Mermin once summarized a popular attitude towards quantum theory as ‘Shut up and calculate.’ We suggest a different slogan: ‘Shut up and contemplate!’ – Lucien Hardy and Rob Spekkens, 2010 This chapter is dedicated to the foundations of quantum theory, or as it's more fashionably called these days, quantum foundations. Here, we will use all of the things we've learned so far to probe some very deep questions: 1. What features of nature are imposed on us by quantum theory? 2. Conversely, what features of a physical theory are imposed on us by (our current understanding of) nature? 3. Which of these features are ‘properly quantum’, in the sense that they have no counterpart in any classical physical theory? We'll address these questions by looking at one of those most celebrated (and historically controversial) properties of quantum theory: quantum non-locality . First, we will give a precise definition of non-locality and prove that it exists within the theory of quantum processes and in fact already within the comparatively tiny subtheory of causal Clifford maps . Then, we will present a new process theory called spek , which has locality built right in. A remarkable thing is that the two theories of Clifford maps and spek are identical in every respect except one: the phase group of a single system. And (another spoiler alert!) it is indeed this one difference that kills the proof of non-locality that works for quantum theory. Quantum Non-locality Quantum non-locality is probably still the least understood of all the new quantum features, in both philosophical and structural terms. Our upbringing in a seemingly ‘classical’ world, and especially our undeniably corrupting ‘classical’ scientific education, tends to make us expect two things from a physical theory: 1. Realism : physical systems have real pre-existing properties, and hence the outcome of ‘measuring’ such a property is fixed in some way prior to the measurement. 2. Locality : it is impossible for one system to affect another distant system instantaneously.
Mermin once summarized a popular attitude towards quantum theory as ‘Shut up and calculate.’ We suggest a different slogan: ‘Shut up and contemplate!’ – Lucien Hardy and Rob Spekkens, 2010 This chapter is dedicated to the foundations of quantum theory, or as it's more fashionably called these days, quantum foundations. Here, we will use all of the things we've learned so far to probe some very deep questions: 1. What features of nature are imposed on us by quantum theory? 2. Conversely, what features of a physical theory are imposed on us by (our current understanding of) nature? 3. Which of these features are ‘properly quantum’, in the sense that they have no counterpart in any classical physical theory? We'll address these questions by looking at one of those most celebrated (and historically controversial) properties of quantum theory: quantum non-locality . First, we will give a precise definition of non-locality and prove that it exists within the theory of quantum processes and in fact already within the comparatively tiny subtheory of causal Clifford maps . Then, we will present a new process theory called spek , which has locality built right in. A remarkable thing is that the two theories of Clifford maps and spek are identical in every respect except one: the phase group of a single system. And (another spoiler alert!) it is indeed this one difference that kills the proof of non-locality that works for quantum theory. Quantum Non-locality Quantum non-locality is probably still the least understood of all the new quantum features, in both philosophical and structural terms. Our upbringing in a seemingly ‘classical’ world, and especially our undeniably corrupting ‘classical’ scientific education, tends to make us expect two things from a physical theory: 1. Realism : physical systems have real pre-existing properties, and hence the outcome of ‘measuring’ such a property is fixed in some way prior to the measurement. 2. Locality : it is impossible for one system to affect another distant system instantaneously.
A key aim of quantum foundations is to characterize the sense in which nature goes beyond classical physics. Understanding nonclassicality is one of our best avenues towards finding a satisfactory interpretation of quantum theory. By determining which classical principles cannot be satisfied in any empirically adequate physical theory, we begin to see which principles can be preserved, which in turn gives us insight into the ontology of the world. These insights then guide us in determining which questions to ask and which experiments to perform next. Furthermore, it is these nonclassical aspects of nature that give rise to new technologies such as the speed-ups of quantum computation or the security of quantum key distribution. \n \nThe gold standard for establishing that a phenomenon is truly nonclassical is to prove that it violates the principle of local causality or the principle of noncontextuality. Much of this thesis reports on my research relating to these two principles. This research primarily involves (i) finding new justifications for our notions of nonclassicality; (ii) refining their fundamental definitions; (iii) quantifying and characterizing their various manifestations; and (iv) finding applications where nonclassical phenomena act as resources for information processing. \n \nUltimately, all of this work on nonclassicality is woven together into a novel framework for physical theories introduced by myself, John Selby, and Rob Spekkens. Its main advantage over preexisting frameworks is that it maintains a clear distinction between which elements of a given physical theory directly describe causal processes, and which refer only to one’s inferences about causal processes. This clarifies a number of confusions in the literature which arose precisely because previous frameworks scrambled causal and inferential concepts. Furthermore, local causality and noncontextuality emerge in this framework as the assumptions that the causal and inferential structures (respectively) that are operationally observed must be respected in the underlying ontology. This work constitutes a first step in developing a new interpretation of quantum theory—the first interpretation designed to satisfy the spirit of both local causality and noncontextuality.
Quantum nonlocality has been a long-standing and fascinating problem in the foundations of quantum theory since its proof by the Bell inequality. It seems to contradict our intuitive picture of space-time and thus urges us to reconsider its foundation. With the advent of quantum information science, it has attracted renewed interest from a fundamental point of view as well as from a quantum information-theoretic one. Another fundamental question on the nature of quantum theory is how we understand what quantum states represent. (Quantum states hereafter refer to pure states.) This is also a long-standing problem since the early days of quantum theory. Roughly speaking, there are two conflicting views on this matter: one is ontology and the other is epistemology. Ontological models of quantum theory are models in which we assume some underlying reality for physical systems and require that all the predictions of quantum theory should be reproduced. The ontological models are divided into two categories, depending on which views the models are based on, ontology or epistemology. In the models based on ontology, quantum states represent the underlying reality in some sense, and different quantum states correspond to different realities. In the models based on epistemology, quantum states do not necessarily correspond to the underlying realities, and even different quantum states can be consistent with a common underlying reality. The former type of ontological models of quantum theory is called 1⁄4-ontic. The latter is called 1⁄4-epistemic. Thus, it might be rather confusing that ontological models are classified further into 1⁄4-ontic models and 1⁄4-epistemic ones. We use these terms here in accordance with the recent literature on quantum foundations. (For details of these notions in the ontological models of quantum theory, see the paper by Harrigan and Spekkens.) Quite recently, the question just mentioned has been answered by Pusey et al., and one of the above models has been refuted. It was shown that 1⁄4-epistemic models are inconsistent with the predictions of quantum theory. In other words, epistemological views on quantum states turned out to be false as long as we assume some underlying reality for physical systems. Therefore, only 1⁄4-ontic models of quantum theory are valid for describing quantum systems. This significantly advances our understanding on the nature of quantum states. It is expected that various conclusions will be drawn from this fact. In this paper, we will discuss a connection with nonlocality. The above two fundamental questions, nonlocality and the nature of quantum states, are intimately connected, and the connection was first implicitly recognised by Einstein in the early days of quantum theory. In a private correspondence with Schrodinger, Einstein constructed a fascinating argument that relates the above two. (That argument is distinctly different from the well-known EPR argument.) Recently, it was revisited and reconstructed in accordance with more rigorous notions in quantum foundations. Nonlocality is usually shown by violations of Bell inequalities. (Bell inequalities hereafter include all the subsequent developments such as the CHSH inequality.) Even if it is not clear whether the models of quantum theory are 1⁄4-ontic or 1⁄4-epistemic, we can prove nonlocality just by showing the violation of Bell inequalities. However, the above argument originated by Einstein implies that if we know the models are 1⁄4-ontic in advance (as we do now), we can prove nonlocality in a simpler way without Bell inequalities. (This argument had been known in quantum foundations before the refutation of 1⁄4-epistemic models.) Nonlocality is rigorously defined as a violation of the notion of local causality defined by Bell. Showing the violation of Bell inequalities is one way of demonstrating the violation of local causality. Since the simpler proof initiated by Einstein directly demonstrates the violation of local causality, it is a more straightforward proof of nonlocality from a conceptual point of view. In the following, we will see that this simple argument of nonlocality without Bell inequalities can be used to show multipartite nonlocality as well. Before moving onto multipartite cases, first let us define the necessary notions for presenting our result. First of all, we assume that there exists a complete specification of the properties of a physical system, which we refer to as the ontic state of that system. The ontic states represent the underlying realities and are not necessarily equal to ordinary quantum states. (In other words, the ontic states correspond to hidden variables.) Ontological models of quantum theory are models that assume ontic states and reproduce all the predictions of quantum theory. An important question is how ontic states correspond to quantum states in quantum theory. In order to frame the question accurately, we need the notion of an epistemic state, which represents the relation between them. Suppose we prepare a physical system in a quantum state 1⁄4. Although the system is uniquely specified as 1⁄4 in quantum theory, there still remains a possibility that it allows a more detailed specification in the ontic state space, in which case many different ontic states turn out to be consistent with 1⁄4. The relation between ontic states and the preparation procedure, P1⁄4, is described by a conditional probability distribution over the ontic state space, pð jP1⁄4Þ, which we refer to as an epistemic state. In general, the preparation P can be a preparation of a physical system in a mixed sate. Unless there exists a one-to-one correspondence between ontic states and quantum states, the probability distribution spreads over . (See Fig. 1.) The question of whether an ontological model is 1⁄4-ontic or 1⁄4-epistemic is concerned with the behaviour of the epistemic states for two distinct preparations of quantum states. Consider two distinct preparations of quantum states, Journal of the Physical Society of Japan 82 (2013) 055003
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. – Max Planck, 1936 The only way quantum theory allows us to interact with quantum systems is via quantum processes. Thus, the only way we can extract information about the state of a quantum system is to apply some non-deterministic process and observe which of the branches happened. For that reason, many people refer to the act of applying certain quantum processes as measurement and sometimes even as observation . A consequence of this unfortunate terminology is that one of the most touted ‘strange’ features of quantum theory is often misleadingly described as follows: ‘The mere act of observing a quantum state changes it.’ ‘Misleading’, since the concept of measurement described in the previous paragraph is far from the passive concept of observation familiar from our macroscopic world, but is rather a non-trivial process that will almost always drastically affect the quantum state. So, the mysterious aspect of quantum theory is not that ‘observation’ alters the quantum state but, rather, that it is impossible, even in principle, to ‘observe’ a quantum system in the classical sense. Given this fundamental restriction on how we can extract information from a quantum system, what can we learn about that system by means of a quantum process? The answer is, in fact, very little! First, performing a particular measurement could, for the vast majority of quantum states, yield any outcome i with a non-zero probability. In that case, obtaining an outcome i doesn't tell us much about what the state of the system was. Second, we say ‘was’, because the measurement will moreover irreversibly change the state according to the outcome. So rather than revealing the state of a system, a quantum measurement typically erases that state from history! Nonetheless, these quantum measurements are crucial to quantum theory since they constitute the only interface between us, in our classical world, and the quantum world. The great insight of the quantum computing community is that the non-deterministic changes induced by these quantum measurements are not a nuisance, but rather an extremely useful resource.
Quantum mechanics is usually defined in terms of some loosely connected axioms and rules. Such a foundation is far from the beauty of, e.g., the `principles' underlying classical mechanics. Motivated, in addition, by notorious interpretation problems, there have been numerous attempts to modify or `complete' quantum mechanics. A first attempt was based on so-called hidden variables; its proponents essentially tried to expel the non-classical nature of quantum mechanics. More recent proposals intend to complete quantum mechanics not within mechanics proper but on a `higher (synthetic) level'; by means of a combination with gravitation theory (R Penrose), with quantum information theory (C M Caves, C A Fuchs) or with psychology and brain science (H P Stapp). I think it is fair to say that in each case the combination is with a subject that, per se , suffers from a very limited understanding that is even more severe than that of quantum mechanics. This was acceptable, though, if it could convincingly be argued that scientific progress desperately needs to join forces. Quantum mechanics of a closed system was a beautiful and well understood theory with its respective state being presented as a point on a deterministic trajectory in Liouville space---not unlike the motion of a classical N -particle system in its 6N -dimensional phase-space. Unfortunately, we need an inside and an outside view, we need an external reference frame, we need an observer. This unavoidable partition is the origin of most of the troubles we have with quantum mechanics. A pragmatic solution is introduced in the form of so-called measurement postulates: one of the various incompatible properties of the system under consideration is supposed to be realized (i.e. to become a fact, to be defined without fundamental dispersion) based on `instantaneous' projections within some externally selected measurement basis. As a result, the theory becomes essentially statistical rather than deterministic; furthermore there is an asymmetry between the observed and the observing. This is the point where consciousness may come in. Complemented by an introduction and several appendices, Henry Stapp's book consists essentially of three parts: theory, implications, and new developments. The theory part gives a very readable account of the Copenhagen interpretation, some aspects of a psychophysical theory, and, eventually, hints towards a quantum foundation of the brain--mind connection. The next part, `implications', summarizes some previous attempts to bridge the gap between the working rules of quantum mechanics and their possible consequences for our understanding of this world (Pauli, Everett, Bohm, Heisenberg). The last section, `new developments', dwells on some ideas about the conscious brain and its possible foundation on quantum mechanics. The book is an interesting and, in part, fascinating contribution to a field that continues to be a companion to `practical' quantum mechanics since its very beginning. It is doubtful whether such types of `quantum ontologies' will ever become (empirically) testable; right now one can hardly expect more than to be offered some consistent `grand picture', which the reader may find more or less acceptable or even rewarding. Many practicing quantum physicists, though, will remain unimpressed. The shift from synthetic ontology to analytic ontology is the foundation of the present work. This means that fundamental wholes are being partitioned into their ontologically subordinate components by means of `events'. The actual event, in turn, is an abrupt change in the Heisenberg state describing the quantum universe. The new state then defines the tendencies associated with the next actual event. To avoid infinite regression in terms of going from one state of tendencies to the next, consciousness is there to give these events a special `feel', to provide a status of `intrinsic actuality'. The brain of an alert human observer is similar in an important way to a quantum detection device: it can amplify small signals to large macroscopic effects. On the other hand, actual events are not postulated to occur exclusively in brains. They are more generally associated with the formation of records. Records are necessarily part of the total state of the universe: it is obvious that the state of the universe cannot undergo a Schrödinger dynamics and at the same time record its own history. `The full universe consists therefore of an exceedingly thin veneer of relatively sluggish, directly observable properties resting on a vast ocean or rapidly fluctuating unobservable ones.' The present ideas also bear on how the world should be seen to develop. While conventional cosmology encounters problems as to how to define the intial conditions, which would enter the governing equations of motion, here `the boundary conditions are set not at some initial time, but gradually by a sequence of acts that imposes a sequence of constraints. After any sequence of acts there remains a collection of possible worlds, some of which will be eliminated by the next act.' Connected with those acts is `meaning': there has always been some speculation about the special significance of local properties in our understanding of the world. One could argue that correlations (even the quantum correlations found, e.g., in the EPR-experiments) were as real as anything else. But also Stapp stresses the special role of locality: the `local observable properties, or properties similar to them are the natural, and perhaps exclusive, carriers of meaning in the quantum universe. From this point of view the quantum universe tends to create meaning.' This sounds like an absolute concept: meaning not with respect to something else, but defined intrinsically---not easy to digest. The role of consciousness in the developing quantum universe requires more attention. `The causal irrelevance of our thoughts within classical physics constitutes a serious deficiency of that theory, construed as a description of reality.' This is taken to be entirely different within quantum mechanics. `The core idea of quantum mechanics is to describe our activities as knowledge-seeking and knowledge-using agents.' `21st century science does not reduce human beings to mechanical automata. Rather it elevates human beings to agents whose free choices can, according to the known laws, actually influence their behaviour.' An example with respect to perception is discussed: `Why, when we look at a triangle, do we experience three lines joined at three points and not some pattern of neuron firings?' The brain `does not convert an actual whole triangle into some jumbled set of particle motions; rather it converts a concatenation of separate external events into the actualization of some single integrated pattern of neural activity that is congruent to the perceived whole triangle.' How convincing is this proposal? It is hard to tell. I think Henry Stapp did a good job, but there are tight limitations to any such endeavour. Quantum mechanics is often strange indeed, but it also gives rise to our classical world around us. For the emergence of classicality jumps and measurement projections (the basic phenomena connected with those fundamental events of choice) are not needed. Therefore, I doubt whether the explanation of the evolution of our world really allows (or requires) that much free choice. On the other hand, most scientist will agree that empirical science was not possible without free will: we could not ask independent questions if this asking was part of a deterministic trajectory. The fact that the result of a quantum measurement is indeterminate (within given probabilities) does certainly not explain free will. How about the type of measurment? The experimentalist will have to assume that he can select the pertinent observable within some limits. But given a certain design the so-called pointer basis (producing stable measurement results) is no longer a matter of free choice. `The main theme of classical physics is that we live in a clocklike universe.' Today it is often assumed that the universe was a big (quantum-) computer or a cellular automaton. Many would be all too happy to leave that rather restrictive picture behind. But where to go? Stapp suggests giving consciousness a prominent role: `The most profound alteration of the fundamental principles was to bring consciousness of human beings into the basic structure of the physical theory.' How far we are able to go in this direction will depend on the amount of concrete research results becoming available to support this view.
In This Remarkable Treatise, Professor Schafer shares his conclusions from a lifelong search for evidence -- from quantum science -- of the existence of a transcendent part of physical reality. With a view as wide as the cosmos and as infinitesimal as the electron, he combines disciplinary thought from science, philosophy, religion, and ethics to address the educated generalist and layman with a profound look at existence. When modern science adopted objectivity and experimental testing as operational principles, the ensuing advances in the physical sciences first clashed with, and then destroyed, the world view of the ruling religious mythologies. A general disorientation resulted and a schism evolved in society, disconnecting the world of facts from the world of values. In a disastrous way the covenant which all religious mythologies claimed existed between humanity and nature was broken, and it seemed to many that life itself was rendered meaningless in the process. However, twentieth-century science has discovered -- in quantum phenomena -- a part of physical reality which, in contrast to the mechanical world of classical science, has all the characteristics of a transcendent reality Now the foundation of the material world is discovered to be nonmaterial, the constituents of real things are found to be not real in the same way as the things that they make, and non-local, faster-than-light influences are found to pervade a universe whose nature is mindlike. Braving controversy, Professor Schafer concludes that the discovery of the phenomena of quantum mechanics has established a new covenant -- between the human mind and the mindlike background of the universe -- one thatprovides a home again to the homeless and meaning to seemingly pointless life. In this new understanding of the world, the universe must be assumed to have a moral as well as a physical order, and facts and values derive, again, from a single source.
Quantum theory is considered to be the most fundamental and most accurate physical theory of today. Although quantum theory is conceptually difficult to understand, its mathematical structure is quite simple. What determines this particularly simple and elegant mathematical structure? In short: Why is quantum theory as it is? Addressing such questions is the aim of investigating the foundations of quantum theory. In the past this field of research was sometimes considered as an academic subject without much practical impact. However, with the emergence of quantum information theory this perception has changed significantly and both fields started to fruitfully influence each other. Today fundamental aspects of quantum theory attract increasing attention and the field belongs to the most exciting subjects of theoretical physics. This thesis is concerned with a particular branch in this field, namely, with so-called Generalized Probabilistic Theories (GPTs), which provide a unified theoretical framework in which classical and quantum theory emerge as special cases. This is used to examine nonlocal features that help to distinguish quantum theory from alternative toy theories. In order to extend the scope of theories that can be examined with the framework, we also introduce several generalizations to the framework itself. We start in Chapter 1 with introducing the standard GPT framework and summarize previous results, based on a review paper of the author [New J. Phys. 13, 063024 (2011)]. To keep the introduction accessible to a broad readership, we follow a constructive approach. Starting from few basic physically motivated assumptions we show how a given set of observations can be manifested in an operational theory. Furthermore, we characterize consistency conditions limiting the range of possible extensions. We point out that non-classical features of single systems can equivalently result from higher dimensional classical theories that have been restricted. Entanglement and non-locality, however, are shown to be genuine non-classical features. We review features that have been found to be specific for quantum theory separably or single and joint systems. Chapter 2 incorporates results published in [J. Phys. A 47(32), pp. 1-32 (2014)] and [Proc. QPL 2011 via EPTCS vol. 95, pp. 183–192 (2012)]. The GPT framework is applied to show how the structure of local state spaces indirectly affects possible nonlocal correlations, which are global properties of a theory. These correlations are stronger than those possible in a classical theory, but happen to show different restrictions that can be linked to the structure of subsystems. We first illustrate the phenomenon with toy theories with particular local state spaces. We than show that a particular class of joint states (inner product states), whose existence depends on geometrical properties of the local subsystems, can only have correlations for a known limited set called Q1. All bipartite correlations of both, quantum and classical correlations, can be mapped to measurement statistics from such joint states. Chapter 3 shows unpublished results on entanglement swapping in GPTs. This protocol, which is well known in quantum information theory, allows to nonlocally transfer entanglement to initially unentangled parties with the help of a third party that shares entanglement with each. We review our approach published in [Proc. QPL 2011 via EPTCS vol. 95, pp. 183–192 (2012)], which mimics the joint systems' structure of quantum theory by modifying a popular toy theory known as boxworld. However, it is illustrated that this approach fails for bigger multipartite systems due to inconsistencies evoked by entanglement swapping. It turns out that the GPT framework does not allow entanglement swapping for general subsystems with two-dimensional state spaces with transitive pure states. Altering the GPT framework to allow completely globally degrees of freedom, however, enables us to construct consistent entanglement swapping for these subsystems. This construction resembles the situation in quantum theory on a real Hilbert space. A questionable assumption usually taken in the standard GPT framework is the so-called no-restriction hypothesis. It states that the measurement that are possible in a theory can be derived from the state space. In fact, this assumption seems to exist for reasons of mathematical convenience, but it seems to lack physical motivation. We generalize the GPT framework to also account for systems that do not obey the no-restriction hypothesis in Chapter 4, which presents results published in [Phys. Rev. A 87, 052131 (2013)] and [Proc. QPL 2013, to be published in EPTCS]. The extended framework includes new classes of probabilistic theories. As an example, we show how to construct theories that include intrinsic noise. We also provide a "self-dualization" procedure that requires the violation of the no-restriction hypothesis. This procedure restricts the measurement of arbitrary theories such that the theories act as if they were self-dual. Self-duality has recently gathered lots of interest, since such theories share many features of quantum theory. For example Tsirelson’s bound holds for correlations on the maximally entangled state in these theories. Finally, we characterize the maximal set of joint states that can be consistently defined for given subsystems. This generalizes the maximal tensor product of the standard GPT framework.