Kochen-Specker Theorem Research Articles

Value-indefinite observables are almost everywhere

Kochen-Specker theorems assure the breakdown of certain types of non-contextual hidden variable theories through the non-existence of global, holistic frame functions; alas they do not allow us to identify where this breakdown occurs, nor the extent of it. It was recently shown [Phys. Rev. A 86, 062109 (2012)] that this breakdown does not occur everywhere; here we show that it is maximal in that it occurs almost everywhere, and thus prove that quantum indeterminacy--often referred to as contextuality or value indefiniteness--is a global property as is often assumed. In contrast to the Kochen-Specker theorem, we only assume the weaker non-contextuality condition that any potential value assignments that may exist are locally non-contextual. Under this assumption, we prove that once a single arbitrary observable is fixed to occur with certainty, almost (i.e. with Lebesgue measure one) all remaining observables are indeterminate.

Primitive nonclassical structures of theN-qubit Pauli group

Several types of nonclassical structures within the $N$-qubit Pauli group that can be seen as fundamental resources for quantum information processing are presented and discussed. Identity Products (IDs), structures fundamentally related to entanglement, are defined and explored. The Kochen-Specker theorem is proved by particular sets of IDs that we call KS sets. We also present a new theorem that we call the $N$-qubit Kochen-Specker theorem, which is proved by particular sets of IDs that we call NKS sets, and whose utility is that it leads to efficient constructions for KS sets. We define the criticality, or irreducibility, of these structures, and its connection to entanglement. All representative critical IDs for up to $N=4$ qubits are presented, and numerous families of critical IDs for arbitrarily large values of $N$ are discussed. The critical IDs for a given $N$ are a finite set of geometric objects that appear to fully characterize the nonclassicality of the $N$-qubit Pauli group. Methods for constructing critical KS sets and NKS sets from IDs are given, and experimental tests of entanglement, contextuality, and nonlocality are discussed. Possible applications and connections to other work are also discussed

Interpreting the Modal Kochen–Specker theorem: Possibility and many worlds in quantum mechanics

In this paper we attempt to physically interpret the Modal Kochen–Specker (MKS) theorem. In order to do so, we analyze the features of the possible properties of quantum systems arising from the elements in an orthomodular lattice and distinguish the use of “possibility” in the classical and quantum formalisms. Taking into account the modal and many worlds non-collapse interpretation of the projection postulate, we discuss how the MKS theorem rules the constraints to actualization, and thus, the relation between actual and possible realms.

Simple Hardy-Like Proof of Quantum Contextuality

Contextuality and nonlocality are two fundamental properties of nature. Hardy's proof is considered the simplest proof of nonlocality and can also be seen as a particular violation of the simplest Bell inequality. A fundamental question is: Which is the simplest proof of contextuality? We show that there is a Hardy-like proof of contextuality that can also be seen as a particular violation of the simplest noncontextuality inequality. Interestingly, this new proof connects this inequality with the proof of the Kochen-Specker theorem, providing the missing link between these two fundamental results, and can be extended to an arbitrary odd number n of settings, an extension that can be seen as a particular violation of the n-cycle inequality.

State-Independent Proof of Kochen—Specker Theorem with Thirty Rank-Two Projectors

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.

Kochen-Specker Sets with a Mixture of 16 Rank-1 and 14 Rank-2 Projectors for a Three-Qubit System

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.

Bounding Temporal Quantum Correlations

Sequential measurements on a single particle play an important role in fundamental tests of quantum mechanics. We provide a general method to analyze temporal quantum correlations, which allows us to compute the maximal correlations for sequential measurements in quantum mechanics. As an application, we present the full characterization of temporal correlations in the simplest Leggett-Garg scenario and in the sequential measurement scenario associated with the most fundamental proof of the Kochen-Specker theorem.

Proofs of the Kochen-Specker theorem based on theN-qubit Pauli group

We present a number of observables-based proofs of the Kochen-Specker (KS) theorem based on the N-qubit Pauli group for N >= 4, thus adding to the proofs that have been presented earlier for the two- and three-qubit groups. These proofs have the attractive feature that they can be presented in the form of diagrams from which they are obvious by inspection. They are also irreducible in the sense that they cannot be reduced to smaller proofs by ignoring some subset of qubits and/or observables in them. A simple algorithm is given for transforming any observables-based KS proof into a large number of projectors-based KS proofs; if the observables-based proof has O observables, with each observable occurring in exactly two commuting sets and any two commuting sets having at most one observable in common, the number of associated projectors-based parity proofs is 2^O. We introduce symbols for the observables- and projectors-based KS proofs that capture their important features and also convey a feeling for the enormous variety of both these types of proofs within the N-qubit Pauli group. We discuss an infinite family of observables-based proofs, whose members apply to all numbers of qubits from two up, and show how it can be used to generate projectors-based KS proofs involving only nine bases (or experimental contexts) in any dimension of the form 2^N for N >= 2. Some implications of our results are discussed.

The coevent formulation of quantum theory

Understanding quantum theory has been a subject of debate from its birth. Many different formulations and interpretations have been proposed. Here we examine a recent novel formulation, namely the coevents formulation. It is a histories formulation and has as starting point the Feynman path integral and the decoherence functional. The new ontology turns out to be that of a coarse-grained history. We start with a quantum measure defined on the space of histories, and the existence of zero covers rules out single-history as potential reality (the Kochen Specker theorem casted in histories form is a special case of a zero cover). We see that allowing coarse-grained histories as potential realities avoids the previous paradoxes, maintains deductive non-contextual logic (alas non-Boolean) and gives rise to a unique classical domain. Moreover, we can recover the probabilistic predictions of quantum theory with the use of the Cournot's principle. This formulation, being both a realist formulation and based on histories, is well suited conceptually for the purposes of quantum gravity and cosmology.

Tests against noncontextual models with measurement disturbances

The testability of the Kochen-Specker theorem is a subject of ongoing controversy. A central issue is that experimental implementations relying on sequential measurements cannot achieve perfect compatibility between the measurements and that therefore the notion of noncontextuality is not well defined. We demonstrate by an explicit model that such compatibility violations may yield a violation of noncontextuality inequalities, even if we assume that the incompatibilities merely originate from context-independent noise. We show, however, that this problem can be circumvented by combining the ideas behind Leggett-Garg inequalities with those of the Kochen-Specker theorem.

Experimental Certification of Random Numbers via Quantum Contextuality

The intrinsic unpredictability of measurements in quantum mechanics can be used to produce genuine randomness. Here, we demonstrate a random number generator where the randomness is certified by quantum contextuality in connection with the Kochen-Specker theorem. In particular, we generate random numbers from measurements on a single trapped ion with three internal levels, and certify the generated randomness by showing a bound on the minimum entropy through observation of violation of the Klyachko-Can-Binicioglu-Shumovsky (KCBS) inequality. Concerning the test of the KCBS inequality, we close the detection efficiency loophole for the first time and make it relatively immune to the compatibility loophole. In our experiment, we generate 1 × 105 random numbers that are guaranteed to have 5.2 × 104 bits of minimum entropy with a 99% confidence level.

Maximally epistemic interpretations of the quantum state and contextuality.

We examine the relationship between quantum contextuality (in both the standard Kochen-Specker sense and in the generalized sense proposed by Spekkens) and models of quantum theory in which the quantum state is maximally epistemic. We find that preparation noncontextual models must be maximally epistemic, and these in turn must be Kochen-Specker noncontextual. This implies that the Kochen-Specker theorem is sufficient to establish both the impossibility of maximally epistemic models and the impossibility of preparation noncontextual models. The implication from preparation noncontextual to maximally epistemic then also yields a proof of Bell's theorem from an Einstein-Podolsky-Rosen-like argument.

Entropic inequalities as a necessary and sufficient condition to noncontextuality and locality

The assumption of local realism in a Bell locality scenario imposes non-trivial conditions on the Shannon entropies of the associated probability distributions, expressed by linear entropic Bell inequalities. In principle, these entropic inequalities provide necessary but not sufficient criteria for the existence of a local hidden variable model reproducing the correlations, as, for example, the paradigmatic nonlocal PR-box is entropically not different from a classically correlated box. In this paper we show that for the n-cycle scenario, entropic inequalities completely characterize the set of local correlations. In particular, every nonsignalling box which violates the CHSH inequality -- including the PR-box -- can be locally modified so that it also violates the entropic version of CHSH inequality. As we show, any nonlocal probabilistic model when appropriately mixed with a local model, violates an entropic inequality, thus evidencing a very peculiar kind of nonlocality. As the n-cycle captures equally well both the notion of local realism introduced by Bell and that of noncontextuality presented by the Kochen-Specker theorem, the results are also valid for noncontextuality scenarios.

GHZ paradoxes based on an even number of qubits

GHZ paradoxes are presented for all even numbers of qubits from four up. They are obtained from proofs of the Kochen–Specker (KS) theorem by showing how the assumption of noncontextuality can be justified on the basis of locality. The nature of the entangled states involved in our paradoxes is discussed. Some multiqubit proofs of the KS theorem are also presented in the form of diagrams from which they are visually obvious. The implications of our results are discussed. ► Presents a new infinite class of GHZ paradoxes. ► Points out a new class of entangled states connected with the paradoxes. ► Presents new proofs of the Kochen–Specker theorem.

A Comparison of Two Topos-Theoretic Approaches to Quantum Theory

The aim of this paper is to compare the two topos-theoretic approaches to quantum mechanics that may be found in the literature to date. The first approach, which we will call the contravariant approach, was originally proposed by Isham and Butterfield, and was later extended by Doring and Isham. The second approach, which we will call the covariant approach, was developed by Heunen, Landsman and Spitters. Motivated by coarse-graining and the Kochen-Specker theorem, the contravariant approach uses the topos of presheaves on a specific context category, defined as the poset of commutative von Neumann subalgebras of some given von Neumann algebra. In particular, the approach uses the spectral presheaf. The intuitionistic logic of this approach is given by the (complete) Heyting algebra of closed open subobjects of the spectral presheaf. We show that this Heyting algebra is, in a natural way, a locale in the ambient topos, and compare this locale with the internal Gelfand spectrum of the covariant approach. In the covariant approach, a non-commutative C*-algebra (in the topos Set) defines a commutative C*-algebra internal to the topos of covariant functors from the context category to the category of sets. We give an explicit description of the internal Gelfand spectrum of this commutative C*-algebra, from which it follows that the external spectrum is spatial. Using the daseinisation of self-adjoint operators from the contravariant approach, we give a new definition of the daseinisation arrow in the covariant approach and compare it with the original version. States and state-proposition pairing in both approaches are compared. We also investigate the physical interpretation of the covariant approach.

Classical, quantum and nonsignalling resources in bipartite games

We study bipartite games that arise in the context of nonlocality with the help of graph theory. Our main results are alternate proofs that deciding whether a no-communication classical winning strategy exists for certain games (called forbidden-edge and covering games) is NP-complete, while the problem of deciding if these games admit a nonsignalling winning strategy is in P. We discuss relations between quantum winning strategies and orthogonality graphs. We also show that every pseudotelepathy game yields both a proof of the Bell–Kochen–Specker theorem and an instance of a two-prover interactive proof system that is classically sound, but that becomes unsound when provers use shared entanglement.

Strong Kochen-Specker theorem and incomputability of quantum randomness

The Kochen-Specker theorem shows the impossibility for a hidden variable theory to consistently assign values to certain (finite) sets of observables in a way that is non-contextual and consistent with quantum mechanics. If we require non-contextuality, the consequence is that many observables must not have pre-existing definite values. However, the Kochen-Specker theorem does not allow one to determine which observables must be value indefinite. In this paper we present an improvement on the Kochen-Specker theorem which allows one to actually locate observables which are provably value indefinite. Various technical and subtle aspects relating to this formal proof and its connection to quantum mechanics are discussed. This result is then utilized for the proposal and certification of a dichotomic quantum random number generator operating in a three-dimensional Hilbert space.

Bell's theorem from Moore's theorem

It is shown that the restrictions of what can be inferred from classically recorded observational outcomes that are imposed by the no-cloning theorem, the Kochen–Specker theorem and Bell's theorem also follow from restrictions on inferences from observations formulated within classical automata theory. Similarities between the assumptions underlying classical automata theory and those underlying universally unitary quantum theory are discussed.

All-or-Nothing-Type Kochen-Specker Experiment with Thermal Lights

Recently, the Kochen—Specker theorem has been demonstrated and quantum contextuality has been observed experimentally by using special and independent quantum states. On the other hand, recent investigations have shown that a thermal light source can mimic the two-photon entangled source to perform the imaging effects. We perform an all-or-nothing—type Kochen-Specker experiment using the thermal light source and obtain the same results as those using quantum sources. The physical origin for such a phenomenon is analyzed.

Five-qubit contextuality, noise-like distribution of distances between maximal bases and finite geometry

Employing five commuting sets of five-qubit observables, we propose specific 160–661 and 160–21 state proofs of the Bell–Kochen–Specker theorem that are also proofs of Bellʼs theorem. A histogram of the ‘Hilbert–Schmidt’ distances between the corresponding maximal bases shows in both cases a noise-like behavior. The five commuting sets are also ascribed a finite-geometrical meaning in terms of the structure of symplectic polar space W(9,2).