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This paper examines no-hidden-variables theorems in quantum mechanics from the point of view of statistical mechanics. It presents a general analysis of the measurement process in the Boltzmannian framework that leads to a characterization of (in)compatible measurements and reproduces several features of quantum probabilities often described as “non-classical”. The analysis is applied to versions of the Kochen–Specker and Bell theorems to shed more light on their implications. It is shown how, once the measurement device and the active role of the measurement process are taken into account, contextuality appears as a natural feature of random variables. This corroborates Bell’s criticism that no-go results of the Kochen–Specker type are based on gratuitous assumptions. In contrast, Bell-type theorems are much more profound, but should be understood as nonlocality theorems rather than no-hidden-variables theorems. Finally, the paper addresses misunderstandings and misleading terminology that have confused the debate about hidden variables in quantum mechanics.

This paper examines no-hidden-variables theorems in quantum mechanics from the point of view of statistical mechanics. It presents a general analysis of the measurement process in the Boltzmannian framework that leads to a characterization of (in)compatible measurements and reproduces several features of quantum probabilities often described as "non-classical". The analysis is applied to versions of the Kochen–Specker and Bell theorems to shed more light on their implications. It is shown how, once the measurement device and the active role of the measurement process are taken into account, contextuality appears as a natural feature of random variables. This corroborates Bell's criticism that no-go results of the Kochen–Specker type are based on gratuitous assumptions. In contrast, Bell-type theorems are much more profound, but should be understood as nonlocality theorems rather than no-hidden-variables theorems. Finally, the paper addresses misunderstandings and misleading terminology that have confused the debate about hidden variables in quantum mechanics. [ABSTRACT FROM AUTHOR]

Abstract All quantum random number generators based on measuring value indefinite observables are at least three-dimensional because the Kochen–Specker Theorem and the Located Kochen–Specker Theorem are false in dimension two. In this article, we construct quantum random number generators based on measuring a three-dimensional value indefinite observable that generates binary quantum random outputs with the same randomness qualities as the ternary ones: the outputs are maximally unpredictable.

All quantum random number generators based on measuring value indefinite observables are at least three-dimensional because the Kochen–Specker Theorem and the Located Kochen–Specker Theorem are false in dimension two. In this article, we construct quantum random number generators based on measuring a three-dimensional value indefinite observable that generates binary quantum random outputs with the same randomness qualities as the ternary ones: the outputs are maximally unpredictable. [ABSTRACT FROM AUTHOR]

I propose a technique for identifying fundamental properties using structures already present in physical theories. I argue that, in conjunction with a particular naturalistic commitment, that I dub 'algebraic naturalism', these structures can be used to generate a standard of metaphysical determinacy. This standard can be used to rule out the possibility of a virulent strain of 'deep' metaphysical indeterminacy that has been imputed to quantum mechanics. [ABSTRACT FROM AUTHOR]

It will be shown that the Peres–Mermin square admits value-definite noncontextual hidden-variable models if the observables associated with the operators can be measured only sequentially but not simultaneously. Namely, sequential measurements allow for noncontextual models in which hidden states update between consecutive measurements. Two recent experiments realizing the Peres–Mermin square by sequential measurements will also be analyzed along with other hidden-variable models accounting for these experiments. [ABSTRACT FROM AUTHOR]

The quantum realms of nonlocality and contextuality are delineated by Bell’s theorem and the Kochen–Specker theorem, respectively, embodying phenomena that surpass the explanatory capacities of classical theories. These realms hold transformative potential for the fields of information and computing technology. In this study, we unveil a “all-versus-nothing” proof that concurrently illustrates the veracity of these two seminal theorems, fostering a more nuanced comprehension of the intricate relationship intertwining quantum nonlocality and contextuality. Leveraging the capabilities of three singlet pairs and a Greenberger–Horne–Zeilinger state analyzer, our proof not only substantiates the conflict between quantum mechanics and hidden-variable theories from another perspective, but can also be readily verifiable utilizing the existing linear optics technology.

We describe an explicitly non-contextual statistical model of hidden variables for the qutrit, which fully reproduces the predictions of quantum mechanics, and thus, bypasses the constraints imposed by the Kochen–Specker theorem and its subsequent reformulations. We notice that these renowned theorems crucially rely on the implicitly assumed existence of an absolute frame of reference with respect to which physically indistinguishable tests related by spurious gauge transformations can supposedly be assigned well-defined distinct identities. We observe that the existence of such an absolute frame of reference is not required by fundamental physical principles, and hence, assuming it is an unnecessarily restrictive demand. [ABSTRACT FROM AUTHOR]

By elaborating on the results presented in Lógica cuántica, Nmatrices y adecuación I, here we discuss the notions of adequacy and truth functionality in quantum logic from the point of view of a non-deterministic semantics based on Nmatrices. We present a proof of the impossibility of providing a functional semantics for the quantum lattice. An advantage of our proof is that it is independent of the number of truth values involved, generalizing previous works. Due to the impossibility of defining adequate interpreting sets for the conjunction and disjunction (as was shown in our previous work), it follows that a homomorphism between the lattice of quantum projections and a Boolean algebra of n-elements cannot exist. [ABSTRACT FROM AUTHOR]

In this paper we discuss the notions of adequacy and truth functionality in quantum logic from the point of view of a non-deterministic semantics. We give a characterization of the degree of non-functionality which is compatible with the propositional structure of quantum theory, showing that having truth-functional connectives, together with some assumptions regarding the relation of logical consequence, commits us to the adequacy of the interpretation sets of these connectives. An advantage of our proof is that it is independent of the number of truth values involved, generalizing previous works. We also show the failure of the adequacy of every Nmatrix that is a model of the non-distributive lattice of quantum propositions. [ABSTRACT FROM AUTHOR]

Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason's theorem. [ABSTRACT FROM AUTHOR]

There are two different and logically independent concepts of noncontextuality in quantum mechanics. First, an ontological (hidden variable) model for quantum mechanics is called noncontextual if every ontic (hidden) state determines the probability of the outcomes of every measurement independently of what other measurements are simultaneously performed. Second, an ontological model is noncontextual if any two measurements which are represented by the same self-adjoint operator or, equivalently, which have the same probability distribution of outcomes in every quantum state also have the same probability distribution of outcomes in every ontic state. I will call the first concept simultaneous noncontextuality, the second measurement noncontextuality. In the paper I will overview and critically analyze some of the most significant accounts of contextuality in the literature and subsume them under these two categories. [ABSTRACT FROM AUTHOR]

The Greenberger–Horne–Zeilinger (GHZ) argument against noncontextual local hidden variables is recast in quantum logical terms of fundamental propositions, states and probabilities. Unlike Kochen–Specker- and Hardy-like configurations, this operator based argument proceeds within four nonintertwining contexts. The nonclassical performance of the GHZ argument is due to the choice or filtering of observables with respect to a particular state. We study the varieties of GHZ games one could play in these four contexts, depending on the chosen state of the GHZ basis. [ABSTRACT FROM AUTHOR]

We describe an explicitly non-contextual statistical model of hidden variables for the qutrit, which fully reproduces the predictions of quantum mechanics, and thus, bypasses the constraints imposed by the Kochen–Specker theorem and its subsequent reformulations. We notice that these renowned theorems crucially rely on the implicitly assumed existence of an absolute frame of reference with respect to which physically indistinguishable tests related by spurious gauge transformations can supposedly be assigned well-defined distinct identities. We observe that the existence of such an absolute frame of reference is not required by fundamental physical principles, and hence, assuming it is an unnecessarily restrictive demand.

The quantum realms of nonlocality and contextuality are delineated by Bell's theorem and the Kochen–Specker theorem, respectively, embodying phenomena that surpass the explanatory capacities of classical theories. These realms hold transformative potential for the fields of information and computing technology. In this study, we unveil a "all-versus-nothing" proof that concurrently illustrates the veracity of these two seminal theorems, fostering a more nuanced comprehension of the intricate relationship intertwining quantum nonlocality and contextuality. Leveraging the capabilities of three singlet pairs and a Greenberger–Horne–Zeilinger state analyzer, our proof not only substantiates the conflict between quantum mechanics and hidden-variable theories from another perspective, but can also be readily verifiable utilizing the existing linear optics technology. [ABSTRACT FROM AUTHOR]

Kolmogorov’s axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason’s theorem.

It is of interest to study how contextual quantum mechanics is, in terms of the violation of Kochen Specker state-independent and state-dependent non-contextuality inequalities. We present state-independent non-contextuality inequalities with large violations, in particular, we exploit a connection between Kochen–Specker proofs and pseudo-telepathy games to show KS proofs in Hilbert spaces of dimension d ⩾ 2 ^17 with the ratio of quantum value to classical bias being $O(\sqrt{d}/\mathrm{log}\,d)$ . We study the properties of this KS set and show applications of the large violation. It has been recently shown that Kochen–Specker proofs always consist of substructures of state-dependent contextuality proofs called 01-gadgets. We show a one-to-one connection between 01-gadgets in ${\mathbb{C}}^{d}$ and Hardy paradoxes for the maximally entangled state in ${\mathbb{C}}^{d}\otimes {\mathbb{C}}^{d}$ . We use this connection to construct large violation 01-gadgets between arbitrary vectors in ${\mathbb{C}}^{d}$ , as well as novel Hardy paradoxes for the maximally entangled state in ${\mathbb{C}}^{d}\otimes {\mathbb{C}}^{d}$ , and give applications of these constructions. As a technical result, we show that the minimum dimension of the faithful orthogonal representation of a graph in ${\mathbb{R}}^{d}$ is not a graph monotone, a result that may be of independent interest.

We study hidden-variable models from quantum mechanics and their abstractions in purely probabilistic and relational frameworks by means of logics of dependence and independence, which are based on team semantics. We show that common desirable properties of hidden-variable models can be defined in an elegant and concise way in dependence and independence logic. The relationship between different properties and their simultaneous realisability can thus be formulated and proven on a purely logical level, as problems of entailment and satisfiability of logical formulae. Connections between probabilistic and relational entailment in dependence and independence logic allow us to simplify proofs. In many cases, we can establish results on both probabilistic and relational hidden-variable models by a single proof, because one case implies the other, depending on purely syntactic criteria. We also discuss the 'no-go' theorems by Bell and Kochen-Specker and provide a purely logical variant of the latter, introducing non-contextual choice as a team-semantical property. [ABSTRACT FROM AUTHOR]