**Abstract**Towards the end of 2011, Pusey, Barrett and Rudolph (PBR) derived a theorem that aimed to show that the quantum state must be ontic (a state of reality) in a broad class of realist approaches to quantum theory. This result attracted a lot of attention and controversy. The aim of this review article is to to review the background to the PBR Theorem, to provide a clear presentation of the theorem itself, and to review related work that has appeared since the publication of the PBR paper. In particular, this review: - Explains what it means for the quantum state to be ontic or epistemic (a state of knowledge). - Reviews arguments for and against an ontic interpretation of the quantum state as they existed prior to the PBR Theorem. - Explains why proving the reality of the quantum state is a very strong constraint on realist theories in that it would imply many of the known no-go theorems, such as Bell's Theorem and the need for an exponentially large ontic state space. - Provides a comprehensive presentation of the PBR Theorem itself, along with subsequent improvements and criticisms of its assumptions. - Reviews two other arguments for the reality of the quantum state: the rst due to Hardy and the second due to Colbeck and Renner, and explains why their assumptions are less compelling than those of the PBR Theorem. - Reviews subsequent work aimed at ruling out stronger notions of what it means for the quantum state to be epistemic and points out open questions in this area. The overall aim is not only to provide the background needed for the novice in this area to understand the current status, but also to discuss often overlooked subtleties that should be of interest to the experts.

**1 Introduction**In 1964, John Bell fundamentally changed the way that we think about quantum theory [1]. Shimony famously referred to tests of Bell's Theorem as \experimental metaphysics" [2], but I disagree with this characterization. What Bell's Theorem really shows us is that the foundations of quantum theory is a bona de eld of physics, in which questions are to be resolved by rigorous argument and experiment, rather than remaining the subject of open-ended debate. In other words, it is a mistake to crudely divide quantum theory into its practical part and its interpretation, and to think of the latter as metaphysics, experimental or otherwise. In the wake of Bell's Theorem, the study of entanglement and nonlocality has become a mainstream field of physics, particularly in light of its practical applications in quantum information science, but Bell's broader lesson|that the interpretation of quantum theory should be approached as a rigorous science|has rather been missed. This is nowhere more evident than in debates about the status of the quantum state. The question of just what type of thing the quantum state, or wavefunction, represents, has been with us since the beginnings of quantum theory. The likes of de Broglie and Schrodinger initially wanted to view the wavefunction as a real physical wave, just like the waves of classical eld theory, with perhaps some additional structure to account for particle-like or \quantum" properties [3]. In contrast, following Born's introduction of the probability rule [4], the Copenhagen interpretation advocated by Bohr, Heisenberg, Pauli et. al. came to view the wavefunction as a \probability wave" and denied the need for a more fundamental reality to underlie it [5]. In modern terms, most realist interpretations of quantum theory; such as many-worlds [6{8], de Broglie-Bohm theory [9{12], spontaneous collapse theories [13, 14], and modal interpretations [15]; view the wavefunction as part of reality, whereas those that follow more Copenhagenish lines [16{26] tend to view it as a representation of knowledge, information, or belief. The big advantage of the latter view is that the notorious collapse of the wavefunction can be explained as the eect of acquiring new information, no more serious than the updating of classical probabilities in the light of new data, rather than as an anomalous physical process that needs to be eliminated or explained away. The question then is whether this is a necessary dichotomy. Is the only way to avoid having this weird multidimensional object as part of reality to give up on reality altogether, or can we reach a compromise in which there is a well-founded reality, but one in which the wavefunction only represents knowledge? This seems like a question that is ripe for attacking with the kind of conceptual rigour that Bell brought to nonlocality, and indeed Pusey, Barrett and Rudolph (PBR) have recently proven a theorem to the eect that the wavefunction must be ontic (i.e. a state of reality), as opposed to epistemic (i.e. a state of knowledge) in a broad class of realist approaches to quantum theory [27]. Since then, there has been much discussion and criticism of the PBR Theorem in both formal [28{39] and informal venues [40{49], as well as a couple of attempts to derive the same conclusion as PBR from dierent assumptions [50, 51]. The PBR Theorem and its successors all employ auxiliary assumptions of varying degrees of reasonableness. Without these assumptions, it has been shown that the wavefunction may be epistemic [52]. Therefore, there has also been subsequent work aimed at ruling out stronger notions of what it means for the wavefunction to be epistemic, without using such auxiliary assumptions [53{58]. The aim of this review article is to provide the background necessary for understanding these results, to provide a comprehensive presentation and criticism of them, and to explain their implications. One of the most intriguing things about proving that the wavefunction must be ontic is that it would imply a large number of existing no-go results, including Bell's Theorem [1] and excess baggage theorems [59{61](i.e. showing that the size of the ontic state space must be innite and must scale exponentially with the number of systems). Therefore, even apart from its foundational signicance, proving the reality of the wavefunction could potentially provide a powerful unication of the known no-go theorems, and may have applications in quantum information theory. My aim is that this review should be accessible to as wide an audience as possible, but I have made three decisions about how to present the material that make my treatment somewhat more involved than those found elsewhere in the literature. Firstly, I adopt rigorous measure theoretic probability theory. It is common in the literature to specialize to nite sample spaces or to adopt a less rigorous approach to continuous spaces, which basically involves proving all results as if you were dealing with smooth and continuous probability densities and then hoping everything still works when you throw in a bunch of Dirac delta functions. Although a measure theoretic approach may reduce accessibility, there are important reasons for adopting it. It would be odd to attempt to prove the reality of the wavefunction within a framework that does not admit a model in which the wavefunction is real in the rst place. Since the wavefunction involves continuous parameters, this means that there is no option of specializing to nite sample spaces. Furthermore, there are several special cases of interest for which the optimistic nonrigorous approach simply does not work, including the case where the wavefunction, and only the wavefunction, is the state of reality. Therefore, in order to cover all the cases of interest, there is really no option other than taking a measure theoretic approach. As an aid to accessibility, I outline how the main denitions and arguments specialize to the case of a nite sample space, which should be sucient for those who do not wish to get embroiled in the technical details. Secondly, it is common in the literature to assume that we are interested in modelling all pure states and all projective measurements on some nite dimensional Hilbert space, and to specialize results to that context. However, some results apply equally to subsets of states and measurements, which I call fragments of quantum theory. In addition, it is known that some fragments of quantum theory, have natural models in which the wavefunction is epistemic [62{64]. Therefore, I think it is important to emphasize the minimal structures in which the various results can be proved, rather than just assuming that we are trying to model all states and measurements on some Hilbert space. The third presentation decision concerns my treatment of preparation contextuality (see x 5.3 for the formal denition). The main issue we are interested in is whether pure quantum states must be ontic, since it is uncontroversial that mixed states can at least sometimes represent lack of knowledge about which of a set of pure states was prepared. It is common in the literature to assume that each pure quantum state is represented by a unique probability measure over the possible states of reality, but I do not make this assumption. In a preparation contextual model, dierent methods of preparing a quantum state may lead to dierent probability measures. In fact, this must occur for mixed states [65], so it seems sensible to allow for the possibility that it might occur for pure states as well. In addition, some of the intermediate results to be discussed hold equally well for mixed states, but this can only be established by adopting denitions that are broad enough to encompass mixed states, which are necessarily preparation contextual. These three presentation decisions mean that the denitions, statements of results, and proofs that appear in this review often dier from those in the existing literature. Generally, this is just a matter of making a few obvious generalizations, without substantively changing the ideas. For this reason, I do not explicitly point out where such generalizations have been made. The review is divided into three parts. - Part I is a general review of the distinction between ontic and epistemic interpretations of the quantum state. It discusses the arguments that had been given for ontic and epistemic interpretations of the wavefunction prior to the discovery of the PBR Theorem. My aim in this part is to convince you that there is some merit to the epistemic interpretation and that previous arguments for the reality of the quantum state are unconvincing. In this part, I also give a formal denition of the class of models to which the PBR Theorem and related results apply, and dene what it means for the quantum state to be ontic or epistemic within this class of models. Following this, I give a detailed discussion of the other no-go theorems that would follow as corollaries of proving the reality of the wavefunction. - Part II reviews the three theorems that attempt to prove the reality of the wavefunction: the PBR Theorem, Hardy's Theorem, and the Colbeck-Renner Theorem. The treatment of the PBR Theorem is the most detailed of the three, since it has attracted the largest literature and has been subject to the largest amount of confusion and criticism. In my view, it makes the strongest case of the three theorems for the reality of the wavefunction, although it is still not bulletproof, so I go to some lengths to sort the silly criticisms from the substantive ones. The assumptions behind the Hardy and Colbeck-Renner Theorems receive a more critical treatment, but the theorems are still presented in detail because they are interestingly related to other arguments about realist interpretations of quantum theory. - Part III deals with attempts to go beyond the rigid distinction between epistemic and ontic interpretations of the wavefunction by positing stronger constraints on epistemic interpretations. One of the aims of doing this is to remove the problematic auxiliary assumptions needed to prove the three main theorems, whilst still arriving at a conclusion that is morally similar. This part is shorter than the other two and mostly just summarizes the known results without proof. The reason for this is that many of the results are only preliminary and will likely be superseded by the time this review is published. My main aim in this part is to point out the most promising directions for future research. Arxiv