Maximal Non-Kochen-Specker Sets and a Lower Bound on the Size of Kochen-Specker Sets

• URL: http://arxiv.org/abs/2403.05230v1
• Date: Fri, 8 Mar 2024 11:38:16 GMT
• Title: Maximal Non-Kochen-Specker Sets and a Lower Bound on the Size of Kochen-Specker Sets
• Authors: Tom Williams and Andrei Constantin
• Abstract summary: A Kochen-Specker (KS) set is a finite collection of vectors on the two-sphere containing no antipodal pairs. The existence of KS sets lies at the heart of Kochen and Specker's argument against non-contextual hidden variable theories.
• Score: 1.5163329671980246
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A Kochen-Specker (KS) set is a finite collection of vectors on the two-sphere containing no antipodal pairs for which it is impossible to assign 0s and 1s such that no two orthogonal vectors are assigned 1 and exactly one vector in every triplet of mutually orthogonal vectors is assigned 1. The existence of KS sets lies at the heart of Kochen and Specker's argument against non-contextual hidden variable theories and the Conway-Kochen free will theorem. Identifying small KS sets can simplify these arguments and may contribute to the understanding of the role played by contextuality in quantum protocols. In this paper we derive a weak lower bound of 10 vectors for the size of any KS set by studying the opposite notion of large non-KS sets and using a probability argument that is independent of the graph structure of KS sets. We also point out an interesting connection with a generalisation of the moving sofa problem around a right-angled hallway on the two-sphere.

Related papers

• Reply to Pavičić's 'Comment on "Optimal conversion of Kochen-Specker sets into bipartite perfect quantum strategies"' (arXiv:2502.13787) [0.0]
  Pavivci'c claims that 10 statements in our paper are wrong. We respond to other claims made by Pavivci'c about our work.
  arXiv  Detail & Related papers  (2025-03-04T20:00:06Z)
• Two fundamental solutions to the rigid Kochen-Specker set problem and the solution to the minimal Kochen-Specker set problem under one assumption [0.0]
  We show that two fundamental structures of quantum theory define two rigid KS sets. The second construction provides a clue to solve the minimal KS problem. We conjecture that 31 is the solution to the minimal KS set problem.
  arXiv  Detail & Related papers  (2025-01-20T18:13:28Z)
• Geometric structure and transversal logic of quantum Reed-Muller codes [51.11215560140181]
  In this paper, we aim to characterize the gates of quantum Reed-Muller (RM) codes by exploiting the well-studied properties of their classical counterparts. A set of stabilizer generators for a RM code can be described via $X$ and $Z$ operators acting on subcubes of particular dimensions.
  arXiv  Detail & Related papers  (2024-10-10T04:07:24Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Generalised Kochen-Specker Theorem for Finite Non-Deterministic Outcome Assignments [0.0]
  We show that the Kochen-Specker (KS) theorem rules out hidden variable theories with outcome assignments in the set $0, p, 1-p, 1$ for $p in [0,1/d) cup (1/d, 1/2]$.
  arXiv  Detail & Related papers  (2024-02-14T14:02:37Z)
• The Schmidt rank for the commuting operator framework [58.720142291102135]
  The Schmidt rank is a measure for the entanglement dimension of a pure bipartite state. We generalize the Schmidt rank to the commuting operator framework. We analyze bipartite states and compute the Schmidt rank in several examples.
  arXiv  Detail & Related papers  (2023-07-21T14:37:33Z)
• A SAT Solver and Computer Algebra Attack on the Minimum Kochen-Specker Problem [14.693394941317843]
  We present a new verifiable proof-producing method based on a combination of a Boolean satisfiability solver and a computer algebra system. Our method shows that a KS system in 3D must contain at least 24 vectors. We also provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a size of 40.3 TiB in order 23.
  arXiv  Detail & Related papers  (2023-06-23T06:42:59Z)
• Connecting classical finite exchangeability to quantum theory [69.62715388742298]
  Exchangeability is a fundamental concept in probability theory and statistics. We show how a de Finetti-like representation theorem for finitely exchangeable sequences requires a mathematical representation which is formally equivalent to quantum theory.
  arXiv  Detail & Related papers  (2023-06-06T17:15:19Z)
• Unextendibility, uncompletability, and many-copy indistinguishable ensembles [77.34726150561087]
  We study unextendibility, uncompletability and analyze their connections to many-copy indistinguishable ensembles. We report a class of multipartite many-copy indistinguishable ensembles for which local indistinguishability property increases with decreasing mixedness.
  arXiv  Detail & Related papers  (2023-03-30T16:16:41Z)
• Reexamination of the Kochen-Specker theorem: Relaxation of the completeness assumption [0.0]
  The Kochen-Specker theorem states that exclusive and complete deterministic outcome assignments are impossible for certain sets of measurements. We show it is possible to construct a joint quasiprobability distribution over any KS set by relaxing the completeness assumption.
  arXiv  Detail & Related papers  (2022-10-13T08:13:16Z)
• Automated Generation of Arbitrarily Many Kochen-Specker and Other Contextual Sets in Odd Dimensional Hilbert Spaces [0.0]
  We give three methods for automated generation of arbitrarily many contextual KS and non-KS sets in any dimension. No explicit vectors for the original Kochen-Specker set were known so far, while we now generate them from 24 vector components.
  arXiv  Detail & Related papers  (2022-02-16T17:15:45Z)
• Annihilating Entanglement Between Cones [77.34726150561087]
  We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
  arXiv  Detail & Related papers  (2021-10-22T15:02:39Z)
• Contextuality in composite systems: the role of entanglement in the Kochen-Specker theorem [0.0]
  The Kochen--Specker (KS) theorem reveals the nonclassicality of single quantum systems. Bell's theorem and entanglement concern the nonclassicality of composite quantum systems.
  arXiv  Detail & Related papers  (2021-09-28T10:10:53Z)
• Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds [70.32518963244466]
  A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant.
  arXiv  Detail & Related papers  (2021-06-10T19:54:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.