Related papers: Kochen-Specker Sets and Generalized Orthoarguesian…
We establish a necessary and sufficient condition for a representation of a lattice ordered semigroup to be regular, in the sense that certain extensions are completely positive definite. This result generalizes a theorem due to Brehmer…
GL_q(N)- and SO_q(N)-covariant deformations of the completely symmetric/antisymmetric projectors with an arbitrary number of indices are explicitly constructed as polynomials in the braid matrices. The precise relation between the…
According to Pavi\v{c}i{\'c}, Kochen and Specker's 117-observable set is not a ``Kochen-Specker set''. By the same reason, in arXiv:2502.13787, Pavi\v{c}i{\'c} claims that 10 statements in our paper ``Optimal conversion of Kochen-Specker…
Associated to every complex reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, and which are obtained by killing all braid words that are "sufficiently long", as well as some…
We construct Zariski-dense surface subgroups in infinitely many commensurability classes of uniform lattices of the split real Lie groups $\operatorname{SL}(n,\mathbb{R})$, $\operatorname{Sp}(2n,\mathbb{R})$, $\operatorname{SO}(k+1,k)$, and…
Motivated by quantum states with zero transition probability, we introduce the notion of ortho-set which is a set equipped with a relation $\neq_\mathrm{q}$ satisfying: $x\neq_\mathrm{q} y$ implies both $x\neq y$ and $y \neq_\mathrm{q} x$.…
The Sum-of-Squares (SoS) hierarchy is a semi-definite programming meta-algorithm that captures state-of-the-art polynomial time guarantees for many optimization problems such as Max-$k$-CSPs and Tensor PCA. On the flip side, a SoS lower…
In this paper, we consider saturation problems related to the celebrated Erd\H{o}s--Szekeres convex polygon problem. For each $n \ge 7$, we construct a planar point set of size $(7/8) \cdot 2^{n-2}$ which is saturated for convex $n$-gons.…
Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This…
GHZ paradoxes are presented for all even numbers of qubits from four up. They are obtained from proofs of the Kochen-Specker (KS) theorem by showing how the assumption of noncontextuality can be justified on the basis of locality. The…
We consider some generalization of the theory of quantum states, which is based on the analysis of long standing problems and unsatisfactory situation with the possible interpretations of quantum mechanics. We demonstrate that the…
Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such…
We investigate a category of quantum posets that generalizes the category of posets and monotone functions. Up to equivalence, its objects are hereditarily atomic von Neumann algebras equipped with quantum partial orders in Weaver's sense.…
The Heisenberg-Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some…
Certain aspects of some unitary quantum systems are well-described by evolution via a non-Hermitian effective Hamiltonian, as in the Wigner-Weisskopf theory for spontaneous decay. Conversely, any non-Hermitian Hamiltonian evolution can be…
In standard measure theory the measure on the base set Omega is normalised to one, which encodes the statement that "Omega happens". Moreover, the rules imply that the measure of any subset A of Omega is strictly positive if and only if A…
We introduce a new class of complex Hadamard matrices which have not been studied previously. We use these matrices to construct a new infinite family of parity proofs of the Kochen-Specker theorem. We show that the recently discovered…
We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the 'usual' nil-Coxeter algebras:…
In this paper we study ideas which have proved useful in topological network theory in the context of lattices of numbers. A number lattice $L_S$ is a collection of row vectors, over $\mathbb{Q}$ on a finite column set $S,$ generated by…
A periodic lattice in Euclidean 3-space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard…