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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. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

When a measurement is compatible with each of two other measurements that are incompatible with one another, these define distinct contexts for the given measurement. The Kochen–Specker theorem rules out models of quantum theory that satisfy a particular assumption of context-independence: that sharp measurements are assigned outcomes both deterministically and independently of their context. This notion of noncontextuality is not suited to a direct experimental test because realistic measurements always have some degree of unsharpness due to noise.However, a generalized notion of noncontextuality has been proposed that is applicable to any experimental procedure, including unsharp measurements, but also preparations as well, and for which a quantumno-go result still holds. According to this notion, the model need only specify a probability distribution over the outcomes of a measurement in a context-independentway, rather than specifying a particular outcome. It also implies novel constraints of context-independence for the representation of preparations. In this article,we describe a general technique for translating proofs of the Kochen–Specker theorem into inequality constraints on realistic experimental statistics, the violation of which witnesses the impossibility of a noncontextual model.We focus on algebraic state-independent proofs, using the Peres–Mermin square as our illustrative example.Our technique yields the necessary and sufficient conditions for a particular set of correlations (between the preparations and the measurements) to admit a noncontextual model. The inequalities thus derived are demonstrably robust to noise.We specify how experimental data must be processed in order to achieve a test of these inequalities.We also provide a criticism of prior proposals for experimental tests of noncontextuality based on the Peres–Mermin square. [ABSTRACT FROM AUTHOR]

A geometric interpretation for quantum correlations and entanglement according to a particular framework of emergent quantum mechanics is developed. The mechanism described is based on two ingredients: 1. At an hypothetical sub-quantum level description of physical systems, the dynamics has a regime where it is partially ergodic and 2. A formal projection from a two-dimensional time mathematical formalism of the emergent quantum theory to the usual one-dimensional time formalism of quantum dynamics. Observable consequences of the theory are obtained. Among them we show that quantum correlations must be instantaneous from the point of view of the spacetime description, but the spatial distance up to which they can be observed must be bounded. It is argued how our mechanism avoids Bell theorem and Kochen-Specken theorem. Evidence for non-signaling faster than the speed of light in our proposal is discussed. [ABSTRACT FROM AUTHOR]

It is well known that in quantum mechanics we cannot always define consistently properties that are context independent. Many approaches exist to describe contextual properties, such as Contextuality by Default (CbD), sheaf theory, topos theory, and non-standard or signed probabilities. In this paper, we propose a treatment of contextual properties that is specific to quantum mechanics, as it relies on the relationship between contextuality and indistinguishability. In particular, we propose that if we assume the ontological thesis that quantum particles or properties can be indistinguishable yet different, no contradiction arising from a Kochen-Specker-type argument appears: when we repeat an experiment, we are in reality performing an experiment measuring a property that is indistinguishable from the first, but not the same. We will discuss how the consequences of this move may help us understand quantum contextuality. [ABSTRACT FROM AUTHOR]

We investigate small geometric configurations that furnish observable-based proofs of the Kochen-Specker theorem. Assuming that each context consists of the same number of observables and each observable is shared by two contexts, it is proved that the most economical proofs are the famous Mermin-Peres square and the Mermin pentagram featuring, respectively, 9 and 10 observables, there being no proofs using less than 9 observables. We also propose a new proof with 14 observables forming a 'magic' heptagram. On the other hand, some other prominent small-size finite geometries, like the Pasch configuration and the prism, are shown not to be contextual. [ABSTRACT FROM AUTHOR]

The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valuation in quantum mechanics as exemplified, in particular, by Kochen-Specker's theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event algebras. We show explicitly that the latter category is equipped with an object of truth values, or classifying object, which constitutes the appropriate tool for assigning truth values to propositions describing the behavior of quantum systems. Effectively, this category-theoretic representation scheme circumvents consistently the semantic ambiguity with respect to truth valuation that is inherent in conventional quantum mechanics by inducing an objective contextual account of truth in the quantum domain of discourse. The philosophical implications of the resulting account are analyzed. We argue that it subscribes neither to a pragmatic instrumental nor to a relative notion of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature. [ABSTRACT FROM AUTHOR]

We study the relation between the Kochen-Specker theorem (the KS theorem) and quantum computing. The KS theorem rules out a realistic theory of the KS type. We consider the realistic theory of the KS type that the results of measurements are either +1 or −1. We discuss an inconsistency between the realistic theory of the KS type and the controllability of quantum computing. We have to give up the controllability if we accept the realistic theory of the KS type. We discuss an inconsistency between the realistic theory of the KS type and the observability of quantum computing. We discuss the inconsistency by using the double-slit experiment as the most basic experiment in quantum mechanics. This experiment can be for an easy detector to a Pauli observable. We cannot accept the realistic theory of the KS type to simulate the double-slit experiment in a significant specific case. The realistic theory of the KS type can not depicture quantum detector. In short, we have to give up both the observability and the controllability if we accept the realistic theory of the KS type. Therefore, the KS theorem is a precondition for quantum computing, i.e., the realistic theory of the KS type should be ruled out. [ABSTRACT FROM AUTHOR]

Contextuality is one of the fundamental deviations of quantum mechanics from classical physics. The Kochen- Specker theorem shows that non-contextual classical physics with hidden variables is inconsistent with the predictions of quantum mechanics. Parity proof is one of the many different approaches applied to prove Kochen-Specker theorem for quantum system with different dimensionality. This method of proof requires Kochen-Specker sets that were composed of an odd number of bases and an even number of projectors. In most of the cases, the number of Kochen-Specker sets produced is large and its production is generally aided by computer calculation. However, manual generation of the Kochen-Specker sets are also reported previously for qutrit and two-qubit systems. We put forward the first and surprisingly simple method to generate manually Kochen-Specker sets, with respect to the Mermin pentagram, that can be used for the parity proof of the Kochen-Specker theorem in a state space of eight-dimensional three-qubit system. [ABSTRACT FROM AUTHOR]

At the heart of the Conway-Kochen Free Will Theorem and Kochen and Specker's argument against non-contextual hidden variable theories is the existence of a Kochen-Specker (KS) system: a set of points on the sphere that has no {0,1}-coloring such that at most one of two orthogonal points are colored 1 and of three pairwise orthogonal points exactly one is colored 1. In public lectures, Conway encouraged the search for small KS systems. At the time of writing, the smallest known KS system has 31 vectors. Arends, Ouaknine and Wampler have shown that a KS system has at least 18 vectors, by reducing the problem to the existence of graphs with a topological embeddability and non-colorability property. The bottleneck in their search proved to be the sheer number of graphs on more than 17 vertices and deciding embeddability. Continuing their effort, we prove a restriction on the class of graphs we need to consider and develop a more practical decision procedure for embeddability to improve the lower bound to 22. [ABSTRACT FROM AUTHOR]

The well known Kochen-Specker's theorem is devoted to the problem of hidden variables in quantum mechanics. The Kochen-Specker theorem says: There is no two-valued probability measure on the real Hilbert space of dimension three. In the paper we present an analogy of Kochen-Specker's theorem in Pontryagin space: A Pontryagin spase H of dimension greater than or equal to three has a two-valued probability measure if and only if H has indefinite rank one: in which case, any such two-valued probability measure on H is unique. [ABSTRACT FROM AUTHOR]

Unpredictability is an important concept throughout physics and plays a central role in quantum information theory. Despite this, little effort has been devoted to studying generalised notions or models of (un)predictability in physics. In this paper, we continue the programme of developing a general, non-probabilistic model of (un)predictability in physics. We present a more refined model that is capable of studying different degrees of "relativised" unpredictability. This model is based on the ability of an agent, acting via uniform, effective means, to predict correctly and reproducibly the outcome of an experiment using finite information extracted from the environment. We use this model to study the degree of unpredictability certified by different quantum phenomena further, showing that quantum complementarity guarantees a form of relativised unpredictability that is weaker than that guaranteed by Kochen--Specker-type value indefiniteness. We exemplify further the difference between certification by complementarity and value indefiniteness by showing that, unlike value indefiniteness, complementarity is compatible with the production of computable sequences of bits. [ABSTRACT FROM AUTHOR]

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We propose a new quantum ontology, in which properties are the fundamental building blocks. In this property ontology physical systems are defined as bundles of type-properties (specified by algebras of observables in a Hilbert space). Not all elements of such bundles are associated with definite case-properties, and this accommodates the Kochen-Specker theorem and contextuality. Moreover, we do not attribute an identity to the type-properties, which gives rise to a novel form of the bundle theory. There are no "particles" in the sense of classical individuals in this ontology, although the behavior of such individuals is mimicked in some circumstances. This picture leads in a natural way to the symmetrization postulates for systems of many "identical particles". [ABSTRACT FROM AUTHOR]

The Kochen-Specker theorem proves the inability to assign, simultaneously, noncontextual definite values to all (of a finite set of) quantum mechanical observables in a consistent manner. If one assumes that any definite values behave noncontextually, one can nonetheless only conclude that some observables (in this set) are value indefinite. In this paper, we prove a variant of the Kochen-Specker theorem showing that, under the same assumption of noncontextuality, if a single one-dimensional projection observable is assigned the definite value 1, then no one-dimensional projection observable that is incompatible (i.e., non-commuting) with this one can be assigned consistently a definite value. Unlike standard proofs of the Kochen- Specker theorem, in order to localise and show the extent of value indefiniteness, this result requires a constructive method of reduction between Kochen-Specker sets. If a system is prepared in a pure state |, then it is reasonable to assume that any value assignment (i.e., hidden variable model) for this system assigns the value 1 to the observable projecting onto the one-dimensional linear subspace spanned by |, and the value 0 to those projecting onto linear subspaces orthogonal to it. Our result can be interpreted, under this assumption, as showing that the outcome of a measurement of any other incompatible one-dimensional projection observable cannot be determined in advance, thus formalising a notion of quantum randomness. [ABSTRACT FROM AUTHOR]

We will consider a convex subset of a metric linear space and a certain group of actionsGon this set, that allow us to define the notion of Haar zero measure on sets that have zero Haar measure for the translation (by adding) invariant HSY prevalence theory. In this way, we will be able to define the meaning ofG-prevalent set according to the pioneering work of Christensen. Our setting considers problems which take into account the convex structure and this is quite different from the previous results on prevalence which consider basically the linear additive structure. In this setting, we will show a kind of quantitative Kupka–Smale theorem, and also we generalize a result about rotation numbers which was first considered by J.-C. Yoccoz (and, also by M. Tsujii). Among other things we present an estimation of the amount of hyperbolicity in a setting that we believe was not considered before. [ABSTRACT FROM PUBLISHER]

In order to claim that one has experimentally tested whether a noncontextual ontological model could underlie certain measurement statistics in quantum theory, it is necessary to have a notion of noncontextuality that applies to unsharp measurements, i.e., those that can only be represented by positive operator-valued measures rather than projection-valued measures. This is because any realistic measurement necessarily has some nonvanishing amount of noise and therefore never achieves the ideal of sharpness. Assuming a generalized notion of noncontextuality that applies to arbitrary experimental procedures, it is shown that the outcome of a measurement depends deterministically on the ontic state of the system being measured if and only if the measurement is sharp. Hence for every unsharp measurement, its outcome necessarily has an indeterministic dependence on the ontic state. We defend this proposal against alternatives. In particular, we demonstrate why considerations parallel to Fine's theorem do not challenge this conclusion. [ABSTRACT FROM AUTHOR]

The well known Kochen-Specker's theorem (Kochen and Specker J. Math. Mech. 17:59-87, ) is devoted to the problem of hidden variables in quantum mechanics. In the paper we present a geometric proof for an indefinite analogy of Kochen-Specker's theorem. On the real three-dimensional Krein space there exists unique two-valued probability measure. [ABSTRACT FROM AUTHOR]

Recently Cator and Landsman made a comparison between Bell's Theorem and Conway and Kochen's Strong Free Will Theorem. Their overall conclusion was that the latter is stronger in that it uses fewer assumptions, but also that it has two shortcomings. Firstly, no experimental test of the Conway-Kochen Theorem has been performed thus far, and, secondly, because the Conway-Kochen Theorem is strongly connected to the Kochen-Specker Theorem it may be susceptible to the finite precision loophole of Meyer, Kent and Clifton. In this paper I show that the finite precision loophole does not apply to the Conway-Kochen Theorem. [ABSTRACT FROM AUTHOR]

This paper describes a simple, causally deterministic model of quantum measurement based on an amplitude threshold detection scheme. Surprisingly, it is found to reproduce many phenomena normally thought to be uniquely quantum in nature. To model an $$N$$ -dimensional pure state, the model uses $$N$$ complex random variables given by a scaled version of the wave vector with additive complex noise. Measurements are defined by threshold crossings of the individual components, conditioned on single-component threshold crossings. The resulting detection probabilities match or approximate those predicted by quantum mechanics according to the Born rule. Nevertheless, quantum phenomena such as entanglement, contextuality, and violations of Bell's inequality under local measurements are all shown to be exhibited by the model, thereby demonstrating that such phenomena are not without classical analogs. [ABSTRACT FROM AUTHOR]

It is shown how the 300 rays associated with the antipodal pairs of vertices of a 120-cell (a four-dimensional regular polytope) can be used to give numerous 'parity proofs' of the Kochen-Specker theorem ruling out the existence of noncontextual hidden variables theories. The symmetries of the 120-cell are exploited to give a simple construction of its Kochen-Specker diagram, which is exhibited in the form of a 'basis table' showing all the orthogonalities between its rays. The basis table consists of 675 bases (a basis being a set of four mutually orthogonal rays), but all the bases can be written down from the few listed in this paper using some simple rules. The basis table is shown to contain a wide variety of parity proofs, ranging from 19 bases (or contexts) at the low end to 41 bases at the high end. Some explicit examples of these proofs are given, and their implications are discussed. [ABSTRACT FROM AUTHOR]

The logic of a physical theory reflects the structure of the propositions referring to the behaviour of a physical system in the domain of the relevant theory. It is argued in relation to classical mechanics that the propositional structure of the theory allows truth-value assignment in conformity with the traditional conception of a correspondence theory of truth. Every proposition in classical mechanics is assigned a definite truth value, either 'true' or 'false', describing what is actually the case at a certain moment of time. Truth-value assignment in quantum mechanics, however, differs; it is known, by means of a variety of 'no go' theorems, that it is not possible to assign definite truth values to all propositions pertaining to a quantum system without generating a Kochen-Specker contradiction. In this respect, the Bub-Clifton 'uniqueness theorem' is utilized for arguing that truth-value definiteness is consistently restored with respect to a determinate sublattice of propositions defined by the state of the quantum system concerned and a particular observable to be measured. An account of truth of contextual correspondence is thereby provided that is appropriate to the quantum domain of discourse. The conceptual implications of the resulting account are traced down and analyzed at length. In this light, the traditional conception of correspondence truth may be viewed as a species or as a limit case of the more generic proposed scheme of contextual correspondence when the non-explicit specification of a context of discourse poses no further consequences. [ABSTRACT FROM AUTHOR]

In this essay, I develop order-theoretic notions of determinism and contextuality on domains and topoi. In the process, I develop a method for quantifying contextuality and show that the order-theoretic sense of contextuality is analogous to the sense embodied in the topos-theoretic statement of the Kochen--Specker theorem. Additionally, I argue that this leads to a relation between the entropy associated with measurements on quantum systems and the second law of thermodynamics. The idea that the second law has its origin in the ordering of quantum states and processes dates to at least 1958 and possibly earlier. The suggestion that the mechanism behind this relation is contextuality, is made here for the first time. [ABSTRACT FROM AUTHOR]

There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs. [ABSTRACT FROM AUTHOR]

The Kochen—Specker theorem states that noncontextual hidden variable theories are incompatible with quantum mechanics. We provide a state-independent proof of the Kochen—Specker theorem using the smallest number of projectors, i.e., thirty rank-2 projectors, associated with the Mermin pentagram for a three-qubit system. [ABSTRACT FROM AUTHOR]

Kochen—Specker (KS) theorem denies the possibility for the noncontextual hidden variable theories to reproduce the predictions of quantum mechanics. A set of projection operators (projectors) and bases used to show the impossibility of noncontextual definite values assignment is named as the KS set. Since one KS set with a mixture of 16 rank-1 projectors and 14 rank-2 projectors was proposed in 1995 [Kernaghan M and Peres A Phys. Lett. A 198 (1995) 1] for a three-qubit system, there have been plenty of the same type KS sets and we propose a systematic way to produce them. We also propose a probabilistic state-dependent proof of the KS theorem that mainly focuses on the values assignment for all the rank-2 projectors. [ABSTRACT FROM AUTHOR]