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 literature, there exists the 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 notion of maximal $ψ$-epistemicity. The implications of our results and their connections to other no-go theorems are discussed.
The conspiracy of ontic states responding to measurements contextually to comply with noncontextual quantum mechanical probabilities is analyzed for general ontological models. A general physical picture of ontological space structure and how ontic states are disturbed by measurement contexts is presented. A common assumption in ontological models called lambda-sufficiency is analyzed and argued to be inconsistent with measurement contextuality for ontological models with certain symmetry in ontic space with respect to quantum states and measurements.
We show that it is possible to construct a preparation non-contextual ontological model that does not exhibit "transformation contextuality" for single qubits in the stabilizer subtheory. In particular, we consider the "blowtorch" map and show that it does not exhibit transformation contextuality under the Grassmann Wigner-Weyl-Moyal (WWM) qubit formalism. Furthermore, the transformation in this formalism can be fully expressed at order $\hbar^0$ and so does not qualify as a candidate quantum phenomenon. In particular, we find that the Grassmann WWM formalism at order $\hbar^0$ corresponds to …
Generalized contextuality refers to our inability of explaining measurement statistics using a context-independent probabilistic and ontological model. On the other hand, measurement statistics can also be modeled using the framework of general probabilistic theories (GPTs). Here, starting from a construction of GPTs based on a Gleason-type theorem, we fully characterize these structures with respect to their permission and rejection of generalized (non)contextual ontological models. It follows that in any GPT construction the three insistence of (i) the no-restriction hypothesis, (ii) the ont…
Beginning with the Bell theorem, cyclic systems of dichotomous random variables have been the object of many foundational findings in quantum mechanics. Here, we ask the question: if one chooses a cyclic system "at random" (uniformly within the hyperbox of all possible systems with given marginals), what are the odds that it will be contextual? We show that the odds of contextuality rapidly tend to zero as the size of the system increases. The result is based on the Contextuality-by-Default theory, in which we do not have to assume that the systems are subject to the no-disturbance/no-signalin…
The most well-known tool for studying contextuality in quantum computation is the n-qubit stabilizer state tableau representation. We provide an extension that describes not only the quantum state, but is also outcome deterministic. The extension enables a value assignment to exponentially many Pauli observables, yet remains quadratic in both memory and computational complexity. Furthermore, we show that the mechanisms employed for contextuality and measurement disturbance are wholly separate. The model will be useful for investigating the role of contextuality in $n$-qubit quantum computation…
Contextuality, a generalization of non-locality, has been proposed as the resource that provides the computational speed-up for quantum computation. For universal quantum computation using qudits, of odd-prime dimension, contextuality has been shown to be a necessary and possibly sufficient resource. However, the role of contextuality in quantum computation with qubits remains open. The $n$-qubit stabilizer formalism, which by itself cannot provide a quantum computer super-polynomial computational advantage over a classical counterpart, is contextual. Therefore contextuality cannot be identifi…
Realist interpretations of quantum mechanics presuppose the existence of elements of reality that are independent of the actions used to reveal them. Such a view is challenged by several no-go theorems that show quantum correlations cannot be explained by non-contextual ontological models, where physical properties are assumed to exist prior to and independently of the act of measurement. However, all such contextuality proofs assume a traditional notion of causal structure, where causal influence flows from past to future according to ordinary dynamical laws. This leaves open the question of …
According to a standard view, quantum mechanics (QM) is a contextual theory and quantum probability does not satisfy Kolmogorov's axioms. We show, by considering the macroscopic contexts associated with measurement procedures and the microscopic contexts (mu-contexts) underlying them, that one can interpret quantum probability as epistemic, despite its non-Kolmogorovian structure. To attain this result we introduce a predicate language L(x), a classical probability measure on it and a family of classical probability measures on sets of mu contexts, each element of the family corresponding to a…
The ontological models framework distinguishes $ψ$-ontic from $ψ$-epistemic wavefunctions. It is, in general, quite straightforward to categorize the wave-function of a certain quantum theory. Nevertheless, there has been a debate about the ontological status of the wave-function in the statistical interpretation of quantum mechanics: is it $ψ$-epistemic and incomplete or $ψ$-ontic and complete? I will argue that the wavefunction in this interpretation is best regarded as $ψ$-ontic and incomplete.
In the framework of ontological models, the inherently nonclassical features of quantum theory always seem to involve properties that are fine tuned, i.e. properties that hold at the operational level but break at the ontological level. Their appearance at the operational level is due to unexplained special choices of the ontological parameters, which is what we mean by a fine tuning. Famous examples of such features are contextuality and nonlocality. In this article, we develop a theory-independent mathematical framework for characterizing operational fine tunings. These are distinct from cau…
A theory is universal contextual if its prediction cannot be reproduced by an ontological model satisfying both preparation and measurement noncontextuality assumptions. In this report, we first generalize the logical proofs of quantum preparation and measurement contextuality for qubit system for any odd number of preparations and measurements. Based on the logical proof, we derive testable universally non-contextual inequalities violated by quantum theory. We then propose a class of two-party communication games and show that the average success probability of winning such games is solely li…
Starting from arbitrary sets of quantum states and measurements, referred to as the prepare-and-measure scenario, an operationally noncontextual ontological model of the quantum statistics associated with the prepare-and-measure scenario is constructed. The operationally noncontextual ontological model coincides with standard Spekkens noncontextual ontological models for tomographically complete scenarios, while covering the non-tomographically complete case with a new notion of a reduced space, which we motivate following the guiding principles of noncontextuality. A mathematical criterion, c…
In this work we analyse the notion of measurement non-contextuality (MNC) and identify contextual scenarios which involve sequential measurements of only a single measurement device. We show that any non-contextual ontological model fails to explain the statistics of outcomes of a single carefully constructed positive operator valued measure (POVM) executed sequentially on a quantum system. The context of measurement arises from the different configurations in which the device can be used. We develop an inequality from the non-contextual (NC) ontic model, and construct a quantum situation invo…
This paper illustrates a direct connection between quantum steering and non-trivial preparation contextuality. In two party-two measurement per party-two outcomes per measurement $(2-2-2)$ Bell scenario, any argument of Bell nonlocality is a proof of trivial preparation contextuality; however, the converse may not hold. If one of the parties (say, Alice) performs the measurements of more than two dichotomic observables, then it is possible to find a set of non-trivial functional relations between Alice's observables. We argue that the existence of a suitable set of such non-trivial relations b…
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 al…
Status of quantum wavefunction is one of the most debated issues in quantum foundations -- whether it corresponds directly to the reality or just represents knowledge or information about some aspect of reality. In this letter we propose a {\it $ψ$-ontology} theorem that excludes a class of ontological explanations where quantum wavefunction is treated as mere information. Our result, unlike the acclaimed Pusey-Barrett-Rudolph's theorem, does not presume the absence of holistic ontological properties for product quantum preparations. At the core of our derivation we utilize the seminal no-go r…
Dimension witness provides a device-independent certification of the minimal dimension required to reproduce the observed data without imposing assumptions on the functioning of the devices used to generate the experimental statistics. In this paper, we provide a family of Bell expressions where Alice and Bob perform $2^{n-1}$ and $n$ number of dichotomic measurements respectively which serve as the device-independent dimension witnesses of Hilbert space of $2^{m}$ dimensions with $m=1,2..2^{\lfloor n/2\rfloor}$. The family of Bell expressions considered here determines the success probability…
Classical probabilistic models of (noisy) quantum systems are not only relevant for understanding the non-classical features of quantum mechanics, but they are also useful for determining the possible advantage of using quantum resources for information processing tasks. A common feature of these models is the presence of inaccessible information, as captured by the concept of preparation contextuality: There are ensembles of quantum states described by the same density operator, and hence operationally indistinguishable, and yet in any probabilistic (ontological) model, they should be describ…
Most scholars maintain that quantum mechanics (QM) is a contextual theory and that quantum probability does not allow an epistemic (ignorance) interpretation. By inquiring possible connections between contextuality and non-classical probabilities we show that a class T of theories can be selected in which probabilities are introduced as classical averages of Kolmogorovian probabilities over sets of (microscopic) contexts, which endows them with an epistemic interpretation. The conditions characterizing T are compatible with classical mechanics (CM), statistical mechanics (SM) and QM, hence we …
