We investigate the limits of epistemic models in reproducing the empirical predictions of general quantum preparations. This involves comparing the common quantum overlap determined by the anti-distinguishability of a set of preparations with the common epistemic overlap of the probability distribution over the ontic states describing these preparations. We show that no epistemic model with non-zero common overlap can reproduce the predictions of quantum theory for mixed preparations, even for qubit systems. We explicitly provide sets of distinct mixed preparations, starting from qutrit systems, that give rise to identical quantum mixed states for which the common epistemic overlap must be zero. Finally, we demonstrate the strongest form of preparation contextuality by presenting pairs of mixed preparations that yield identical quantum mixed states while their epistemic overlap must vanish.
We investigate the limits of epistemic models in reproducing the empirical predictions of general quantum preparations. This involves comparing the common quantum overlap determined by the anti-distinguishability of a set of preparations with the common epistemic overlap of the probability distribution over the ontic states describing these preparations. We show that no epistemic model with non-zero common overlap can reproduce the predictions of quantum theory for mixed preparations, even for qubit systems. We explicitly provide sets of distinct mixed preparations, starting from qutrit systems, that give rise to identical quantum mixed states for which the common epistemic overlap must be zero. Finally, we demonstrate the strongest form of preparation contextuality by presenting pairs of mixed preparations that yield identical quantum mixed states while their epistemic overlap must vanish.
According to a recent no-go theorem [M. Pusey, J. Barrett and T. Rudolph, Nat. Phys. 8, 475 (2012)], models in which quantum states correspond to probability distributions over the values of some underlying physical variables must have the following feature: the distributions corresponding to distinct quantum states do not overlap. In such a model, it cannot coherently be maintained that the quantum state merely encodes information about underlying physical variables. The theorem, however, considers only models in which the physical variables corresponding to independently prepared systems are independent, and this has been used to challenge the conclusions of that work. Here we consider models that are defined for a single quantum system of dimension d, such that the independence condition does not arise, and derive an upper bound on the extent to which the probability distributions can overlap. In particular, models in which the quantum overlap between pure states is equal to the classical overlap between the corresponding probability distributions cannot reproduce the quantum predictions in any dimension d ≥ 3. Thus any ontological model for quantum theory must postulate some extra principle, such as a limitation on the measurability of physical variables, to explain the indistinguishability of quantum states. Moreover, we show that as d→∞, the ratio of classical and quantum overlaps goes to zero for a class of states. The result is noise tolerant, and an experiment is motivated to distinguish the class of models ruled out from quantum theory.
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.
We ask whether the operational quantum description is complete at the level of preparations: can the empirically accessible properties of a finite preparation set be reproduced exactly by a hidden-variable description, or must every such completion contain additional structure that is not operationally accessible? We formalize this through epistemic completeness, a preparation-side notion of classicality requiring exact preservation of empirical preparation-properties by the corresponding ontic quantities obtained by conditioning on the ontic state and allowing all response schemes compatible with positivity and normalization. For the canonical family of set-distinguishability tasks, we prove that every epistemically complete theory satisfies an equality: for every finite preparation set, the average pairwise distinguishability equals the average set-distinguishability. Any nonzero deviation certifies epistemic incompleteness and lower-bounds the excess ontic communication power that every ontic completion must conceal. Because unrestricted classical communication models, and more generally commuting quantum theories, are epistemically complete, every nonzero deviation also yields a quantum communication advantage and witnesses of coherence and measurement incompatibility. We formulate semidefinite-programming relaxations and see-saw lower bounds, and show that quantum theory violates the equality in both directions. The trine and tetrahedral qubit ensembles are numerically certified maximizers for the n=3 and n=4 equalities; their violations persist for arbitrarily low positive visibility and arbitrarily large leakage short of complete disclosure; and the Kochen--Specker ψ-epistemic model exactly saturates the hidden ontic excess for these maximal positive violations. Numerics suggest that the maximal positive deviation increases with the number of preparations.
Maximally ψ−epistemic models cannot explain gambling with two qubits
We investigate the minimal proof for ruling out maximally ψ−epistemic interpretations of quantum theory, in which the indistinguishable nature of two quantum states is fully explained by the epistemic overlap of their corresponding distributions over ontic states. To this end, we extend the standard notion of epistemic overlap by considering a penalized distinguishability game involving two states and three possible answers, named as Quantum Gambling. In this context, using only two pure states and their equal mixture, we present an experimentally robust no-go theorem for maximally ψ−epistemic models, showing that qubit states achieve the maximal difference between quantum and epistemic overlaps.
We study the extent to which ψ-epistemic models for quantum measurement statistics-models where the quantum state does not have a real, ontic status-can explain the indistinguishability of nonorthogonal quantum states. This is done by comparing the overlap of any two quantum states with the overlap of the corresponding classical probability distributions over ontic states in a ψ-epistemic model. It is shown that in Hilbert spaces of dimension d≥4, the ratio between the classical and quantum overlaps in any ψ-epistemic model must be arbitrarily small for certain nonorthogonal states, suggesting that such models are arbitrarily bad at explaining the indistinguishability of quantum states. For dimensions d=3 and 4, we construct explicit states and measurements that can be used experimentally to put stringent bounds on the ratio of classical-to-quantum overlaps in ψ-epistemic models, allowing one in particular to rule out maximally ψ-epistemic models more efficiently than previously proposed.
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.