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Sonia L'Innocente

Publications and source records attributed to Sonia L'Innocente.

14 recordsLinked to original sources
arXiv2026

A dynamical approach to Schur's Theorem

A classical result of Schur of 1904 shows that an abstract group with finite central quotient has finite derived subgroup. Schur's Theorem has many important consequences and generalizations, which have been extensively investigated in the literature. We develop a new dynamical interpretation of Schur's Theorem for locally compact groups, using the notion of topological entropy of Adler, Konheim and McAndrew. We first consider groups with compact central quotient, introduced and called $\mathsf{Z}$-groups by Grosser and Moskowitz in the 1960s, proving that if $G$ is a connected group such that $G/Z(G)$ is compact and with continuous endomorphisms of finite topological entropy, then also $\overline{[G,G]}$ is compact and with continuous endomorphisms of finite topological entropy. The connectedness assumption is essential, since its absence allows us to construct a profinite group as counterexample. Furthermore, we study the Heisenberg groups $\mathbb{H}_n(R)$ on certain locally compact rings $R$ as a framework in which a dynamical Schur-Type Theorem persists, even though the central quotient need not be compact. In particular, we find new formulas for the $p$-rank of these Heisenberg groups.

math.GR↗
arXiv2026

Composing $p$-adic qubits: from representations of SO(3)$_p$ to entanglement and universal quantum logic gates

In the context of $p$-adic quantum mechanics, we investigate composite systems of $p$-adic qubits and $p$-adically controlled quantum logic gates. We build on the notion of a single $p$-adic qubit as a two-dimensional irreducible representation of the compact $p$-adic special orthogonal group SO(3)$_p$. We show that the classification of these representations reduces to the finite case, as they all factorise through some finite quotient SO(3)$_p$ mod $p^k$. Then, we tackle the problem of $p$-adic qubit composition and entanglement, fundamental for a $p$-adic formulation of quantum information processing. We classify the representations of SO(3)$_p$ mod $p$, and analyse tensor products of two $p$-adic qubit representations lifted from SO(3)$_p$ mod $p$. We solve the Clebsch-Gordan problem for such systems, revealing that the coupled bases decompose into singlet and doublet states. We further study entanglement arising from those stable subsystems. For $p=3$, we construct a set of gates from $4$-dimensional irreducible representations of SO(3)$_p$ mod $p$ that we prove to be universal for quantum computation.

quant-ph↗
arXiv2024

Characterising the Haar measure on the $p$-adic rotation groups via inverse limits of measure spaces

We determine the Haar measure on the compact $p$-adic special orthogonal groups of rotations $\mathrm{SO}(d)_p$ in dimension $d=2,3$, by exploiting the machinery of inverse limits of measure spaces, for every prime $p>2$. We characterise $\mathrm{SO}(d)_p$ as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each $\mathrm{SO}(d)_p$. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on $\mathrm{SO}(d)_p$. Our results pave the way towards the study of the irreducible projective unitary representations of the $p$-adic rotation groups, with potential applications to the recently proposed $p$-adic quantum information theory.

math-ph↗
arXiv2024

Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups

We provide a general expression of the Haar measure $-$ that is, the essentially unique translation-invariant measure $-$ on a $p$-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the $p$-adic special orthogonal groups in dimension two, three and four (for every prime number $p$). In particular, the Haar measure on $\mathrm{SO}(2,\mathbb{Q}_p)$ is obtained by a direct application of our general formula. As for $\mathrm{SO}(3,\mathbb{Q}_p)$ and $\mathrm{SO}(4,\mathbb{Q}_p)$, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain $p$-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field $\mathbb{Q}_p$ and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the $p$-adic special orthogonal groups, with potential applications in $p$-adic quantum mechanics and in the recently proposed $p$-adic quantum information theory.

math-ph↗
arXiv2024

Irreducibility in generalized power series

A classical tool in the study of real closed fields are the fields $K((G))$ of generalized power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian group $G$. In this paper we enlarge the family of ordinals $α$ of non-additively principal Cantor degree for which $K((\mathbb{R}^{\le 0}))$ admits irreducibles of order type $α$ far beyond $α=ω^2 $ and $α= ω^3$ known prior to this work.

math.AC↗
arXiv2024

A factorisation theory for generalised power series and omnific integers

We prove that in every ring of generalised power series with non-positive real exponents and coefficients in a field of characteristic zero, every series admits a factorisation into finitely many irreducibles of infinite support, the number of which can be bounded in terms of the order type of the series, and a unique product, up to multiplication by a unit, of factors of finite support. We deduce analogous results for the ring of omnific integers within Conway's surreal numbers, using a suitable notion of infinite product. In turn, we solve Gonshor's conjecture that the omnific integer $ω^{\sqrt{2}} + ω+ 1$ is prime. We also exhibit new classes of irreducible and prime generalised power series and omnific integers, generalising previous work of Berarducci and Pitteloud.

math.LO↗
arXiv2018

Bézout domains and lattice-valued modules

Let B be a commutative Bézout domain B and let MSpec(B) be the maximal spectrum of B. We obtain a Feferman-Vaught type theorem for the class of B-modules. We analyse the definable sets in terms, on one hand, of the definable sets in the classes of modules over the localizations of B by the maximal ideals of B, and on the other hand, of the constructible subsets of MSpec(B). When B has good factorization, it allows us to derive decidability results for the class B-modules, in particular when B is the ring of algebraic integers or its intersection with real numbers or p-adic numbers.

math.LO↗
arXiv2017

Factorisation of germ-like series

A classical tool in the study of real closed fields are the fields $K((G))$ of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian group $G$. A fundamental result of Berarducci ensures the existence of irreducible series in the subring $K((G^{\leq 0}))$ of $K((G))$ consisting of the generalised power series with non-positive exponents. It is an open question whether the factorisations of a series in such subring have common refinements, and whether the factorisation becomes unique after taking the quotient by the ideal generated by the non-constant monomials. In this paper, we provide a new class of irreducibles and prove some further cases of uniqueness of the factorisation.

math.LO↗
arXiv2009

Quantum state transfer in a q-deformed chain

We investigate the quantum state transfer in a chain of particles satisfying q-deformed oscillators algebra. This general algebraic setting includes the spin chain and the bosonic chain as limiting cases. We study conditions for perfect state transfer depending on the number of sites and excitations on the chain. They are formulated by means of irreducible representations of a quantum algebra realized through Jordan-Schwinger maps. Playing with deformation parameters, we can study the effects of nonlinear perturbations or interpolate between the spin and bosonic chain.

quant-ph↗