The Hadwiger number (G) of a graph G is the largest integer h such that the complete graph on h nodes Kh is a minor of G. Equivalently, (G) is the largest integer such that any graph on at most (G) nodes is a minor ofG. The Hadwiger’s conjecture states that for any graph G, (G) (G), where (G) is the chromatic number of G. It is well-known that for any connected undirected graph G, there exists ...

When the state of a quantum system belongs to a N-dimensional Hilbert space, with N the power of a prime number, it is possible to associate to the system a finite field (Galois field) with N elements. In this paper, we introduce generalized Bell states that can be intrinsically expressed in terms of the field operations. These Bell states are in one to one correspondence with the N 2 elements ...

Phase-locking governs the phase noise in classical clocks through effects described in precise mathematical terms. We seek here a quantum counterpart of these effects by working in a finite Hilbert space. We use a coprimality condition to define phase-locked quantum states and the corresponding Pegg-Barnett type phase operator. Cyclotomic symmetries in matrix elements are revealed and related t...

We construct a Π1-class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relationship between upper box dimension and packing dimension for Π1-classes.

We develop a systematic theory for the construction of quantum codes from classical Reed–Solomon over \({\mathbb {F}}_{p^k}\), where p is prime and \(k \in {\mathbb {Z}}^{+}\). Based on two \([n,n-d_m+1,d_m]\) we provide an \([[n,n+n_e-2(d_m-1),d_m]]\) entanglement-assisted code qudits dimension \(d=p^k\) that saturates Singleton bound needs \(n_e\) entangled qudits, which involves obtaining ex...

Abstract. A complete set of d + 1 mutually unbiased bases exists in a Hilbert spaces of dimension d, whenever d is a power of a prime. We discuss a simple construction of d+1 disjoint classes (each one having d−1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. Such a construction works properly for prime dimension. We investigate an alternative construc...

We construct a novel, real-valued, expansion of any Hermitian operator defined in a Hilbert space of finite dimension N , where N is a prime number or an integer power of a prime. The expansion has a direct interpretation in terms of the operator expectation values for a set of complementary bases. The expansion can be said to be the complement to the discrete Wigner function. We subsequently u...

We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be cons...

Classical LBP such as complexity and high dimensions of feature vectors that make it necessary to apply dimension reduction processes. In this paper, we introduce an improved LBP algorithm to solve these problems that utilizes Fast PCA algorithm for reduction of vector dimensions of extracted features. In other words, proffer method (Fast PCA+LBP) is an improved LBP algorithm that is extracted ...