As a counterexample to
ψ-ontology theorems we consider a
ψ-epistemic interpretation of the wave function in the configuration space representation with a configuration space trajectory defining the ontology. This shows that
ψ-ontology theorems appear unable to prove that the wave function cannot be interpreted as epistemic. We identify the criterion used to decide if
ψ is epistemic or ontological as the misleading part. Different states having overlaps is only sufficient for being epistemic. In an epistemic interpretation the information which completely identifies the wave function also has an objective base in reality - the preparation procedure, which has to be really executed to prepare a state. Naming the
λ∈Λ "hidden variables" and considering them as part of the state of the system we also identify as misleading, because it hides the possibility that they may be visible and external to the system. This leads to further misconceptions about the consequences of the theorems.
Abstract An ontological model is termed as maximally ψ -epistemic if the overlap between any two quantum states is fully accounted for by the overlap of their respective probability distributions of ontic states. However, in the literature, there exist two different mathematical definitions (termed here as 1M ψ E and 2M ψ E) that capture the equivalent notion of maximal ψ -epistemicity. In this work, we provide three theorems to critically examine the connections between preparation non-contextuality and the aforementioned two definitions of maximal ψ -epistemicity. In theorem 1, we provide a simple and direct argument of an existing proof to demonstrate that the mixed state preparation non-contextuality implies the first definition of maximal ψ -epistemicity. In theorem 2, we prove that the second definition of maximal ψ -epistemicity implies pure-state preparation non-contextuality. If both the definitions capture the equivalent notion of maximal ψ -epistemicity, then from the aforementioned two theorems one infers that the mixed-state preparation non-contextuality implies pure-state preparation non-contextuality. But, in theorem 3, we demonstrate that the mixed-state preparation non-contextuality in an ontological model implies pure-state contextuality and vice versa. This leads one to conclude that 1M ψ E and 2M ψ E capture inequivalent notions of maximal ψ -epistemicity. The implications of our results and their connections to other no-go theorems are discussed.
Since the famous debates of Einstein & Bohr, physicists have argued about the nature of the quantum state. Is it best thought of as describing the-way-things-are out there in the world, or merely as a description of our knowledge of such things? In more recent years, this has been termed the ψ-epistemic/ψ-ontic debate, and that distinction has been given a mathematical definition within the ontological models formalism. This formalism is a framework for describing a large class of interpretations of quantum mechanics. Here we show that consideration of sequential measurements, and the fact that the quantum state changes during measurements, has been a neglected topic in this area as it places nontrivial restrictions on the structure of ψ-epistemic ontological models. \n \nWe do this by finding a general restriction on the structure of ψ-epistemic models, although not one that is strong enough to rule them out categorically. We then apply this restriction to all of the known examples of ψ-epistemic ontological models and show that they can’t represent sequential measurements. We also present a new version of the ontological models formalism, based on hidden Markov models, which we develop briefly and describe how it may be useful for either proving a no-go theorem (i.e. ‘ψ-epistemic models can’t exist’) or for learning more about the structure of ψ-epistemic models and how to construct them.
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.
The main question under investigation in this thesis was the reality of the quantum state. We approached this problem by studying a recent no-go result by Pusey, Barrett and Rudolph, namely the PBR theorem. In Section 1 we provided a short introduction to quantum theory. A few relevant examples were covered, namely decompositions of quantum states, impossibility of the ignorance interpretation and the distinguishability of pure quantum states. Section 2 developed measure theoretic tools needed in the framework of ontological models. The Radon-Nikodym theorem was proven in order to justify the use of an alternative expression of the variational distance of probability measures. In Section 3 we developed a general framework for ontological models. In short, the ontological models framework can be considered the most general framework currently available for hidden variable theories and no-go results. We used the measure theoretic tools of Section 2 to make the formalism rigorous. Original research included in Section 3 was the characterization of a property called antidistinguishability for qubit states. A necessary and sufficient condition for antidistinguishability was given. It was then proven that any set of pure qubit states that is not antidistinguishable can be made antidistinguishable by adding a single carefully chosen state into the set. In the end of Section 3, some original observations were presented. Namely, a specific connection with contextuality was proven for sometimes ψ-ontic and never ψ-ontic ontological models. However, it was then noted that these observations were not enough to exclude never ψ-ontic ontological models, a feat that would have provided a new no-go theorem on its own. Contextuality in the ontological models framework was studied in Section 4. The main results were establishing preparation contextuality from the failure of maximal ψ-epistemicity and a proof of Bell's theorem based on preparation contextuality established this way. We concluded that ψ-ontic models cannot be Bell local based on these results. Finally in Section 5 the implications of ψ-ontology were discussed. By careful analysis of the results, it was noted that accepting the reality of the quantum state exposes explanatory gaps in the formalism. Possible solutions to the challenges introduced by these explanatory gaps were discussed through interpretations of quantum mechanics and the so-called measurement problem.
Abstract Harrigan and Spekkens (Found Phys 40:125–157, 2010) provided a categorization of quantum ontological models classifying them as
ψ ψ -ontic or
ψ ψ -epistemic if the quantum state
ψ ψ describes respectively either a physical reality or mere observers’ knowledge. Moreover, they claimed that Einstein—who was a supporter of the statistical interpretation of quantum mechanics—endorsed an epistemic view of
ψ. ψ. In this essay we critically assess such a classification and some of its consequences by proposing a twofold argumentation. Firstly, we show that Harrigan and Spekkens’ categorization implicitly assumes that a complete description of a quantum system (its ontic state,
λ λ ) only concernssingle,individualsystems instantiatingabsolute,intrinsicproperties. Secondly, we argue that such assumptions conflict with some current interpretations of quantum mechanics, which employ different ontic states as a complete description of quantum systems. In particular, we will show that, since in the statistical interpretation ontic states describe ensembles rather than individuals, such a view cannot be considered
ψ ψ -epistemic. As a consequence, the authors misinterpreted Einstein’s view concerning the nature of the quantum state. Next, we will focus onrelational quantum mechanicsandperspectival quantum mechanics, which in virtue of their relational and perspectival metaphysics employ ontic states
λ λ dealing with relational properties. We conclude that Harrigan and Spekkens’ categorization is too narrow and entails an inadequate classification of the mentioned interpretations of quantum theory. Hence, any satisfactory classification of quantum ontological models ought to take into account the variations of
λ λ across different interpretations of quantum mechanics.