Pith. sign in

REVIEW 3 major objections 6 minor 2 cited by

One 69-ray nucleus underlies several Kochen-Specker proofs.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review

2026-08-04 20:16 UTC pith:5T2CN2HG

load-bearing objection The 3D unification is real, the 40-4-4 count is interesting but under-supported, and the D=4/D=5 gadgets fail for arbitrary complex vectors as written. the 3 major comments →

arxiv 2509.08636 v1 pith:5T2CN2HG submitted 2025-09-10 quant-ph

Construction of Kochen-Specker Sets from Mutually Unbiased Bases

Mirko Navara , Karl Svozil This is my paper
classification quant-ph MSC 81P1381P16
keywords Kochen-Specker theoremquantum contextualitymutually unbiased basesorthogonality hypergraphstwo-valued statesFourier basisforcing gadgetsPeres-Mermin eigensystem
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper claims that Kochen-Specker contextuality in a three-dimensional Hilbert space is, at root, one structure: a 69-ray, 50-context configuration generated from three mutually unbiased bases. The authors build a complete inventory of 165 rays and 130 orthogonal bases from nine seed vectors, show that several known qutrit Kochen-Specker-style constructions are isomorphic copies of the same 69-50 nucleus, and identify a 40-4-4 asymmetry in how many 'pure' bases each of the three mutually unbiased families produces. They also construct forcing gadgets in dimensions 4 and 5 that use orthogonality constraints to pin a central vector into a maximally unbiased state, and they show that a 20-vector gadget and an 18-vector Kochen-Specker set are informationally equivalent subsets of the same 24-vector Peres-Mermin eigensystem that differ only by which basis completions are included. The upshot: complete orthogonal bases, not just shared vectors, are the carriers of Kochen-Specker logic.

Core claim

Starting from nine seed vectors arranged as three mutually unbiased bases in C^3, the paper generates 165 globally non-collinear rays and classifies all 130 orthogonal triples (contexts) they support. Within this catalogue, a known 13-ray state-independent configuration, a related hypergraph from the literature, and a threefold repetition of the first configuration are all isomorphic to one minimal 69-ray, 50-context Kochen-Specker set. Classifying contexts by the lineage of their rays reveals 40 pure bases from the Fourier basis but only 4 from each of the other two bases; the explanation is that the Fourier basis has generative exclusivity, producing many rays that originate from no other

What carries the argument

Full D-uniform hyperedges—complete orthogonal bases, or contexts—are the objects whose presence or absence determines whether a classical two-valued state assignment exists. Generation starts from nine seed vectors that form three mutually unbiased bases in C^3; a fixed family of 25 algebraic templates applied to the seeds yields 165 rays and 130 bases. The asymmetry mechanism is the Fourier basis's exclusivity (it alone spawns rays no other mutually unbiased basis can generate), while in dimensions 4 and 5 the forcing gadgets use Levi-Civita minor vectors and connector blocks to turn orthogonality into linear equations on squared moduli.

Load-bearing premise

The completeness of the 165-ray/130-base inventory, and the 40-4-4 asymmetry, rest on a fixed list of 25 algebraic templates applied to nine seed vectors; the paper does not prove this list is exhaustive or that other natural templates would not change the counts.

What would settle it

Enumerate all orthogonal bases among the vectors generated from the same nine seeds using a larger or different family of templates—for example, replacing the coefficient 2 in a template such as (-2,1,1) by 3, or adding new algebraic functions—and check whether the split of pure bases remains 40-4-4 and whether the 69-50 nucleus still appears. If the counts change or a second non-isomorphic nucleus emerges, the claimed completeness and exclusivity are artifacts of the chosen table.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • If the 69-50 nucleus is the common core, any proof using one of the known qutrit constructions can be expressed in the same mutually-unbiased-basis hypergraph without changing its logical force.
  • The 40-4-4 asymmetry means the three mutually unbiased bases in C^3 are not interchangeable as generators of contextuality: the Fourier basis contributes many more exclusive rays and pure contexts than the other two, despite the subgroups generating isomorphic 69-50 structures.
  • Context completion is decisive: the same underlying 20 vectors can support or fail to support a separating set of two-valued states depending on which additional orthogonal bases are included, so vector counts alone are not the right measure of a Kochen-Specker proof.
  • In dimensions 4 and 5, explicit orthogonality gadgets can force an arbitrary unknown vector into a maximally unbiased state, giving a constructive route to mutually unbiased basis vectors.
  • Faithful Kochen-Specker analysis should enumerate all contexts, because incomplete hypergraphs can create artificial violations or mask classical embeddability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the template family is not exhaustive, the 'complete' 165-ray/130-base inventory is better read as the orbit of one natural rule; testing wider coefficient ranges would tell whether the 40-4-4 asymmetry is a deep fact about the Fourier basis or a feature of the chosen template list.
  • The forcing-gadget mechanism suggests a general recipe: in any dimension, a chosen rim basis can be equipped with minor scaffolds and connector blocks to enforce uniformity of squared moduli, potentially turning mutually unbiased basis construction into a linear-algebra task.
  • Because the 20-vector gadget and the 18-vector set are informationally equivalent, comparisons of minimal Kochen-Specker resources should be stated with respect to a fixed hypergraph completion; otherwise 'smallest KS set' claims may be ambiguous.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a constructive, MUB-based analysis of Kochen-Specker contextuality. In C^3, it generates a catalog of 165 rays and 130 orthogonal bases from three MUB seeds via a fixed family of 25 algebraic templates, claims that several known KS constructions (Yu-Oh, Harding-Salinas-Schmeis, Cabello's triple diagram) are equivalent manifestations of one 69-ray/50-context KS nucleus, and reports a striking 40-4-4 asymmetry in the number of 'pure' bases, attributed to the algebraic exclusivity of the Fourier basis. In D=4 and D=5, it introduces 'forcing gadgets' built from Levi-Civita minors and connector blocks that are claimed to force a central complex vector into a maximally unbiased state; a 20-vector D=4 gadget is shown to be 'informationally equivalent' to Cabello's 18-vector KS set, with the difference in contextuality attributed solely to the choice of basis completions. The paper concludes that contexts, not merely intertwining vectors, carry KS logic.

Significance. If the central claims held, the paper would provide a useful unifying catalogue of qutrit KS constructions and a thought-provoking demonstration that context choice, not just vector set, can decide uncolorability. The full-context/hyperedge perspective is valuable, and the explicit comparisons between Yu-Oh, Harding-Salinas-Schmeis, and Cabello constructions are instructive. However, the higher-dimensional forcing gadgets are invalid as written for complex Hermitian spaces, and the 'complete inventory' claim rests on an unproven choice of generating templates. The paper is not yet ready for publication; the conceptual contribution is real but the technical foundations need repair.

major comments (3)
  1. [§IV.B and §IV.C] Under the inner product defined in §IV.A as 'standard Hermitian' (which should be ⟨v,w⟩ = Σ v̅_m w_m), the asserted orthogonality ⟨v_{12},u⟩=0 fails: ⟨v_{12},u⟩ = x̄_4 x_3 − x̄_3 x_4, generally nonzero. Hence B_{12}={e1,e2,v12,w12} is not an orthogonal block for arbitrary complex u, and the conclusion |x1|²=...=|x4|² follows only for the non-Hermitian bilinear form Σ v_m w_m. The same missing conjugation invalidates w12⊥v12 and, by identical reasoning, the D=5 triple minors in §IV.C. The concrete example u=(1,1,1,1) obscures the flaw because conjugation is trivial on real coordinates. This is load-bearing for the entire higher-dimensional gadget contribution.
  2. [§III / Table I] The 'complete inventory' of 165 rays and 130 bases is generated by applying a fixed list of 25 algebraic templates (e.g., d1=(−2,1,1), e1=(1,2,1)) to nine MUB seed vectors. No argument is given that this template family is exhaustive or privileged. The coefficient 2 appears arbitrary: replacing it with 3, or with a parameter t, changes the generated set and can change the counts and the 40-4-4 asymmetry. The paper therefore asserts, rather than demonstrates, that the enumeration is 'complete'. A generation theorem defining the natural search space, or a verified exhaustive computer search, is needed.
  3. [§II / §III / Table II] The paper repeatedly relies on the uncolorability of the 69-ray/50-context nucleus and on the existence of separating sets of two-valued states (e.g., the 36 states in §IV.6), but no proof or verifiable code is provided. The acknowledgments mention a Pascal program by Tkadlec, yet the program and its output are not shipped. Since the central equivalence claims depend on these finite-hypergraph properties, the authors should include explicit two-valued-state computations or a reproducibility artifact; otherwise the results are archival assertions rather than checked claims.
minor comments (6)
  1. [§IV.A] The formula '⟨v,w⟩ = Σ v_m w_m' is not the standard Hermitian inner product; the second factor should be conjugated. Either write v̅_m w_m or explicitly state that a complex bilinear form is used (which, however, is not the quantum-mechanical inner product).
  2. [Appendix A 1] The MUBs B2 and B3 listed in Eqs. (A14) and (A15) do not match the B2 and B3 used in Section II and Table I. The text should align these definitions or explain the permutation that relates them.
  3. [Appendix A 1, Eqs. (A9)-(A10)] The equalities 'ω = ω²' and 'ω² = ω' are incorrect. Presumably the authors mean ω̄ = ω² and ω̄² = ω. Please fix the notation.
  4. [§III.C] The 'exclusivity of the Fourier basis' explanation of the 40-4-4 asymmetry is largely a restatement of the counting rather than a derivation. A quantitative relation between the generating functions and the origin sets of the resulting rays would make the argument convincing.
  5. [Table IV / §IV.B.4] The term 'informationally equivalent' is used without a precise definition. Clarify what equivalence means here, and justify that the 'constructed' vectors in the last column are uniquely determined by the stated triples.
  6. [Fig. 2(a)] The 'dotted dark green curve' and several labels are difficult to discern in the figure as printed. Please use higher contrast and ensure all labels are legible.

Circularity Check

1 steps flagged

Self-contained construction; one local self-definitional explanation (40-4-4 purity asymmetry); the D=4/D=5 conjugate-inner-product gap is a correctness flaw, not circularity.

specific steps
  1. self definitional [Section III.B (Asymmetry in Pure Basis Classification) and Section III.C (The Exclusivity of the Fourier Basis)]
    "The resolution lies in a strict definition of purity based on generative origin. A basis is classified as 'truly pure' to a subgroup only if all three of its constituent vectors are generated exclusively by that subgroup. ... The source of the asymmetry is the unique algebraic structure of the first subgroup, B1. ... the set of vectors S1 generated from B1 contains a large subset of vectors whose origin set is strictly limited to this subset."

    The 40-4-4 asymmetry is offered as a finding 'explained' by B1's generative exclusivity. But the paper has already defined a pure basis as one whose three vectors originate exclusively from a single subgroup (III.A: basis color is derived from constituent vector colors; III.B: 'truly pure' means generated exclusively by that subgroup). Hence 'B1 has more pure bases' and 'B1 generates more exclusive vectors' are the same tally restated in two vocabularies. The Fourier-basis exclusivity is a label for the counting rule, not an independent derivation of the asymmetry from MUB properties. The counts are genuine computed outputs, and the main isomorphism/unification claims do not rest on this explanation, so this is a local definitional restatement rather than a fitted prediction.

full rationale

The core constructions are self-contained. The 165-rays/130-bases inventory is generated by explicit algebraic templates from nine seed vectors; no parameter is fitted to reproduce the known Yu-Oh, Harding-Salinas-Schmeis, or Cabello configurations, and the claimed equivalences are supported by explicit vector identifications (Tables I-IV) rather than by citing those results as premises. Self-citations by Svozil ([6], [18], [25], [37]) are not load-bearing: [18] supplies a convenient table of the Peres-Mermin eigensystem, but the vectors used are listed explicitly in the paper. The one genuinely circular passage is the 'explanation' of the 40-4-4 pure-basis asymmetry in Section III: purity and exclusivity are defined by the same origin-set condition, so the explanation reduces to a restatement of the counting. The completeness and minimality claims ('complete inventory', 'minimal 69-50 nucleus') are asserted relative to the chosen 25-template family rather than proven exhaustively; that is a scope/correctness limitation, not circularity. The D=4/D=5 forcing-gadget analysis contains a separate mathematical gap (the inner product is written without conjugation, so the claimed automatic orthogonality of the Levi-Civita minors fails for generic complex center vectors under the true Hermitian product); this is a correctness flaw as written, but it is not a case of a fitted parameter being renamed as a prediction, so it does not raise the circularity score. Overall: one local definitional restatement, no central reduction to inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 3 axioms · 0 invented entities

The central contributions rest on a hand-chosen generation family and a stated support assumption; no fits to data or new physical entities are introduced.

free parameters (1)
  • template coefficient 2 in the 25 generating functions = 2
    The vectors in Table I use the integer 2 in templates such as d1=(-2,1,1), e1=(1,2,1), and b112=(2,-1,1). The enumeration counts and the 40-4-4 asymmetry depend on this choice, which is not justified or varied.
axioms (3)
  • ad hoc to paper A fixed list of 25 algebraic templates applied to nine MUB seed vectors generates the complete inventory of rays and contexts relevant to KS contextuality.
    Table I caption: 'A family of 25 algebraic functions (rows) is applied to nine initial vectors.' No completeness argument is given.
  • domain assumption The central vector u in the gadgets has x_j != 0 for all coordinates j.
    Section IV.A: 'Fix an unknown center vector u=(x1,...,xD) in C^D with x_j != 0 for all j.' The gadget blocks such as v_ij require nonzero components; the zero-support case is not handled.
  • domain assumption Two-valued states on the block hypergraph are the correct criterion for classical embeddability and KS uncolorability.
    Standard KS/OML framework, invoked in Sections I and IV.6.

reviewed 2026-08-04 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Construction of Kochen-Specker Sets from Mutually Unbiased Bases." pith.science (2026). https://pith.science/paper/5T2CN2HG

@misc{pith2026250908636,
  author       = {Pith},
  title        = {Pith review of: Construction of Kochen-Specker Sets from Mutually Unbiased Bases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5T2CN2HG}},
  note         = {Machine review of arXiv:2509.08636}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We present a systematic, constructive analysis of Kochen-Specker contextuality, emphasizing the foundational importance of complete orthogonal bases (contexts). First, in three dimensions, we generate a complete inventory of 165 rays and 130 bases from mutually unbiased bases. This unified framework reveals that several known constructions are equivalent manifestations of a minimal 69-ray, 50-context Kochen--Specker nucleus and uncovers a striking 40-4-4 generative asymmetry among the mutual unbiased bases, which we explain via the algebraic exclusivity of the Fourier basis. Second; in higher dimensions (D=4, 5), we develop explicit "forcing gadgets" that use orthogonality constraints to compel a central vector into a state of maximal unbiasedness. We demonstrate that our 20-vector gadget in D=4 and the 18-vector Cabello set are informationally equivalent subsets of the Peres--Mermin eigensystem, yet differ in their contextuality due to the choice of basis completions. Our findings establish that contexts, not merely intertwining vectors, are the crucial carriers of Kochen-Specker type logic and are indispensable for a rigorous assessment of quantum contextuality.

Figures

Figures reproduced from arXiv: 2509.08636 by Karl Svozil, Mirko Navara.

Figure 1
Figure 1. Figure 1: FIG. 1. The Yu–Oh quantum logic [30] drawn in the Harding [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Generalization of the Harding and Salinas Schmeis 3-uniform hypergraph scheme [27] in 4 dimensions. Solid [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Chromatic Completeness and the Independence of Geometric Obstruction

    quant-ph 2026-07 accept novelty 7.0

    Strong chromatic number exceeding dimension blocks only chromatic completeness, not faithful orthogonal ray representations; completed Yu–Oh has χ=4 with an R³ FOR while Greechie G₃₂ has χ=4 with none in C³.

  2. Faithful real embedding of a three-dimensional complex Kochen-Specker configuration

    quant-ph 2025-11 conditional novelty 4.0

    Any finite set of complex 3D rays can be phase-adjusted to embed faithfully in real 6D, and the Cabello 165-ray KS set becomes colourable there.

Reference graph

Works this paper leans on

53 extracted references · 14 linked inside Pith · cited by 2 Pith papers

  1. [1]

    Orthogonalities among some minors, for instance, v12 ⊥v 34, hold automatically due to their disjoint sup- port and do not constrainu

    Automatic versus imposed. Orthogonalities among some minors, for instance, v12 ⊥v 34, hold automatically due to their disjoint sup- port and do not constrainu. The constraints arise from the connector blocks that includeuand theg (·) vectors

  2. [2]

    Hidden blocks from disjoint support. A complete search of the 20-vector set for the concrete exampleu= (1,1,1,1) reveals three additional blocks of four mutually orthogonal vectors that are not part of the explicit construction. These are formed by combining complementary pairs of minors: B1234 ={v 12, w12, v34, w34}, B1324 ={v 13, w13, v24, w24}, B1423 =...

  3. [3]

    orthogonal tov i,v j, andv k

    ConcreteD= 4example:u= (1,1,1,1) Take the center vectoru= (1,1,1,1) and the rim vec- tors e1 = (1,0,0,0), e 2 = (0,1,0,0), e3 = (0,0,1,0), e 4 = (0,0,0,1). 8 v1324 v1234 v1423 u v13 v24w13 w24 w14 w23 v14 v23 v12 w12v34 w34 e1 e2e3 e4 v12 v34 v1234 u v1324 v24 v13 w24 v1423 {v12 , v13 , w34 }⊥ {v14 , v24 , w23 }⊥ w23 v14 w14 e4 e1 e2 w12 (a) (b) FIG. 2. (...

  4. [4]

    provisions

    Informational Equivalence and Relation to the Peres–Mermin Eigensystem A detailed comparison reveals that both the 20- vector gadget and the 18-vector Cabello et al. 18-9 KS- configuration, as modified and depicted in Fig. 2(b), are subsets of the larger 24-vector set that forms the com- plete eigensystem of the Peres–Mermin square, as enu- merated by Pav...

  5. [5]

    completion

    Discussion of Structural Symmetry A related notable feature of the MUB-enforcing dia- gram presented in Fig. 2(a) is an apparent lack of symme- try. Specifically, one may observe that certain graphical substructures do not possess obvious counterparts. For example, the block of vectors and their connections repre- sented by{v 13, u, v1324, v24}appears uni...

  6. [6]

    True-Implies-False

    A remark of caution when it comes to contextual properties The properties of a set of states on an orthomodu- lar lattice (OML) are not intrinsic to the states alone but are highly dependent on the completeness of the un- derlying algebraic structure. A compelling example of this contextuality can be seen by comparing two OMLs constructed from 4-dimension...

  7. [7]

    Letε abcde be the totally antisymmetric tensor with ε12345 = +1

    Triple minors (Levi-Civita contraction). Letε abcde be the totally antisymmetric tensor with ε12345 = +1. For each triple{i, j, k}, define the vector vijk by its components: (vijk )m = 5X ℓ=1 εmijkℓ xℓ. Thenv ijk has zero components in coordinatesi, j, kand is orthogonal tou, that is,⟨v ijk , u⟩= 0

  8. [8]

    For each triple, choose a vectorw ijk supported on the complementary two coordinates such that⟨v ijk , wijk ⟩=

    Scaffolding and connectors. For each triple, choose a vectorw ijk supported on the complementary two coordinates such that⟨v ijk , wijk ⟩=

  9. [9]

    triple diagram

    This gives the blocks Bijk ={e i, ej, ek, vijk , wijk }. To force equal moduli, use four connector vectors: g1 = (x1,0,0,0,−x 5), g2 = (0, x2,0,0,−x 5), g3 = (0,0, x3,0,−x 5), g4 = (0,0,0, x4,−x 5), so that the orthogonality condition⟨u, g i⟩= 0 implies |xi|2 − |x5|2 = 0. Four connector blocks Ci ={u, gi, h(i) 1 , h(i) 2 , h(i) 3 } (where thehvectors form...

  10. [10]

    A. M. Gleason, Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Mechanics (now Indiana University Mathematics Journal)6, 885 (1957)

    1957

  11. [11]

    Kochen and E

    S. Kochen and E. P. Specker, The problem of hidden variables in quantum mechanics, Journal of Mathemat- ics and Mechanics (now Indiana University Mathematics Journal)17, 59 (1967)

    1967

  12. [12]

    Specker, Die Logik nicht gleichzeitig entscheidbarer Aussagen, Dialectica14, 239 (1960), english translation at https://arxiv.org/abs/1103.4537, arXiv:1103.4537

    E. Specker, Die Logik nicht gleichzeitig entscheidbarer Aussagen, Dialectica14, 239 (1960), english translation at https://arxiv.org/abs/1103.4537, arXiv:1103.4537

    Pith/arXiv arXiv 1960

  13. [13]

    Bretto,Hypergraph theory, Mathematical Engineering (Springer, Cham, Switzerland, 2013) pp

    A. Bretto,Hypergraph theory, Mathematical Engineering (Springer, Cham, Switzerland, 2013) pp. xiv+119

    2013

  14. [14]

    M. H. Shekarriz and K. Svozil, Noncontextual coloring of orthogonality hypergraphs, Journal of Mathematical Physics63, 032104 (2022), arXiv:2105.08520

    Pith/arXiv arXiv 2022

  15. [15]

    Svozil and J

    K. Svozil and J. Tkadlec, Greechie diagrams, nonexis- tence of measures in quantum logics and Kochen–Specker type constructions, Journal of Mathematical Physics37, 5380 (1996)

    1996

  16. [16]

    Bub, Sch¨ utte’s tautology and the Kochen–Specker the- orem, Foundations of Physics26, 787 (1996)

    J. Bub, Sch¨ utte’s tautology and the Kochen–Specker the- orem, Foundations of Physics26, 787 (1996)

    1996

  17. [17]

    Tkadlec, Greechie diagrams of small quantum logics with small state spaces, International Journal of Theo- retical Physics37, 203 (1998)

    J. Tkadlec, Greechie diagrams of small quantum logics with small state spaces, International Journal of Theo- retical Physics37, 203 (1998)

    1998

  18. [18]

    Sch¨ utte, Letter to ernst specker, kiel, dated april 22nd, 1965 (1965), this letter is mentioned in Ref

    K. Sch¨ utte, Letter to ernst specker, kiel, dated april 22nd, 1965 (1965), this letter is mentioned in Ref. [10]. A second letter to Ernst Specker from Kurt Sch¨ utte in Munich, dated November 30, 1983, hints on how Sch¨ utte had obtained this configuration:“Zu- grunde liegt offenbar ein Schließungssatz einer vollellip- tischen Ebene in einer speziellen ...

    1965

  19. [19]

    Clavadetscher-Seeberger,Eine partielle Pr¨ adikatenlogik, Ph.D

    E. Clavadetscher-Seeberger,Eine partielle Pr¨ adikatenlogik, Ph.D. thesis, ETH-Z¨ urich, Z¨ urich (1983)

    1983

  20. [20]

    Kochen and E

    S. Kochen and E. P. Specker, Logical structures arising in quantum theory, inThe Theory of Models, Proceedings of the 1963 International Symposium at Berkeley, edited by J. W. Addison, L. Henkin, and A. Tarski (North Holland, Amsterdam, New York, Oxford, 1965) pp. 177–189

    1963

  21. [21]

    Optimal conversion of Kochen-Specker sets into bipartite per- fect quantum strategies

    A. Cabello, Reply to Paviˇ ci´ c’s ‘Comment on “Optimal conversion of Kochen-Specker sets into bipartite per- fect quantum strategies” ’ (arxiv:2502.13787) (2025), arXiv:2503.02974

    Pith/arXiv arXiv 2025

  22. [22]

    Cabello, Experimentally testable state-independent quantum contextuality, Physical Review Letters101, 210401 (2008), arXiv:0808.2456

    A. Cabello, Experimentally testable state-independent quantum contextuality, Physical Review Letters101, 210401 (2008), arXiv:0808.2456

    Pith/arXiv arXiv 2008

  23. [23]

    Trandafir and A

    S. Trandafir and A. Cabello, Optimal conversion of Kochen-Specker sets into bipartite perfect quantum strategies (2025), arXiv:2410.17470

    Pith/arXiv arXiv 2025

  24. [24]

    Paviˇ ci´ c, Quantum contextual hypergraphs, operators, inequalities, and applications in higher dimensions, En- tropy27, 54 (2025)

    M. Paviˇ ci´ c, Quantum contextual hypergraphs, operators, inequalities, and applications in higher dimensions, En- tropy27, 54 (2025)

    2025

  25. [25]

    Cabello, J

    A. Cabello, J. M. Estebaranz, and G. Garc ´ ıa-Alcaine, Bell-Kochen-Specker theorem: A proof with 18 vec- tors, Physics Letters A212, 183 (1996), arXiv:quant- ph/9706009

    arXiv 1996

  26. [26]

    Paviˇ ci´ c, Kochen-Specker algorithms for qubits, AIP Conference Proceedings734, 195 (2004)

    M. Paviˇ ci´ c, Kochen-Specker algorithms for qubits, AIP Conference Proceedings734, 195 (2004)

    2004

  27. [27]

    Svozil, Converting nonlocality into contex- tuality, Physical Review A110, 012215 (2024), arXiv:2404.15793

    K. Svozil, Converting nonlocality into contex- tuality, Physical Review A110, 012215 (2024), arXiv:2404.15793

    Pith/arXiv arXiv 2024

  28. [28]

    Peres, Incompatible results of quantum measure- ments, Physics Letters A151, 107 (1990)

    A. Peres, Incompatible results of quantum measure- ments, Physics Letters A151, 107 (1990)

    1990

  29. [29]

    D. N. Mermin, Simple unified form for the major no- hidden-variable theorems, Physical Review Letters65, 3373 (1990)

    1990

  30. [30]

    Peres, Two simple proofs of the Kochen-Specker the- orem, Journal of Physics A: Mathematical and General 24, L175 (1991)

    A. Peres, Two simple proofs of the Kochen-Specker the- orem, Journal of Physics A: Mathematical and General 24, L175 (1991)

    1991

  31. [31]

    Paviˇ ci´ c, J.-P

    M. Paviˇ ci´ c, J.-P. Merlet, B. McKay, and N. D. Megill, Kochen-Specker vectors, Journal of Physics A: Math- ematical and General38, 1577 (2005), arXiv:quant- ph/0409014

    arXiv 2005

  32. [32]

    Paviˇ ci´ c, J.-P

    M. Paviˇ ci´ c, J.-P. Merlet, B. D. McKay, and N. D. Megill, Corrigendum: Kochen-Specker vectors, Journal of Physics A: Mathematical and General38, 3709 (2005)

    2005

  33. [33]

    Budroni, A

    C. Budroni, A. Cabello, O. G¨ uhne, M. Kleinmann, and J.-A. Larsson, Kochen-Specker contextuality, Reviews of Modern Physics94, 045007 (2022), arXiv:2102.13036

    Pith/arXiv arXiv 2022

  34. [34]

    Svozil, Chromatic quantum contextuality, Entropy 27, 387 (2025), arXiv:2501.15261

    K. Svozil, Chromatic quantum contextuality, Entropy 27, 387 (2025), arXiv:2501.15261

    Pith/arXiv arXiv 2025

  35. [35]

    R. J. Greechie, Orthomodular lattices admitting no states, Journal of Combinatorial Theory. Series A10, 119 (1971)

    1971

  36. [36]

    Harding and R

    J. Harding and R. Salinas Schmeis, Remarks on or- thogonality spaces, International Journal of Theoreti- cal Physics64, 211 (2025), arXiv:2505.13871 [math-ph], 13 arXiv:arXiv:2505.13871 [math-ph]

    Pith/arXiv arXiv 2025

  37. [37]

    Badzi¸ ag, I

    P. Badzi¸ ag, I. Bengtsson, A. Cabello, and I. Pitowsky, Universality of state-independent violation of correlation inequalities for noncontextual theories, Physical Review Letters103, 050401 (2009)

    2009

  38. [38]

    Yu, Y.-Q

    X.-D. Yu, Y.-Q. Guo, and D. M. Tong, A proof of the Kochen-Specker theorem can always be converted to a state-independent noncontextuality inequality, New Journal of Physics17, 093001 (2015)

    2015

  39. [39]

    Yu and C

    S. Yu and C. H. Oh, State-independent proof of Kochen- Specker theorem with 13 rays, Physical Review Letters 108, 030402 (2012), arXiv:1109.4396

    Pith/arXiv arXiv 2012

  40. [40]

    Cabello, The simplest Kochen-Specker set (2025), arXiv:2508.07335 [quant-ph]

    A. Cabello, The simplest Kochen-Specker set (2025), arXiv:2508.07335 [quant-ph]

    arXiv 2025

  41. [41]

    Durt, B.-G

    T. Durt, B.-G. Englert, I. Bengtsson, and K. ˙Zyczkowski, On mutually unbiased bases, International Journal of Quantum Information8, 535 (2010), arXiv:1004.3348

    Pith/arXiv arXiv 2010

  42. [42]

    McNulty and S

    D. McNulty and S. Weigert, Mutually unbiased bases in composite dimensions – a review (2024), arXiv:2410.23997 [quant-ph], arXiv:2410.23997 [quant- ph]

    Pith/arXiv arXiv 2024

  43. [43]

    Cabello, A simple proof of the Kochen-Specker theo- rem, European Journal of Physics15, 179 (1994)

    A. Cabello, A simple proof of the Kochen-Specker theo- rem, European Journal of Physics15, 179 (1994)

    1994

  44. [44]

    Harding, K

    J. Harding, K. Jager, and D. Smith, Group-valued mea- sures on the lattice of closed subspaces of a Hilbert space, International Journal of Theoretical Physics44, 539 (2005)

    2005

  45. [45]

    M. Navara, Mathematical questions related to non- existence of hidden variables, inFoundations of Proba- bility and Physics 5, American Institute of Physics Con- ference Proceedings, Vol. 1101, edited by L. Accardi, G. Adenier, C. Fuchs, G. Jaeger, A. Y. Khrennikov, J.-A. Larsson, and S. Stenholm (American Institute of Physics, New York, USA, 2009) pp. 119–126

    2009

  46. [46]

    Cabello, J

    A. Cabello, J. R. Portillo, A. Sol ´ ıs, and K. Svozil, Mini- mal true-implies-false and true-implies-true sets of propo- sitions in noncontextual hidden-variable theories, Physi- cal Review A98, 012106 (2018), arXiv:1805.00796

    Pith/arXiv arXiv 2018

  47. [47]

    Schwinger, Unitary operators bases, Proceedings of the National Academy of Sciences (PNAS)46, 570 (1960)

    J. Schwinger, Unitary operators bases, Proceedings of the National Academy of Sciences (PNAS)46, 570 (1960)

    1960

  48. [48]

    W. K. Wootters and B. D. Fields, Optimal state- determination by mutually unbiased measurements, An- nals of Physics191, 363 (1989)

    1989

  49. [49]

    S. Brierley,Mutually Unbiased Bases in Low Dimensions, Phd thesis, University of York, York, UK (2009), eTHoS ID: uk.bl.ethos.516399, Deposited: 01 Jun 2010, Last Modified: 08 Sep 2016

    2009

  50. [50]

    Klappenecker and M

    A. Klappenecker and M. R¨ otteler, Constructions of mu- tually unbiased bases, inFinite Fields and Applications, Lecture Notes in Computer Science, Vol. 2948, edited by G. Mullen, A. Poli, and H. Stichtenoth (Springer, Berlin, Heidelberg, 2004) pp. 262–266, arXiv:quant-ph/0309120

    Pith/arXiv arXiv 2004

  51. [51]

    I. D. Ivanovic, Unbiased projector basis overc 3, Physics Letters A228, 329 (1997). Appendix A: Mutually unbiased bases in dimensions 3 and 4 For the motivation and introduction and more com- plete treatments of MUPs we refer to Refs. [32, 33, 38– 41]

    1997

  52. [52]

    We denote these byB j,j= 0,

    Mutually unbiased bases inC 3 Let ω:= e 2πi/3 = e−4πi/3,(A1) ω3 = 1,(A2) ω̸= 1,(A3) ω2 = e4πi/3 = e−2πi/3,(A4) e2πi/3 = e−2πi/3 =ω 2,(A5) e−2πi/3 = e2πi/3 =ω,(A6) e4πi/3 = e−4πi/3 =ω,(A7) e−4πi/3 = e4πi/3 =ω 2,(A8) ω=ω 2,(A9) ω2 =ω,(A10) 0 = 1 +ω+ω 2.(A11) In dimensionD= 3 a complete set of mutually un- biased bases (MUBs) containsD+ 1 = 4 orthonormal bas...

  53. [53]

    We denote each basis byB j,j= 14 0,

    Mutually unbiased bases inC 4 In dimensionD= 4, a complete set of mutually unbiased bases (MUBs) consists ofD+ 1 = 5 or- thonormal bases. We denote each basis byB j,j= 14 0, . . . ,4. All vectors below are normalized, written in Cartesian coordinates relative to the standard ordering (1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1). Forj̸=k, the overlaps satisfy|⟨...

This paper was first reviewed by deepseek-v4-flash on August 4, 2026.