18-Vector Cabello Set in ℂ⁴

• 18-Vector Cabello Set is a minimal Kochen–Specker configuration in ℂ⁴ that organizes 18 rays into 9 complete orthonormal contexts, producing a parity proof.
• It provides a state-independent contradiction against noncontextual hidden-variable assignments, thereby settling the Peres conjecture in dimension 4.
• The set also relates to the 24-vector Peres–Mermin eigensystem, highlighting that contextuality critically depends on basis completions rather than ray content alone.

Searching arXiv for recent and foundational papers on the 18-vector Cabello set. Tool unavailable in this environment; proceeding with the arXiv papers already provided in the source block and citing them directly. The 18-vector Cabello set is the $18$-ray, $9$-context Kochen–Specker configuration introduced by Cabello, Estebaranz, and García-Alcaine in dimension $4$. In the modern literature it is usually treated as the canonical finite, state-independent contradiction between quantum mechanics and noncontextual hidden-variable assignments: it consists of $18$ vectors in $\mathbb C^4$, organized into $9$ complete contexts, and yields an immediate parity proof of the Kochen–Specker theorem. A central result of later work is that this construction is not merely the smallest known example in dimension $4$, but the minimal Kochen–Specker vector set in any Hilbert-space dimension, thereby settling the Peres conjecture (Xu et al., 2020). More recent analysis places the same configuration inside the $24$-vector Peres–Mermin eigensystem and emphasizes that its contextuality depends essentially on the selected basis completions, not on ray content alone (Navara et al., 10 Sep 2025).

1. Formal setting and definition

In the standard Kochen–Specker framework, one considers ideal measurements and their outcomes, called events. In quantum mechanics, rank-$1$ events are represented by vectors, or projectors onto them, and orthogonal vectors correspond to mutually exclusive events. A context is a set of compatible mutually exclusive events; a complete context is one in which exactly one event occurs with certainty for every state, which in the vector setting means an orthonormal basis of the full space (Xu et al., 2020).

A noncontextual hidden-variable model must assign to each event a predetermined value $0$ or $9$0, independent of the context in which that event is considered. For a vector set $9$1, the assignment $9$2 would have to satisfy

$9$3

and for every complete context $9$4,

$9$5

A Kochen–Specker set is a set of vectors for which no such assignment exists (Xu et al., 2020).

In the $9$6 discussion of later work, the same objects are treated projectively: vectors represent the one-dimensional subspaces they span, they may be freely rescaled when forming blocks, and overall signs are ignored because they represent the same ray. In that sense, the 18-vector Cabello set is equally an $9$7-ray configuration (Navara et al., 10 Sep 2025).

2. Explicit $9$8-vector, $9$9-context configuration

The Cabello–Estebaranz–García-Alcaine configuration lives in a four-dimensional Hilbert space and consists of $4$0 vectors arranged into $4$1 complete contexts, each context being a basis of $4$2 (Xu et al., 2020). Using the labels $4$3, the nine contexts are

$4$4

$4$5

$4$6

Up to normalization, the vectors are

$4$7

Later work does not reprint the set as a standalone ordered list, but gives a correspondence table showing how the Cabello-labeled rays sit inside a comparison with a $4$8-vector gadget. In that presentation, examples include

$4$9

together with the convention that these are rays and must be normalized to form literal orthonormal bases (Navara et al., 10 Sep 2025).

3. Parity proof and state-independent contradiction

The 18-vector Cabello set is the paradigmatic parity proof of the Kochen–Specker theorem. Because the $18$0 contexts are complete, any noncontextual assignment must assign exactly one $18$1 in each context, giving a total of $18$2 ones when counted context-by-context. But every vector appears in exactly two contexts. Therefore the total number of assigned ones, counted over all table entries, must be even (Xu et al., 2020).

The contradiction is immediate:

• completeness implies an odd count, equal to $18$3;
• context-independence together with double occurrence implies an even count.

Hence no noncontextual assignment can satisfy the required constraints. This is the standard parity-based Kochen–Specker contradiction for the configuration, and it is used as a canonical example of a state-independent Kochen–Specker proof (Xu et al., 2020).

Historically, the significance of this construction is sharpened by comparison with larger earlier proofs. The same line of work contrasts it with Kochen and Specker’s original $18$4-vector proof in $18$5, as well as later simplifications by Peres, Kernaghan, and others, and notes that the Cabello configuration had long been regarded as the simplest known Kochen–Specker proof before being shown to be provably minimal (Xu et al., 2020).

4. Reduction to a GHZ-type proof

A central technical advance is the conversion of a Kochen–Specker set into a stronger GHZ-type contradiction with fewer events. The general method is: choose one vector as the quantum state and delete that vector together with all vectors orthogonal to it. Applied to the Cabello $18$6-vector set, this yields a $18$7-event GHZ-type proof using $18$8 contexts (Xu et al., 2020).

For the Cabello example, the chosen state is

$18$9

Then certain probabilities are forced: $\mathbb C^4$0 and the remaining relevant contexts imply

$\mathbb C^4$1

$\mathbb C^4$2

$\mathbb C^4$3

For a deterministic noncontextual assignment $\mathbb C^4$4, summing these equations gives

$\mathbb C^4$5

so $\mathbb C^4$6 must be even. Since $\mathbb C^4$7 and $\mathbb C^4$8 are exclusive, $\mathbb C^4$9, and thus any noncontextual model must satisfy

$9$0

Quantum mechanically, however, for the state $9$1,

$9$2

This is a GHZ-type contradiction in the sense used there: under the imposed context constraints, classical noncontextual models force an event sum to vanish, while quantum mechanics makes it equal to one with certainty (Xu et al., 2020).

The same work also defines a Hardy-type proof by the implication

$9$3

while quantum mechanics allows $9$4. If quantum mechanics achieves

$9$5

the contradiction is GHZ-type. The Cabello-derived $9$6-event construction belongs to this stronger class (Xu et al., 2020).

Graph-theoretically, the scenario is encoded by an exclusivity graph $9$7: vertices are events, adjacent vertices are mutually exclusive events, and contexts correspond to cliques. Classical noncontextual assignments form

$9$8

while quantum assignments lie in

$9$9

An exhaustive search over all non-isomorphic graphs with fewer than $4$0 vertices shows that no GHZ-type proof exists with fewer than $4$1 events. Therefore the $4$2-event proof derived from the Cabello set is minimal (Xu et al., 2020).

5. Proof of the Peres conjecture

The decisive theorem is stated as follows: $4$3 This establishes that the Cabello set is the smallest possible Kochen–Specker vector set in any dimension (Xu et al., 2020).

The proof combines three ingredients. First, any Kochen–Specker set can be reduced to a GHZ-type proof with fewer events. Second, exhaustive graph search shows that every GHZ-type proof needs at least $4$4 events. Third, dimension-dependent combinatorial arguments constrain how complete contexts may overlap in a hypothetical smaller Kochen–Specker set (Xu et al., 2020).

A preliminary lemma states: $4$5 Thus a Kochen–Specker set with every vector appearing only once must have at least $4$6 vectors (Xu et al., 2020).

For $4$7 and $4$8, earlier lower bounds already gave $4$9 and $24$0, respectively, so the new work concerns $24$1. For $24$2, if no two complete contexts overlap, then

$24$3

Hence any minimal candidate must contain overlapping contexts $24$4 sharing some $24$5. Since distinct complete contexts cannot differ in only one vector,

$24$6

and therefore

$24$7

Choosing $24$8 as the state and removing all vectors in $24$9 leaves a GHZ-type proof with at most

$1$0

events. Because every GHZ-type proof needs at least $1$1 events,

$1$2

so

$1$3

For $1$4, this implies $1$5 (Xu et al., 2020).

The $1$6 case is handled separately by additional combinatorial restrictions. If two complete contexts share one or two vectors, then their union has at least $1$7 vectors, and the same reduction gives

$1$8

Hence, in any hypothetical $1$9 Kochen–Specker set with fewer than $0$0 vectors, intersections of complete contexts can only have size $0$1 or $0$2. A second restriction excludes the presence of a vector orthogonal to the shared state outside the union of two such contexts, since that would again force $0$3. From these restrictions one derives a three-context overlap pattern with common vector $0$4; the only remaining candidate overlap scheme is then shown to be impossible. Consequently, no $0$5-vector Kochen–Specker set exists in $0$6 either (Xu et al., 2020).

The Cabello set therefore occupies three simultaneous roles: it is the canonical $0$7-vector parity Kochen–Specker proof, the source of the minimal $0$8-event GHZ-type contradiction, and the witness that the lower bound $0$9 is attainable (Xu et al., 2020).

6. Relation to the Peres–Mermin eigensystem and the role of contexts

Later work situates the 18-vector Cabello set inside the larger $9$00-$9$01 Peres–Mermin eigensystem and compares it to a newly constructed $9$02-vector forcing gadget in $9$03. The central claim is that the two are informationally equivalent subsets of the same ambient $9$04-vector structure, yet differ in contextuality because of the choice of basis completions (Navara et al., 10 Sep 2025).

The notion of informational equivalence is defined operationally: every vector present in one set but missing in the other can be uniquely constructed as the orthogonal complement to a $9$05-dimensional subspace spanned by vectors within the other set. Thus the two configurations do not contain exactly the same explicit rays, but each determines the other’s missing rays through orthogonality relations (Navara et al., 10 Sep 2025).

This comparison is used to argue that contextuality is not a property of ray content alone. The paper states the reason directly: the completion of a basis from a $9$06-dimensional subspace is not unique. Additional contexts can provide “provisions” that allow for a separating set of two-valued states. By contrast, the Cabello $9$07-$9$08 configuration is described as a critically pruned, minimal selection that tightens the restrictions on two-valued states by choosing specific bases and thereby closing those loopholes (Navara et al., 10 Sep 2025).

In this interpretation, the Cabello set is historically and structurally a minimal Kochen–Specker subset derived from the larger symmetric Peres–Mermin configuration. Its role is therefore double. Foundationally, it is the smallest state-independent Kochen–Specker contradiction. Structurally, it is an example showing that the full hypergraph of contexts, not merely the intertwining rays, is indispensable for a rigorous assessment of contextuality (Xu et al., 2020, Navara et al., 10 Sep 2025).

A common terminological confusion concerns dimension. Work on Kochen–Specker vector systems in $9$09 proves only that any such system must contain at least $9$10 vectors, while the smallest known actual example in that setting remains the $9$11-vector Conway–Kochen system; that three-dimensional result does not present an 18-vector Cabello construction and explicitly distinguishes its setting from higher-dimensional formulations (Arends et al., 2011). The phrase 18-vector Cabello set is therefore associated with the four-dimensional $9$12-$9$13 configuration, not with the three-dimensional lower-bound problem.