Vector space constructions of Kochen-Specker sets using Hadamard matrices

Petr Lisoněk

Michael Ben-Zvi, Alexander Ma, Manuel L. Reyes +1 more

Norman D. Megill, Mladen Pavičić

Mladen Pavičić, Jean‐Pierre Merlet, Brendan D. McKay +1 more

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.

Jayashree Karmakar, Biswadeep Chatterjee, Rafiuddin Gazi +5 more

Logical Kochen-Specker (KS) contextuality is widely regarded as an intrinsic property of specially constructed measurement configurations. We show instead that it can emerge from KS-colorable vector sets through a constructive procedure we call the Hamilton extension. Defined for four-dimensional vector sets, the Hamilton extension associates each real vector with a measurement context while inducing additional measurement contexts among Hamilton-extended children of distinct parent vectors. These emergent contexts fundamentally alter the compatibility structure, transforming KS-colorable configurations into KS-uncolorable ones and recovering logical contextuality lost under apex-vertex augmentation. We establish a sharp and optimal threshold -- the Hamilton extension of every five-vector parent set remains KS-colorable, whereas suitably chosen six-vector parent sets already generate logical KS contradictions. Thus, six vectors constitute the smallest parent set capable of generating KS contradiction through this mechanism. Our results reveal a new structural route to contextuality, provide a systematic framework for constructing compact KS sets, and have implications for contextuality-based quantum information protocols and graph-theoretic approaches to nonclassicality.

Mladen Pavičić, Jean‐Pierre Merlet, Brendan McKay +1 more

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 an n -dimensional Hilbert space, , 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 , 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 four-dimensional KS vector systems containing up to 24 vectors were generated and described, all three-dimensional vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.