Quantum superposition states are behind many of the curious phenomena exhibited by quantum systems, including Bell non-locality, quantum interference, quantum computational speed-up, and the measurement problem. At the same time, many qualitative properties of quantum superpositions can also be observed in classical probability distributions leading to a suspicion that superpositions may be explicable as probability distributions over less problematic states; that is, a suspicion that superpositions are epistemic. Here, it is proved that, for any quantum system of dimension $$d>3$$ , this cannot be the case for almost all superpositions. Equivalently, any underlying ontology must contain ontic superposition states. A related question concerns the more general possibility that some pairs of non-orthogonal quantum states $$|\psi \rangle ,|\phi \rangle $$ could be ontologically indistinct (there are ontological states which fail to distinguish between these quantum states). A similar method proves that if $$|\langle \phi |\psi \rangle |^{2}\in (0,\frac{1}{4})$$ , then $$|\psi \rangle ,|\phi \rangle $$ must approach ontological distinctness as $$d\rightarrow \infty $$ . The robustness of these results to small experimental error is also discussed.
Quantum superposition states are behind many of the curious phenomena exhibited by quantum systems, including Bell non-locality, quantum interference, quantum computational speed-up, and the measurement problem. At the same time, many qualitative properties of quantum superpositions can also be observed in classical probability distributions leading to a suspicion that superpositions may be explicable as probability distributions over less problematic states; that is, a suspicion that superpositions are epistemic. Here, it is proved that, for any quantum system of dimension d>3 , this cannot be the case for almost all superpositions. Equivalently, any underlying ontology must contain ontic superposition states. A related question concerns the more general possibility that some pairs of non-orthogonal quantum states ∣ψ⟩,∣ϕ⟩ could be ontologically indistinct (there are ontological states which fail to distinguish between these quantum states). A similar method proves that if ∣⟨ϕ∣ψ⟩∣2∈(0,41) , then ∣ψ⟩,∣ϕ⟩ must approach ontological distinctness as d→∞ . The robustness of these results to small experimental error is also discussed.
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
Lying at the heart of quantum mechanics, the notion of quantum superposition causes enormous difficulty in understanding the problem of quantum measurement, which concerns the Einstein-Bohr debate on the conceptual foundations of the quantum theory. According to Bell’s theorem, Einstein’s viewpoint in his debate with Bohr seems to be wrong. In the present paper, a new principle (referred to as the general principle of measurements) is introduced and proved as a mathematical theorem. Based on this principle, various forms of quantum superposition (including quantum entanglement) and the corresponding experiments are scrutinized. The main findings are as follows. (a) Einstein’s viewpoint is correct and Bell’s theorem is problematic. (b) Measurement outcomes of any actual experiment with individual microscopic objects described in a form of quantum superposition, in particular, the experimental results of testing Bell inequalities, are all erroneously explained. (c) Bell inequalities failed to capture the essence of the Einstein-Bohr debate, i.e., whether quantum superposition is legitimate for describing individual microscopic objects. (d) All kinds of quantum superposition violate the general principle of measurements and hence are illegitimate. (e) Quantum mechanics can be completed by using disjunction (“or”) as the logical relation between the orthonormal vectors that span an arbitrarily given Hilbert space for describing a single microscopic object, and the mathematical setting will remain essentially unchanged; hidden-variable theories are irrelevant to the real world. (f) After completing quantum mechanics as above, the difficulty in understanding the problem of quantum measurement disappears naturally. Implications of the findings are also discussed.
Formalizing an old desire of Einstein, ``$\ensuremath{\psi}$-epistemic theories'' try to reproduce the predictions of quantum mechanics, while viewing quantum states as ordinary probability distributions over underlying objects called ``ontic states.'' Regardless of one's philosophical views about such theories, the question arises of whether one can cleanly rule them out by proving no-go theorems analogous to the Bell inequality. In the 1960s, Kochen and Specker (who first studied these theories) constructed an elegant $\ensuremath{\psi}$-epistemic theory for Hilbert space dimension d=2, but also showed that any deterministic $\ensuremath{\psi}$-epistemic theory must be ``measurement contextual'' in dimensions 3 and higher. Last year, the topic attracted renewed attention, when Pusey, Barrett, and Rudolph (PBR) showed that any $\ensuremath{\psi}$-epistemic theory must ``behave badly under tensor product.'' In this paper, we prove that even without the Kochen-Specker or PBR assumptions, there are no $\ensuremath{\psi}$-epistemic theories in dimensions $d\ensuremath{\ge}3$ that satisfy two reasonable conditions: (1) symmetry under unitary transformations and (2) ``maximum nontriviality'' (meaning that the probability distributions corresponding to any two nonorthogonal states overlap). This no-go theorem holds if the ontic space is either the set of quantum states or the set of unitaries. The proof of this result, in the general case, uses some measure theory and differential geometry. On the other hand, we also show the surprising result that without the symmetry restriction, one can construct maximally nontrivial $\ensuremath{\psi}$-epistemic theories in every finite dimension d.
Euclidean theorems are indisputable in flat spaces, but do not hold in curved spaces. Likewise, Bell’s Theorem is true for jointly distributed variables in Kolmogorov probability spaces. Yet, quantum spin variables are not jointly distributed and cannot coexist in Kolmogorov spaces. They have different qualities and operate by different rules. Therefore, Bell’s Theorem does not entail that quantum theory is non-local. The question remains: what is the origin of quantum contextuality? Other theories (not quantum theory) need nonlocality or super-determinism to make similar predictions, because they cannot violate Bell-type inequalities, but why is quantum theory different? The answer is found in the analysis of quantum superposition, in the context of a much older debate about the ontology of linear wave superposition.
Quantum theory describes our universe incredibly successfully. To our classically-inclined brains, however, it is a bizarre description that requires a reimagining of what fundamental reality, or “ontology”, could look like. This thesis examines different ontological features in light of the success of quantum theory, what it requires, and what it rules out. While these investigations are primarily foundational, they also have relevance to quantum information, quantum communication, and experiments on quantum systems. The way that quantum theory describes the state of a system is one of its most unintuitive features. It is natural, therefore, to ask whether a similarly strange description of states is required on an ontological level. This thesis proves that almost all quantum superposition states for d > 3 dimensions must be real—that is, present in the ontology in a well-defined sense. This is a strong requirement which prevents intuitive explanations of the many quantum phenomena which are based on superpositions. A new theorem is also presented showing that quantum theory is incompatible with macro-realist ontologies, where certain physical quantities must always have definite values. This improves on the Leggett-Garg argument, which also aims to prove incompatibility with macro-realism but contains loopholes. Variations on both of these results that are error-tolerant (and therefore amenable to experimentation) are presented, as well as numerous related theorems showing that the ontology of quantum states must be somewhat similar to the quantum states themselves in various specific ways. Extending these same methods to quantum communication, a simple proof is found showing that an exponential number of classical bits are required to communicate a linear number of qubits. That is, classical systems are exponentially bad at storing quantum data. Causal influences are another part of ontology where quantum theory demands a revision of our classical notions. This follows from the outcomes of Bell experiments, as rigorously shown in recent analyses. Here, the task of constructing a native quantum framework for reasoning about causal influences is tackled. This is done by first analysing the simple example of a common cause, from which a quantum version of Reichenbach’s principle is identified. This quantum principle relies on an identification of quantum conditional independence which can be defined in four ways, each naturally generalising a corresponding definition for classical conditional independence. Not only does this allow one to reason about common causes in a quantum experiments, but it can also be generalised to a full framework of quantum causal models (mirroring how classical causal models generalise Reichenbach’s principle). This new definition of quantum causal models is illustrated by examples and strengthened by it’s foundation on a robust quantum Reichenbach’s principle. An unusual, but surprisingly fruitful, setting for considering quantum ontology is found by considering time travel to the past. This provides a testbed for different ontological concepts in quantum theory and new ways to compare classical and quantum frameworks. It is especially useful for comparing computational properties. In particular, time travel introduces non-linearity to quantum theory, which brings (sometimes implicit) ontological assumptions to the fore while introducing strange new abilities. Here, a model for quantum time travel is presented which arguably has fewer objectionable features than previous attempts, while remaining similarly well-motivated. This model is discussed and compared with previous quantum models, as well as with the classical case. Together, these threads of investigation develop a better understanding of how quantum theory affects possible ontologies and how ontological prejudices influence quantum theory.
What does quantum physics tell us about the nature of reality, specifically the parts of reality we do not directly perceive called hidden variables? One may think it could tell us a lot because of our enhanced technological sensing abilities that delve into the realms that quantum physics covers so well. Surprisingly, it seems to surround us in a deeper mystery rather than reveal more of nature's secrets. It seems that we cannot escape from philosophical consideration when dealing with what is hidden in quantum physics. In Part I we will look at how Epistemology and Ontology bear upon Hidden Variables. In Part II we will consider Hidden Variables in the light of Contextuality, and Non-Classicality. Inevitably questions of subjectivity and objectivity arise in dealing with states of observation. How should we think about these states? Perhaps it is a question of the meaning associated with our knowledge of a state-that is, a question of ontology or epistemology. The issue of ontic and epistemic states is particularly important when considering hidden variables in quantum physics because, as one may argue, the interpretation of quantum states as either ontic or epistemic will naturally lead to different assumptions about how reality is constructed; if it is constructed or not. It also raises the question of what attributes we are able to observe simultaneously and that brings contextuality into the discussion. If it turns out that reality is constructed contextually what does that imply about ontological realism? If on the other hand reality is constructed non-contextually what does that imply about ontological realism? Many implications can arise when considering these questions from a quantum physical point of view. In this paper I shall discuss how quantum physics provides some answers to these questions by considering quantum physical states and their measurements.