Kochen-Specker Vectors

Abstract

We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hilbert space of any dimension as well as of all the remaining vectors of the space. KS vectors are elements of any set of orthonormal states, i.e., vectors in n-dim Hilbert space, H^n, n>3 to which it is impossible to assign 1s and 0s in such a way that no two mutually orthogonal vectors from the set are both assigned 1 and that not all mutually orthogonal vectors are assigned 0. Our constructive definition of such KS vectors is based on algorithms that generate MMP diagrams corresponding to blocks of orthogonal vectors in R^n, on algorithms that single out those diagrams on which algebraic 0-1 states cannot be defined, and on algorithms that solve nonlinear equations describing the orthogonalities of the vectors by means of statistically polynomially complex interval analysis and self-teaching programs. The algorithms are limited neither by the number of dimensions nor by the number of vectors. To demonstrate the power of the algorithms, all 4-dim KS vector systems containing up to 24 vectors were generated and described, all 3-dim vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.

19 pages, 6 figures, title changed, introduction thoroughly rewritten, n-dim rotation of KS vectors defined, original Kochen-Specker 192 (117) vector system translated into MMP diagram notation with a new graphical representation, results on Tkadlec's dual diagrams added, several other new results added, journal version: to be published in J. Phys. A, 38 (2005). Web page: http://m3k.grad.hr/pavicic

References in corpus (4)

• Finite-precision measurement does not nullify the Kochen-Specker theorem
• The generalized Kochen-Specker theorem
• Kochen-Specker Theorem for Finite Precision Spin One Measurements
• A geometric proof of the Kochen-Specker no-go theorem

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• Experimental implementation of a Kochen-Specker set of quantum tests
• Applying the simplest Kochen-Specker set for quantum information processing
• Necessary and sufficient condition for quantum state-independent contextuality
• Kochen-Specker set with seven contexts
• Quantum state-independent contextuality requires 13 rays
• Converting contextuality into nonlocality
• Proofs of the Kochen-Specker theorem based on a system of three qubits
• Proofs of the Kochen-Specker theorem based on the N-qubit Pauli group
• On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits
• Recursive proof of the Bell-Kochen-Specker theorem in any dimension $n>3$
• New Kochen-Specker Sets in Four Dimensions
• Parity proofs of the Bell-Kochen-Specker theorem based on the 600-cell
• Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions
• Proof of the Peres conjecture for contextuality
• Quantum Contextuality
• Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number
• Parity proofs of the Kochen-Specker theorem based on the Lie algebra E8
• Graph Approach to Quantum Systems
• Probabilistic Generation of Quantum Contextual Sets
• Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres
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• Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces
• Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets
• Parity proofs of the Kochen-Specker theorem based on the 120-cell
• Quantum social networks
• Extensions of Hardy-type true-implies-false gadgets to classically obtain indistinguishability
• Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem
• New Class of 4-Dim Kochen-Specker Sets
• Certifying sets of quantum observables with any full-rank state
• Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem
• Automated generation of Kochen-Specker sets
• Axiom System and Completeness Expression for Quantum Mechanics
• Hypergraph Contextuality
• Contextuality of Identical Particles
• Hilbert Lattice Equations
• Simplest bipartite perfect quantum strategies
• Generalized Greenberger-Horne-Zeilinger arguments from quantum logical analysis
• Noncontextual coloring of orthogonality hypergraphs
• Automated Generation of Arbitrarily Many Kochen-Specker and Other Contextual Sets in Odd Dimensional Hilbert Spaces
• New Examples of Kochen-Specker Type Configurations on Three Qubits
• Vector Generation of Contextual Sets
• Generalised Kochen-Specker Theorem for Finite Non-Deterministic Outcome Assignments
• Non-Kochen-Specker Contextuality
• Converting nonlocality into contextuality
• Optimal conversion of Kochen-Specker sets into bipartite perfect quantum strategies
• Two fundamental solutions to the rigid Kochen-Specker set problem and the solution to the minimal Kochen-Specker set problem under one assumption
• Conway-Kochen and the Finite Precision Loophole
• Maximal Non-Kochen-Specker Sets and a Lower Bound on the Size of Kochen-Specker Sets
• Generation of Kochen-Specker contextual sets in higher dimensions by dimensional upscaling whose complexity does not scale with dimension and their applications
• Multisetting Bell inequalities for $N$ spins-1 avoiding KS contradiction