Related papers: Quantum groups under very strong axioms
In earlier work we gave a characterisation of pregeometries which are `basic' (that is, admit no `non-degenerate' quotients) relative to two different kinds of quotient operations, namely imprimitive quotients and normal quotients. Each…
Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} +…
The recent proposal (M Planat and M Kibler, Preprint 0807.3650 [quantph]) of representing Clifford quantum gates in terms of unitary reflections is revisited. In this essay, the geometry of a Clifford group G is expressed as a BN-pair, i.e.…
In 2009, Banica and Speicher began to study the compact quantum subgroups of the free orthogonal quantum group containing the symmetric group S_n. They focused on those whose intertwiner spaces are induced by some partitions. These…
The quantum orthogonal arrays define remarkable classes of multipartite entangled states called $k$-uniform states whose every reductions to $k$ parties are maximally mixed. We present constructions of quantum orthogonal arrays of strength…
Quantum computers hold promise to improve the efficiency of quantum simulations of materials and to enable the investigation of systems and properties more complex than tractable at present on classical architectures. Here, we discuss…
In this paper, we introduce the notion of a normal form of one qubit quantum circuits over the basis $\{H, P, T\}$, where $H$, $P$ and $T$ denote the Hadamard, Phase and $\pi/8$ gates, respectively. This basis is known as the {\it standard…
We establish several new topological generation results for the quantum permutation groups $S^+_N$ and the quantum reflection groups $H^{s+}_N$. We use these results to show that these quantum groups admit sufficiently many "matrix models".…
For every $n \geq 1$, let $(\mathrm{FW}_n)$ denote the fixed-point property for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube complexes of dimension $n$). In this article, we construct explicit examples of groups…
Quantum reference frame transformations have been proposed to provide a means by which to translate descriptions of quantum systems relative to each other. At present, there are several differing frameworks for describing quantum reference…
To a representation of $\O_N$ (the Cuntz algebra with $N$ generators) we associate a projection valued measure and we study the case when this measure has atoms. The main technical tool are the spaces invariant for all the operators…
We present an approach for teaching quantum physics at high school level based on the simplest quantum system - the single quantum bit (qubit). We show that many central concepts of quantum mechanics, including the superposition principle,…
We give explicit expression of recurrency formulae of canonical realization for quantum enveloping algebras $U_{q}(sl(n+1,C))$. In these formulas the generators of the algebra $U_{q}(sl(n+1,C))$ are expressed by means of n-canonical q-boson…
We present a new characterization of quantum theory in terms of simple physical principles that is different from previous ones in two important respects: first, it only refers to properties of single systems without any assumptions on the…
This paper extends earlier work on quantum theory representations of natural numbers N, integers I, and rational numbers Ra to describe a space of these representations and transformations on the space. The space is parameterized by 4-tuple…
Quantum mechanics in Hilbert spaces of finite dimension $N$ is reviewed from the number theoretic point of view. For composite numbers $N$ possible quantum kinematics are classified on the basis of Mackey's Imprimitivity Theorem for finite…
The general stable quantum memory unit is a hybrid consisting of a classical digit with a quantum digit (qudit) assigned to each classical state. The shape of the memory is the vector of sizes of these qudits, which may differ. We determine…
This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the…
Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems. However, this difficulty may be overcome by using some controllable quantum system to study another less controllable…
We study the projections of an arbitrary stably Gelfand quantale $Q$ and show that each projection determines a pseudogroup $S\subset Q$ (and a corresponding localic \'etale groupoid $G$) together with a map of involutive quantales…