We consider ontological models of a quantum system, assuming that not all probability distributions over the space $Λ$ of ontic states are preparable, only those belonging to a certain set C. We assume further that every POVM with a finite value space can be measured and that for every density matrix there exists a distribution in C whose outcome statistics is given by the density matrix. We show that this mapping from C to the set of density matrices must be many-to-one, that is, that there must be empirically indistinguishable distributions in C. This shows that there must be limitations to knowledge in the sense of facts in nature that cannot be discovered empirically.
We consider ontological models of a quantum system, assuming that not all probability distributions over the space Λ of ontic states are preparable, only those belonging to a certain set C. We assume further that every POVM with a finite value space can be measured and that for every density matrix there exists a distribution in C whose outcome statistics is given by the density matrix. We show that this mapping from C to the set of density matrices must be many-to-one, that is, that there must be empirically indistinguishable distributions in C. This shows that there must be limitations to knowledge in the sense of facts in nature that cannot be discovered empirically.
Ontological models are attempts to quantitatively describe the results of a probabilistic theory, such as Quantum Mechanics, in a framework exhibiting an explicit realism-based underpinning. Unlike either the well known quasi-probability representations, or the "r-p" vector formalism, these models are contextual and by definition only involve positive probability distributions (and indicator functions). In this article we study how the ontological model formalism can be used to describe arbitrary statistics of a system subjected to a finite set of preparations and measurements. We present three models which can describe any such empirical data and then discuss how to turn an indeterministic model into a deterministic one. This raises the issue of how such models manifest contextuality, and we provide an explicit example to demonstrate this. In the second half of the paper we consider the issue of finding ontological models with as few ontic states as possible.
In this paper, we explore realist models of quantum theory that does not fit into the standard definitions of ontological models. The models here go beyond standard definition of ontological models in the sense that quantum states do not correspond to distributions over the ontic state space and a system prepared in a quantum state is not in an ontic state. Instead, a system in a quantum state is always in a process, i.e. moving around in the ontic state space. Also, quantum measurement outcomes are not direct measurement of the ontic state, but depend probabilistically on the entire path the system takes during the measurement process. Consequently, we explain how, in our model, quantum states can neither be classified as ontic nor epistemic in the sense of knowledge about an underlying reality. In our model, quantum probabilities describes our (objective) knowledge about measurement outcomes. We also look at two hybrid models where either the preparation or measurement do follow the definitions in standard ontological models. Lastly, we propose a form of generalised ontological model that reduces to the standard PBR model when the underlying process reduces to a point in ontic space.
Certain concrete "ontological models" for quantum mechanics (models in which measurement outcomes are deterministic and quantum states are equivalent to classical probability distributions over some space of `hidden variables') are examined. The models are generalizations of Kochen and Specker's such model for a single 2-dimensional system - in particular a model for a three dimensional quantum system is considered in detail. Unfortunately, it appears the models do not quite reproduce the quantum mechanical statistics. They do, however, come close to doing so, and in as much as they simply involve probability distributions over the complex projective space they do reproduce pretty much everything else in quantum mechanics. The Kochen-Specker theorem is examined in the light of these models, and the rather mild nature of the manifested contextuality is 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
We present a new psi-ontology theorem demonstrating that the quantum wave function is ontic (real) rather than epistemic (representing knowledge) in single-world unitary quantum theories (SUQTs). By leveraging a protocol of repeated reversible measurements on a single quantum system, we show that any two distinct quantum states produce different statistical distributions of (erased) measurement outcomes. This theoretical distinguishability implies that different quantum states correspond to different physical realities, supporting the ontic nature of the quantum state. Unlike previous psi-ontology theorems, such as the Pusey-Barrett-Rudolph theorem, our proof relies solely on the unitary evolution and Born rule of SUQTs, without additional assumptions like preparation independence. This strengthens its implications for quantum foundations, particularly in restricting non-psi-ontic interpretations like QBism without assuming an underlying ontic state and its dynamics. The theorem applies to any pair of distinct states in a finite-dimensional Hilbert space, with extensions to infinite-dimensional systems, offering a robust and general argument for the reality of the quantum state.
Ontological models of quantum theory seek to realize quantum theory as a statistical theory over underlying degrees of freedom, whose measurement statistics are described by classical probability theory. Harrigan and Spekkens (2010) proposed a distinction between ontological models in which pure quantum states represent states of reality (ψ-ontic) or knowledge about reality (ψ-epistemic) depending on whether their associated probability distributions overlap, subsequently formalized into measure-theoretic form by Leifer (2014). In this thesis, I examine whether this ψ-ontic/epistemic distinction truly captures the intended difference between ontic and epistemic conceptions of pure states. I consider some mathematical alternatives relating to different notions of non-overlap between probability measures and compare them. I discuss the implications of these results for quantum ontology theorems – theorems that rule out ψ-epistemic readings of pure states under certain assumptions about the ontological model – such as the Pusey-Barrett-Rudolph Theorem (2012).