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Fabienne Chouraqui

Publications and source records attributed to Fabienne Chouraqui.

At least 19 recordsLinked to original sources
arXiv2024

On the Realization of quantum gates coming from the Tracy-Singh product

The Tracy-Singh product of matrices permits to construct a new gate $c \boxtimes c'$ from two $2$-qudit gates $c$ and $c'$. If $c$ and $c'$ are both Yang-Baxter gates, then $c \boxtimes c'$ is also a Yang-Baxter gate, and if at least one of them is entangling, then $c \boxtimes c'$ is also entangling. A natural question arises about the realisation of these gates, $c \boxtimes c'$, in terms of local and universal gates. In this paper, we consider this question and describe this realisation.

quant-ph↗
arXiv2024

The Yang-Baxter equation, Quantum computing and Quantum entanglement

We present a method to construct infinite families of entangling $2$-qudit gates, and amongst them entangling $2$-qudit gates which satisfy the Yang-Baxter equation. We show that, given $2$-qudit gates $c$ and $d$, if $c$ or $d$ is entangling, then their Tracy-Singh product $c \boxtimes d$ is also entangling and we can provide non-entangled states which become entangled after the application of $c \boxtimes d$.

math.GR↗
arXiv2024

When the Tracy-Singh product of matrices represents a certain operation on linear operators

Given two linear transformations, with representing matrices $A$ and $B$ with respect to some bases, it is not clear, in general, whether the Tracy-Singh product of the matrices $A$ and $B$ corresponds to a particular operation on the linear transformations. Nevertheless, it is not hard to show that in the particular case that each matrix is a square matrix of order of the form $n^2$, $n>1$, and is partitioned into $n^2$ square blocks of order $n$, then their Tracy-Singh product, $A \boxtimes B$, is similar to $A \otimes B$, and the change of basis matrix is a permutation matrix. In this note, we prove that in the special case of linear operators induced from set-theoretic solutions of the Yang-Baxter equation, the Tracy-Singh product of their representing matrices is the representing matrix of the linear operator obtained from the direct product of the set-theoretic solutions.

math.CO↗
arXiv2022

Yang-Baxter equation and cryptography

We find a method to construct iteratively from a non-degenerate involutive set-theoretic solution of the Yang-Baxter equation an infinite family of very large non-degenerate involutive set-theoretic solutions. In case the initial solution is irretractable, all the induced solutions are also irretractable. In case the initial solution is indecomposable, we give a criterion to decide whether all the induced solutions are also indecomposable. Besides the interest in the construction of large (indecomposable) solutions of the Yang-Baxter equation, this construction may have some applications in cryptography. Indeed, we suggest a public key encryption method and a signature method based on our construction, and examine their strengths and weaknesses.

math.GR↗
arXiv2022

The Yang-Baxter equation and Thompson's group $F$

In analogy with non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation and braces, we define non-degenerate involutive partial set-theoretic solutions and partial braces. We define the structure group and the structure inverse monoid of such a solution and prove that if the partial solution is square-free, then its structure inverse monoid embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there exists a square-free, non-degenerate involutive partial solution with structure group isomorphic to Thompson's group $F$.

math.GR↗
arXiv2022

A note on Garside monoids and Braces

A left brace is a triple $(\mathcal{B},+,\cdot)$, where $(\mathcal{B},+)$ is an abelian group, $(\mathcal{B},\cdot)$ is a group, and there is a left-distributivity-like axiom that relates between the two operations in $\mathcal{B}$. In analogy with a left brace, we define a left $\mathscr{M}$-brace to be a triple $(\mathcal{B},+,\cdot)$, where $(\mathcal{B},+)$ is a commutative monoid, $(\mathcal{B},\cdot)$ is a monoid, and the axiom of left distributivity holds. A lcm-monoid $M$ is a left-cancellative monoid such that $1$ is the unique invertible element in $M$, and every pair of elements in $M$ admit a lcm with respect to left-divisibility. The class of lcm-monoids contains the Gaussian, quasi-Garside and Garside monoids. We show that every lcm-monoid induces a left $\mathscr{M}$-brace. Furthermore, we show that every Gaussian group induces a partial left brace.

math.GR↗
arXiv2022

Garsideness properties of structure groups of set-theoretic solutions of the Yang-Baxter equation

There exists a multiplicative homomorphism from the braid group B to the Temperley-Lieb algebra TL. Moreover, the homomorphic images in TL of the simple elements form a basis for the vector space underlying TL. In analogy with the case of B, there exists a multiplicative homomorphism from the structure group G of a non-degenerate, involutive set-theoretic solution to an algebra, which extends to a homomorphism of algebras. We construct a finite basis of the underlying vector space of the image of G using the Garsideness properties of the solution.

math.GR↗
arXiv2020

Herzog-Schonheim conjecture, vanishing sums of roots of unity and convex polygons

Let $G$ be a group and $H_1$,\ldots,$H_s$ be subgroups of $G$ of indices $d_1,\ldots,d_s$ respectively. In 1974, M. Herzog and J. Schönheim conjectured that if $\{H_iα_i\}_{i=1}^{i=s}$, $α_i\in G$, is a coset partition of $G$, then $d_1,\ldots,d_s$ cannot be distinct. In this paper, we present the conjecture as a problem on vanishing sum of roots of unity and convex polygons and prove some results using this approach.

math.GR↗
arXiv2020

An approach to the Herzog-Schönheim conjecture using automata

Let $G$ be a group and $H_1$,...,$H_s$ be subgroups of $G$ of indices $d_1$,...,$d_s$ respectively. In 1974, M. Herzog and J. Schönheim conjectured that if $\{H_iα_i\}_{i=1}^{i=s}$, $α_i\in G$, is a coset partition of $G$, then $d_1$,..,$d_s$ cannot be distinct. In this paper, we present a new approach to the Herzog-Schönheim conjecture based on automata and present a translation of the conjecture as a problem on automata.

math.GR↗
arXiv2019

About an extension of the Davenport-Rado result to the Herzog-Schonheim conjecture for free groups

Let $G$ be a group and $H_1$,...,$H_s$ be subgroups of $G$ of indices $d_1$,...,$d_s$ respectively. In 1974, M. Herzog and J. Schönheim conjectured that if $\{H_iα_i\}_{i=1}^{i=s}$, $α_i\in G$, is a coset partition of $G$, then $d_1$,..,$d_s$ cannot be distinct. We consider the Herzog-Schönheim conjecture for free groups of finite rank and propose a new approach, based on an extension of the Davenport-Rado result for $G=\mathbb{Z}$.

math.GR↗
arXiv2018

The Herzog-Schonheim conjecture for finitely generated groups

Let $G$ be a group and $H_1$,...,$H_s$ be subgroups of $G$ of indices $d_1$,...,$d_s$ respectively. In 1974, M. Herzog and J. Schönheim conjectured that if $\{H_iα_i\}_{i=1}^{i=s}$, $α_i\in G$, is a coset partition of $G$, then $d_1$,..,$d_s$ cannot be distinct. We consider the Herzog-Schönheim conjecture for free groups of finite rank and develop a new combinatorial approach, using covering spaces. We define $Y$ the space of coset partitions of $F_n$ and show $Y$ is a metric space with interesting properties. We give some sufficient conditions on the coset partition that ensure the conjecture is satisfied and moreover has a neighborhood $U$ in $Y$ such that all the partitions in $U$ satisfy also the conjecture.

math.GR↗
arXiv2018

The space of coset partitions of $F_n$ and Herzog-Schönheim conjecture

Let $G$ be a group and $H_1$,...,$H_s$ be subgroups of $G$ of indices $d_1$,...,$d_s$ respectively. In 1974, M. Herzog and J. Schönheim conjectured that if $\{H_iα_i\}_{i=1}^{i=s}$, $α_i\in G$, is a coset partition of $G$, then $d_1$,..,$d_s$ cannot be distinct. We consider the Herzog-Schönheim conjecture for free groups of finite rank. We define $Y$ the space of coset partitions of $F_n$ and show $Y$ is a metric space with interesting properties. In a previous paper, we gave some sufficient conditions on the coset partition of $F_n$ that ensure the conjecture is satisfied. Here, we show that each coset partition of $F_n$, which satisfies one of these conditions, has a neighborhood $U$ in $Y$ such that all the partitions in $U$ satisfy also the conjecture.

math.GR↗
arXiv2016

About left orders in Garside groups

We consider the structure group of a non-degenerate symmetric (non-trivial) set-theoretical solution of the quantum Yang-Baxter equation. This is a Bieberbach group and also a Garside group. We show this group is not bi-orderable, that is it does not admit a total order which is invariant under left and right multiplication. Regarding the existence of a left invariant total ordering, there is a great diversity. There exist structure groups with space of left orders homeomorphic to the Cantor set and all left orders Conradian, while there exist others that are even not unique product groups.

math.GR↗
arXiv2016

The Zappa-Szep product of left-orderable groups

It is well-known that the direct product of left-orderable groups is left-orderable and that, under a certain condition, the semi-direct product of left-orderable groups is left-orderable. We extend this result and show that, under a similar condition, the Zappa-Szep product of left-orderable groups is left-orderable. Moreover, we find conditions that ensure the existence of a partial left and right invariant ordering (bi-order) in the Zappa-Szep product of bi-orderable groups and prove some properties.

math.GR↗
arXiv2014

Construction of a group of automorphisms for an infinite family of Garside groups

The structure groups of non-degenerate symmetric set-theoretical solutions of the quantum Yang-Baxter equation provide an infinite family of Garside groups with many interesting properties. Given a non-degenerate symmetric solution, we construct for its structure group a group of automorphisms. Moreover, we show this group of automorphisms admits a subgroup that preserves the Garside properties of the structure group. In some cases, we could also prove the group of automorphisms obtained is an outer automorphism group.

math.GR↗
arXiv2013

Finite quotients of groups of I-type

To every group of $I$-type, we associate a finite quotient group that plays the role that Coxeter groups play for Artin-Tits groups. Since groups of I-type are examples of Garside groups, this answers a question of D. Bessis in the particular case of groups of I-type. Groups of $I$-type are related to finite set theoretical solutions of the Yang-Baxter equation.

math.GR↗
arXiv2011

Garside groups and Yang-Baxter equation

We establish a one-to-one correspondence between a class of Garside groups admitting a certain presentation and the structure groups of non-degenerate, involutive and braided set-theoretical solutions of the quantum Yang-Baxter equation. We also characterize indecomposable solutions in terms of $Δ$-pure Garside groups.

math.GR↗