Kochen Specker Research Articles

A Kochen-Specker system has at least 22 vectors (extended abstract)

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.

What does an experimental test of quantum contextuality prove or disprove?

The possibility of experimentally testing the Bell–Kochen–Specker theorem is investigated critically, following the demonstrations by Meyer, Kent, and Clifton–Kent that the predictions of quantum mechanics are indistinguishable (up to arbitrary precision) from those of a non-contextual model, and the subsequent debate about the extent to which these models are actually classical or non-contextual. The present analysis starts from a careful consideration of these ‘finite-precision’ approximations. A stronger condition for non-contextual models, dubbed ontological faithfulness, is exhibited. It is shown that this allows us to approximately formulate the constraints in Bell–Kochen–Specker theorems, such as to render the usual proofs robust. Consequently, one can experimentally test to finite precision ontologically faithful non-contextuality, and thus experimentally refute explanations from this smaller class. We include a discussion of the relation of ontological faithfulness to other proposals to overcome the finite precision objection.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.

Non-contextual chocolate balls versus value indefinite quantum cryptography

Some quantum cryptographic protocols can be implemented with specially prepared metaphorical chocolate balls representing local hidden variables, others protected by value indefiniteness cannot. This latter feature, which follows from Bell– and Kochen–Specker type arguments, is only present in systems with three or more mutually exclusive outcomes. Conversely, there exist local hidden variable models based on chocolate ball configurations utilizable for cryptography which cannot be realized by quantum systems. The possibility that quantum cryptography supported by value indefiniteness (contextuality) has practical advantages over more conventional quantum cryptographic protocols remains highly speculative.

Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell

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.

Applying the Simplest Kochen-Specker Set for Quantum Information Processing

Kochen-Specker (KS) sets are key tools for proving some fundamental results in quantum theory and also have potential applications in quantum information processing. However, so far, their intrinsic complexity has prevented experimentalists from using them for any application. The KS set requiring the smallest number of contexts has been recently found. Relying on this simple KS set, here we report an input state-independent experimental technique to certify whether a set of measurements is actually accessing a preestablished quantum six-dimensional space encoded in the transverse momentum of single photons.

Experimental implementation of an eight-dimensional Kochen-Specker set and observation of its connection with the Greenberger-Horne-Zeilinger theorem

For eight-dimensional quantum systems there is a Kochen-Specker (KS) set of 40 quantum yes-no tests that is related to the Greenberger-Horne-Zeilinger (GHZ) proof of Bell's theorem. Here we experimentally implement this KS set using an eight-dimensional Hilbert space spanned by the transverse momentum of single photons. We show that the experimental results of these tests violate a state-independent noncontextuality inequality. In addition, we show that, if the system is prepared in states that are formally equivalent to a three-qubit GHZ and $W$ states, then the results of a subset of 16 tests violate a noncontextuality inequality that is formally equivalent to the three-party Mermin's Bell inequality, but for single eight-dimensional quantum systems. These experimental results highlight the connection between quantum contextuality and nonlocality for eight-dimensional quantum systems.

Minimal state-dependent proof of measurement contextuality for a qubit

We show that three unsharp binary qubit measurements are enough to violate a generalized noncontextuality inequality, the Liang-Spekkens-Wiseman inequality, in a state-dependent manner. For the case of trine spin axes we calculate the optimal quantum violation of this inequality. In addition, we show that unsharp qubit measurements do not allow a state-independent violation of this inequality. We thus provide a minimal state-dependent proof of measurement contextuality requiring one qubit and three unsharp measurements. Our result rules out generalized noncontextual models of these measurements which were previously conjectured to exist. More importantly, this class of generalized noncontextual models includes the traditional Kochen-Specker (KS) noncontextual models as a proper subset, so our result rules out a larger class of models than those ruled out by a violation of the corresponding KS inequality in this scenario.

Kochen-Specker set with seven contexts

The Kochen-Specker (KS) theorem is a central result in quantum theory and has applications in quantum information. Its proof requires several yes-no tests that can be grouped in contexts or subsets of jointly measurable tests. Arguably, the best measure of simplicity of a KS set is the number of contexts. The smaller this number is, the smaller the number of experiments needed to reveal the conflict between quantum theory and noncontextual theories and to get a quantum vs classical outperformance. The original KS set had 132 contexts. Here we introduce a KS set with seven contexts and prove that this is the simplest KS set that admits a symmetric parity proof.

A quantum random number generator certified by value indefiniteness

In this paper we propose a quantum random number generator (QRNG) that uses an entangled photon pair in a Bell singlet state and is certified explicitly by value indefiniteness. While ‘true randomness’ is a mathematical impossibility, the certification by value indefiniteness ensures that the quantum random bits are incomputable in the strongest sense. This is the first QRNG setup in which a physical principle (Kochen–Specker value indefiniteness) guarantees that no single quantum bit that is produced can be classically computed (reproduced and validated), which is the mathematical form of bitwise physical unpredictability.We discuss the effects of various experimental imperfections in detail: in particular, those related to detector efficiencies, context alignment and temporal correlations between bits. The analysis is very relevant for the construction of any QRNG based on beam-splitters. By measuring the two entangled photons in maximally misaligned contexts and using the fact that two bitstrings, rather than just one, are obtained, more efficient and robust unbiasing techniques can be applied. We propose a robust and efficient procedure based onXORing the bitstrings together – essentially using one as a one-time-pad for the other – to extract random bits in the presence of experimental imperfections, as well as a more efficient modification of the von Neumann procedure for the same task. We also discuss some open problems.

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.

Correspondence Truth and Quantum Mechanics

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.

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.

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.

On the problem of contextuality in macroscopic magnetization measurements

Abstract We show that sharp measurements of total magnetization cannot be used to reveal contextuality in macroscopic many-body systems of spins of arbitrary dimension. We decompose each such measurement into set of projectors corresponding to well-defined value of total magnetization. We then show that such sets of projectors are too restricted to construct Kochen–Specker sets.

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.

Local realism does not impose the commutative algebraic structure into all quantum observables

Malley and Fine (Phys. Rev. A 2006 73 066102) discussed that all quantum observables would commute simultaneously if we accept a realistic theory of the Bell type (a local realistic theory) for quantum events, provided that all quantum events, including every quantum state and every observable (including every projector), are reproduced by a realistic theory of the Bell type for quantum events. We point out that their claim leads us to a contradiction. We study the relation between a local realistic theory and commutativity of quantum observables in a finite-dimensional space. We show that a realistic theory of the Bell type for quantum events does not impose the commutative algebraic structure into the set of all quantum observables if all quantum events are reproduced by a local realistic theory. We discuss that a violation of Bell locality is derived within a realistic theory of the Kochen–Specker type within quantum events.

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.

Experimental Implementation of a Kochen-Specker Set of Quantum Tests

The conflict between classical and quantum physics can be identified through a series of yes-no tests on quantum systems, without it being necessary that these systems be in special quantum states. Kochen-Specker (KS) sets of yes-no tests have this property and provide a quantum-versus-classical advantage that is free of the initialization problem that affects some quantum computers. Here, we report the first experimental implementation of a complete KS set that consists of 18 yes-no tests on four-dimensional quantum systems and show how to use the KS set to obtain a state-independent quantum advantage. We first demonstrate the unique power of this KS set for solving a task while avoiding the problem of state initialization. Such a demonstration is done by showing that, for 28 different quantum states encoded in the orbital-angular-momentum and polarization degrees of freedom of single photons, the KS set provides an impossible-to-beat solution. In a second experiment, we generate maximally contextual quantum correlations by performing compatible sequential measurements of the polarization and path of single photons. In this case, state independence is demonstrated for 15 different initial states. Maximum contextuality and state independence follow from the fact that the sequences of measurements project any initial quantum state onto one of the KS set's eigenstates. Our results show that KS sets can be used for quantum-information processing and quantum computation and pave the way for future developments.

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.

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.