the probability that is obtained at times the probability that is the result at given at . Let denote the projector onto . Using that , we have Tr, reflecting the fact that all of the eigenstates are equally probable in a maximally mixed state. If was obtained at , the state is reduced to . Then, with and , we have Tre. Therefore,

(1.41)

To continue we perform a discrete Fourier transform with “frequencies”  and :

(1.42)

where the four rotations were expressed as a single rotation e around a vector about an angle of the size and the trace was evaluated in the basis where is diagonal. Following firmly Reference [75], it is enough to treat the rotation matrices as as if , since does not affect the geometrical meaning. If one equates the product of the four matrices in (1.42) to e (with the vector of Pauli matrices), then one obtains the dependence of on , and :

(1.43)

Due to noise or coarse-graining of measurements, respectively, the high frequency components () are not experimentally observable. Because of high frequency components correspond to a sharp measurement resolution . The low frequency limit () of the expression for in the first non-trivial order reads

(1.44)

Now consider a classical spin vector at time with length , where . In analogy to the quantum case, the probability for measuring the classical spin vector’s -component as at and at is

(1.45)

The first probability is given by , where is the unit vector in -direction and is Dirac’s delta function. The classical Newtonian time evolution of the spin components () is given by the Poisson brackets (1.30), where the (classical) Hamilton function is given by (1.29). As is a constant of motion, only the term governs the time evolution. As we already know, the solutions correspond to a rotation around the -axis. Thus, the spin at reads , with the rotation matrix about the -axis by an angle . But for the scalar product in it does not matter whether we rotate around by or around by . Hence, we can write

(1.46)

where is the unit vector with polar angle and azimuthal angle . The remaining probability is

(1.47)

i.e. it depends only on the initial spin and the elapsed time but not on anything that happened at because of the non-invasiveness of classical measurements. In other words, there is no reduction at and the condition “ was the outcome at ” is unnecessary. Here, is the unit vector with the polar angle and azimuthal angle . Hence, eq. (1.45) becomes

(1.48)

with . The Fourier transform reads

(1.49)

with . This implies

(1.50)

where is the angle between and , in total agreement with the low frequency limit (1.44). The classical correlation must be averaged over all possible initial directions of . Mimicking the mixed quantum state, the distribution is isotropic and thus

(1.51)

with d the infinitesimal solid angle element, the vector in as the polar integration axis and . This is the limiting value of the quantum correlation (1.42). We note that both the quantum and the classical correlation for measurements of a rotating spin in time, eqs. (1.42) and (1.51), have a similar form as in the case of a generalized singlet state and measurements in space [75].

The quantum to classical transition. Let us now evaluate under which circumstances the quantum correlation (1.42) becomes classical (1.51), i.e.

(1.52)

First, we note that the allowed frequencies and are independent and thus have to be small in the same order, i.e. , whereas higher frequencies are cut off. In the following, for the sake of a short and intuitive notation, we consider all quantities as positive and ignore any factors of the order of 1 whenever we write . Comparing the exact expression for , obtained from eq. (1.43), with the low frequency classical limit , eq. (1.50), the leading order of the error is :

(1.53)

Since due to eq. (1.50), we have

(1.54)

Thus, is fulfilled if and only if , or equivalently, . Two formal conditions—(i) for the arguments in numerators and (ii) for the denominators in eqs. (1.42) and (1.51), respectively—have to be met to guarantee :

(1.55)
(1.56)

Using eq. (1.54), the left-hand side of condition (1.55) becomes