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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]

We investigate the possibility of constructing Kochen–Specker uncolorable sets of idempotent matrices whose entries lie in various rings, including the rational numbers, the integers, and finite fields. Most notably, we show that there is no Kochen–Specker coloring of the n×n idempotent integer matrices for n≥3, thereby illustrating that Kochen–Specker contextuality is an inherent feature of pure matrix algebra. We apply this to generalize recent no-go results on noncommutative spectrum functors, showing that any contravariant functor from rings to sets (respectively, topological spaces or locales) that restricts to the Zariski prime spectrum functor for commutative rings must assign the empty set (respectively, empty space or locale) to the matrix ring Mn(R) for any integer n≥3 and any ring R. An appendix by Alexandru Chirvasitu shows that Kochen–Specker colorings of idempotents in partial subalgebras of M3(F) for a perfect field F can be extended to partial algebra morphisms into the algebraic closure of F.

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.

The Suppes-Zanotti inequalities involving the joint expectations of just three binary quantum observables are (re-)derived by the hull computation of the respective correlation polytope. A min-max calculation reveals its maximal quantum violations correspond to a generalized Tsirelson bound. Notions of "contextuality" motivated by such violations are critically reviewed. [ABSTRACT FROM AUTHOR]

It is a widespread belief that the Kochen-Specker theorem imposes a contextuality constraint on the ontology of beables in quantum hidden variables theories. On the other hand, after Bell's influential critique, the importance of von Neumann's wrongly called 'impossibility proof' has been severely questioned. However, Max Jammer, Jeffrey Bub and Dennis Dieks have proposed insightful reassessments of von Neumann's theorem: what it really shows is that hidden variables theories cannot represent their beables by means of Hermitian operators in Hilbert space. Hereby I show that i) the very same constraint can be derived from Gleason's theorem, and that ii) if we consider the import of von Neumann's and Gleason's theorems, the relevance of the Kochen-Specker theorem for hidden variables theories gets substantially weakened: it does not force them to be contextual in any interesting sense of the term. [ABSTRACT FROM AUTHOR]

I will argue that the Peres-Mermin square does not necessarily rule out a value-definite (deterministic) noncontextual hidden variable model if the operators are not given a physical interpretation satisfying the following two requirements: (i) each operator is uniquely realized by a single physical measurement; (ii) commuting operators are realized by simultaneous measurements. To underpin this claim, I will construct three hidden variable models for three different physical realizations of the Peres-Mermin square: one violating (i), another violating (ii), and a third one violating both (i) and (ii). [ABSTRACT FROM AUTHOR]

In this paper we present a new categorical approach which attempts to provide an original understanding of QM. Our logos categorical approach attempts to consider the main features of the quantum formalism as the standpoint to develop a conceptual representation that explains what the theory is really talking about —rather than as problems that need to be bypassed in order to allow a restoration of a classical "common sense" understanding of what there is. In particular, we discuss a solution to Kochen-Specker contextuality through the generalization of the meaning of global valuation. This idea has been already addressed by the so called topos approach to QM —originally proposed by Isham, Butterfiled and Döring— in terms of sieve-valued valuations. The logos approach to QM presents a different solution in terms of the notion of intensive valuation. This new solution stresses an ontological (rather than epistemic) reading of the quantum formalism and the need to restore an objective (rather than classical) conceptual representation and understanding of quantum physical reality. [ABSTRACT FROM AUTHOR]

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.

Partition logics often allow a dual probabilistic interpretation: a classical one for which probabilities lie on the convex hull of the dispersion-free weights and another one, suggested independently from the quantum Born rule, in which probabilities are formed by the (absolute) square of the inner product of state vectors with the faithful orthogonal representations of the respective graph. Two immediate consequences are the demonstration that the logico-empirical structure of observables does not determine the type of probabilities alone and that complementarity does not imply contextuality. [ABSTRACT FROM AUTHOR]

Amid the wide variety of interpretations of quantum mechanics, the notion of a fully coherent ontological interpretation has seen a promising evolution over the last few decades. Despite this progress, however, the old dualistic categorical constraints of subjectivity and objectivity, correlate with the metrically restricted definition of local and global, have remained largely in place – a reflection of the broader, persistent inheritance of these comfortable strictures throughout the evolution of modern science. If one traces this inheritance back to its ancient roots in Plato and Aristotle, it is clear that the coherence, scientific utility, and historical durability of the various natural philosophies that followed have been directly proportional to their commitment, tacit or explicit, to object-oriented realism. As Simondon might put it, it is a commitment to the primacy of individuated substance over the process of individuation. For Whitehead, it is the misplaced assimilation of becoming to being, of potentiality to actuality. Quantum mechanics has challenged this commitment with a combined breadth and depth of force that far exceeds any other theory in the history of science. The theory's objectively demonstrable empirical application is manifest only by way of a fundamentally subjective, context-dependent mechanism of measurement. More compelling still, actual system states are always actualizations of locally contextualized yet globally conditioned potential system states, such that "globally objective" reality is no longer merely the object of local measurement, but also its product. Thus, any coherent, ontological interpretation of quantum theory must include a conceptual framework by which objectivity and subjectivity, actuality and potentiality, global and local, being and becoming, individuated fact and process of individuation, are no longer understood as merely epistemic, mutually exclusive category pairs descriptive of an already extant, closed reality – but rather as mutually implicative ontological categories explicative of an ontogenetic, open reality-in-process. [ABSTRACT FROM AUTHOR]

If quantum mechanics is taken for granted, the randomness derived from it may be vacuous or even delusional, yet sufficient for many practical purposes. “Random” quantum events are intimately related to the emergence of both space-time as well as the identification of physical properties through which so-called objects are aggregated. We also present a brief review of the metaphysics of indeterminism.

This study aims to provide an analysis of the complementarity principle in quantum theory through the establishment of partial structural congruence relations between the quantum and Boolean kinds of event structure. Specifically, on the basis of the existence of a categorical adjunction between the category of quantum event algebras and the category of presheaves of variable Boolean event algebras, we establish a twofold complementarity scheme consisting of a generalized/global and a restricted/local conceptual dimension, where the latter conception is subordinate to and constrained by the former. In this respect, complementarity is not only understood as a relation between mutually exclusive experimental arrangements or contexts of comeasurable observables, as envisaged by the original conception, but it is primarily comprehended as a reciprocal relation concerning information transfer between two hierarchically different structural kinds of event structure that can be brought into partial congruence by means of the established adjunction. It is further argued that the proposed category-theoretic framework of complementarity naturally advances a contextual realist conceptual stance towards our deeper understanding of the microphysical nature of reality. [ABSTRACT FROM AUTHOR]

The totalitarian principle establishes that 'anything not forbidden is compulsory'. The problem of quantum correlations is explaining what selects the set of quantum correlations for a Bell and Kochen- Specker (KS) contextuality scenario. Here, we show that two assumptions and a version of the totalitarian principle lead to the quantum correlations. The assumptions are that there is a non-empty set of correlations for any KS contextuality scenario and a statistically independent realization of any two KS experiments. The version of the totalitarian principle says that any correlation not forbidden by these assumptions can be produced. This paper contains a short version of the proof (presented elsewhere) and explores some implications of the result. [ABSTRACT FROM AUTHOR]

We argue that the usual Bloch sphere is insufficient in various aspects for the representation of qubits in quantum information theory. For example, spin flip operations with the quaternions I J K = e 2 π i 2 = − 1 and J I K = + 1 cannot be distinguished on the Bloch sphere. We show that a simple knot theoretic extension of the Bloch sphere representation is sufficient to track all unitary operations for single qubits. Next, we extend the Bloch sphere representation to entangled states using knot theory. As applications, we first discuss contextuality in quantum physics—in particular the Kochen-Specker theorem. Finally, we discuss some arguments against many-worlds theories within our knot theoretic model of entanglement. The key ingredients of our approach are symmetries and geometric properties of the unitary group.

This is an elaboration of the “extra” advantage of the performance of quantized physical systems over classical ones, both in terms of single outcomes as well as probabilistic predictions. From a formal point of view, it is based on entities related to (dual) vectors in (dual) Hilbert spaces, as compared to the Boolean algebra of subsets of a set and the additive measures they support.

In the history of quantum physics several no-go theorems have been proved, and many of them have played a central role in the development of the theory, such as Bell’s or the Kochen-Specker theorem. A recent paper by F. Laudisa has raised reasonable doubts concerning the strategy followed in proving some of these results, since they rely on the standard framework of quantum mechanics, a theory that presents several ontological problems. The aim of this paper is twofold: on the one hand, I intend to reinforce Laudisa’s methodological point by critically discussing Malament’s theorem in the context of the philosophical foundation of quantum field theory; secondly, I rehabilitate Gisin’s theorem showing that Laudisa’s concerns do not apply to it. [ABSTRACT FROM AUTHOR]

Non-locality and contextuality are among the most counterintuitive aspects of quantum theory. They are difficult to study using classical logic and probability theory. In this paper we start with an effect algebraic approach to the study of non-locality and contextuality. We will see how different slices over the category of set valued functors on the natural numbers induce different settings in which non-locality and contextuality can be studied. This includes the Bell, Hardy and Kochen–Specker-type paradoxes. We link this to earlier sheaf theoretic approaches by defining a fully faithful embedding of the category of effect algebras in this presheaf category over the natural numbers. [ABSTRACT FROM AUTHOR]

We develop the point of view where quantum mechanics results from the interplay between the quantized number of 'modalities' accessible to a quantum system, and the continuum of 'contexts' that are required to define these modalities. We point out the specific roles of 'extracontextuality' and 'extravalence' of modalities, and relate them to the Kochen-Specker and Gleason theorems. This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'. [ABSTRACT FROM AUTHOR]

We study nonnegative solutions of parabolic–parabolic Keller–Segel minimal-chemotaxis-growth systems with prototype given by { u t = ∇ ⋅ ( d 1 ∇ u − χ u ∇ v ) + κ u − μ u 2 , x ∈ Ω , t > 0 , v t = d 2 Δ v − β v + α u , x ∈ Ω , t > 0 in a smooth bounded smooth but not necessarily convex domain Ω ⊂ R n ( n ≥ 3 ) with nonnegative initial data u 0 , v 0 and homogeneous Neumann boundary data, where d 1 , d 2 , α , β , μ > 0 , χ , κ ∈ R . We provide quantitative and qualitative descriptions of the competition between logistic damping and other ingredients, especially, chemotactic aggregation to guarantee boundedness and convergence. Specifically, we first obtain an explicit formula μ 0 = μ 0 ( n , d 1 , d 2 , α , χ ) for the logistic damping rate μ such that the system has no blow-ups whenever μ > μ 0 . In particular, for Ω ⊂ R 3 , we get a clean formula for μ 0 : μ 0 ( 3 , d 1 , d 2 , α , χ ) = { 3 4 d 1 α χ , if d 1 = d 2 , χ > 0 and Ω is convex , 3 10 − 2 ( 1 d 1 + 2 d 2 ) α | χ | , otherwise . This offers a quantized effect of the logistic source on the prevention of blow-ups. Our result extends the fundamental boundedness principle by Winkler [42] with d 1 = 1 , d 2 = α = β : = 1 / τ , Ω being convex and sufficiently large values of μ beyond a certain number not explicitly known (except the simple case τ = 1 and χ > 0 ) and quantizes the qualitative result of Yang et al. [52] . Besides, in non-convex domains, since μ 0 ( 3 , 1 , 1 , 1 , χ ) = ( 7.743416 ⋯ ) χ , the recent boundedness result, μ > 20 χ , of Mu and Lin [25] is greatly improved. Then we derive another explicit formula: μ 1 = μ 1 ( d 1 , d 2 , α , β , κ , χ ) = α | χ | 4 κ + d 1 d 2 β for the logistic damping rate so that convergence of bounded solutions is ensured and the respective convergence rates are explicitly calculated out whenever μ > μ 1 . Recent convergence results of He and Zheng [9] are therefore complemented and refined. [ABSTRACT FROM AUTHOR]

Based on the fact that both nonlocality and contextuality are resource theories, it is natural to ask how to amplify them more efficiently. In this paper, we present a contextuality distillation protocol which produces an n-cycle box B ∗ B′ from two given n-cycle boxes B and B′. It works efficiently for a class of contextual n-cycle (n ≥ 4) boxes which we termed as “the generalized correlated contextual n-cycle boxes”. For any two generalized correlated contextual n-cycle boxes B and B′, B ∗ B′ is more contextual than both B and B′. Moreover, they can be distilled toward to the maximally contextual box CHn as the times of iteration goes to infinity. Among the known protocols, our protocol has the strongest approximate ability and is optimal in terms of its distillation rate. What is worth noting is that our protocol can witness a larger set of nonlocal boxes that make communication complexity trivial than the protocol in Brunner and Skrzypczyk (Phys. Rev. Lett. 102, 160403 2009), this might be helpful for exploring the problem that why quantum nonlocality is limited. [ABSTRACT FROM AUTHOR]