arXiv subjects

Fernando Lledó

Publications and source records attributed to Fernando Lledó.

At least 19 recordsLinked to original sources

arXiv2025

Finite dimensional approximations for Hilbert space operators and applications in Quantum Mechanics

In this work, we develop a unified framework for quasidiagonal and Følner-type approximations of linear operators on Hilbert spaces. These approximations (originally formulated for bounded operators and operator algebras) involve sequences of non-zero finite rank orthogonal projections that asymptotically commute with the operator -- either in norm (quasidiagonal) or in mean (Følner). Such structures guarantee spectral approximation results in terms of their finite sections. We extend this theory to unbounded, densely defined closable operators, establishing a generalization of Halmos' classical result: every closable quasidiagonal operator is a compact perturbation of a closable block-diagonal operator on the same domain. Likewise, we introduce sparse Følner sequences and establish an interplay between quasidiagonal approximations and the existence of sparse Følner sequences. The theoretical developments are illustrated with explicit examples using different types of weighted shifts and applied to quantum mechanical models, including a detailed treatment of the Weyl algebra and its Schrödinger representation.

math.FA↗

arXiv2024

Asymptotic robustness of entanglement in noisy quantum networks and graph connectivity

Quantum networks are promising venues for quantum information processing. This motivates the study of the entanglement properties of the particular multipartite quantum states that underpin these structures. In particular, it has been recently shown that when the links are noisy two drastically different behaviors can occur regarding the global entanglement properties of the network. While in certain configurations the network displays genuine multipartite entanglement (GME) for any system size provided the noise level is below a certain threshold, in others GME is washed out if the system size is big enough for any fixed non-zero level of noise. However, this difference has only been established considering the two extreme cases of maximally and minimally connected networks (i.e. complete graphs versus trees, respectively). In this article we investigate this question much more in depth and relate this behavior to the growth of several graph theoretic parameters that measure the connectivity of the graph sequence that codifies the structure of the network as the number of parties increases. The strongest conditions are obtained when considering the degree growth. Our main results are that a sufficiently fast degree growth (i.e. $\Omega(N)$, where $N$ is the size of the network) is sufficient for asymptotic robustness of GME, while if it is sufficiently slow (i.e. $o(\log N)$) then the network becomes asymptotically biseparable. We also present several explicit constructions related to the optimality of these results.

quant-ph↗

arXiv2024

Geometric and spectral analysis on weighted digraphs

In this article we give a geometrical description of the (in general non-selfadjoint) in/out Laplacian $\mathcal{L}^{+/-} = (d^{+/-})^* d$ and adjacency matrix on digraphs with arbitrary weights, where $(d^{+/-})^*$ is the adjoint of the evaluation map $d^{+/-}$ on the terminal/initial vertex of each arc and $d = d^+ + d^-$ denotes the discrete gradient. We prove that the multiplicity of the zero eigenvalue of $\mathcal{L}^{+/-} = (d^{+/-})^* d$ coincides with the number of sources/sinks of the digraph. We also show that for an acyclic digraph with combinatorial weights the spectrum is contained in the set of non-zero integers. The geometrical perspective allows to interpret the set of circulations $\mathcal{C}$ of a weighted digraph as coclosed forms on the arcs, i.e. as the kernel of the discrete divergence $d^*$. Moreover, $\mathcal{C}$ is perpendicular to the set of discrete gradients of functions on the vertices. We also give formulas to compute the capacity of a cut and the value of a flow in terms of $\mathcal{L}^-$ and $d$. We illustrate the results with many concrete examples.

math.CO↗

arXiv2023

A geometric construction of isospectral magnetic graphs

We present a geometrical construction of families of finite isospectral graphs labelled by different partitions of a natural number $r$ of given length $s$ (the number of summands). Isospectrality here refers to the discrete magnetic Laplacian with normalised weights (including standard weights). The construction begins with an arbitrary finite graph $G$ with normalised weight and magnetic potential as a building block from which we construct, in a first step, a family of so-called frame graphs $(F_a)_{a \in \mathbb{N}}$. A frame graph $F_a$ is constructed contracting $a$ copies of $G$ along a subset of vertices $V_0$. In a second step, for any partition $A=(a_1,\dots,a_s)$ of length $s$ of a natural number $r$ (i.e., $r=a_1+\dots+a_s$) we construct a new graph $F_A$ contracting now the frames $F_{a_1},\dots,F_{a_s}$ selected by $A$ along a proper subset of vertices $V_1\subset V_0$. All the graphs obtained by different $s$-partitions of $r\geq 4$ (for any choice of $V_0$ and $V_1$) are isospectral and non-isomorphic. In particular, we obtain increasing finite families of graphs which are isospectral for given $r$ and $s$ for different types of magnetic Laplacians including the standard Laplacian, the signless standard Laplacian, certain kinds of signed Laplacians and, also, for the (unbounded) Kirchhoff Laplacian of the underlying equilateral metric graph. The spectrum of the isospectral graphs is determined by the spectrum of the Laplacian of the building block $G$ and the spectrum for the Laplacian with Dirichlet conditions on the set of vertices $V_0$ and $V_1$ with multiplicities determined by the numbers $r$ and $s$ of the partition.

math.SP↗

arXiv2022

A note on commutation relations and finite dimensional approximations

In this article we show that the main C*-algebras describing the canonical commutation relations of quantum physics, i.e., the Weyl and resolvent algebras, are in the class of Følner C*-algebras, a class of C*-algebras admitting a kind of finite approximations of Følner type. In particular, we show that the tracial states of the resolvent algebra are uniform locally finite dimensional.

math.OA↗

arXiv2022

Isospectral graphs via spectral bracketing

In this article, we develop a perturbative technique to construct families of non-isomorphic discrete graphs which are isospectral for the standard (also called normalised) Laplacian and its signless version. We use vertex contractions as a graph perturbation and spectral bracketing with auxiliary graphs which have certain eigenvalues with high multiplicity. There is no need to know explicitly the eigenvalues or eigenfunctions of the corresponding graphs. We illustrate the method by presenting several families of examples of isospectral graphs including fuzzy complete bipartite graphs and subdivision graphs obtained from the previous examples. All the examples constructed turn out to be also isospectral for the standard (Kirchhoff) Laplacian on the associated equilateral metric graph.

math.CO↗

arXiv2021

The uniform Roe algebra of an inverse semigroup

Given a discrete and countable inverse semigroup $S$ one can study, in analogy to the group case, its geometric aspects. In particular, we can equip $S$ with a natural metric, given by the path metric in the disjoint union of its Schützenberger graphs. This graph, which we denote by $Λ_S$, inherits much of the structure of $S$. In this article we compare the C*-algebra $\mathcal{R}_S$, generated by the left regular representation of $S$ on $\ell^2(S)$ and $\ell^\infty(S)$, with the uniform Roe algebra over the metric space, namely $C^*_u(Λ_S)$. This yields a chacterization of when $\mathcal{R}_S = C^*_u(Λ_S)$, which generalizes finite generation of $S$. We have termed this by finite labeability (FL), since it holds when the $Λ_S$ can be labeled in a finitary manner. The graph $Λ_S$, and the FL condition above, also allow to analyze large scale properties of $Λ_S$ and relate them with C*-properties of the uniform Roe algebra. In particular, we show that domain measurability of $S$ (a notion generalizing Day's definition of amenability of a semigroup, cf., [5]) is a quasi-isometric invariant of $Λ_S$. Moreover, we characterize property A of $Λ_S$ (or of its components) in terms of the nuclearity and exactness of the corresponding C*-algebras. We also treat the special classes of F-inverse and E-unitary inverse semigroups from this large scale point of view.

math.OA↗

arXiv2020

Representation of non-semibounded quadratic forms and orthogonal additivity

A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability of the Hermitian quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including the position operator in quantum mechanics and quadratic forms invariant under a unitary representation of a separable locally compact group. The case of invariance under a compact group is also discussed in detail.

math.FA↗

arXiv2020

Matching number, Hamiltonian graphs and discrete magnetic Laplacians

In this article, we relate the spectrum of the discrete magnetic Laplacian (DML) on a finite simple graph with two structural properties of the graph: the existence of a perfect matching and the existence of a Hamiltonian cycle of the underlying graph. In particular, we give a family of spectral obstructions parametrised by the magnetic potential for the graph to be matchable (i.e., having a perfect matching) or for the existence of a Hamiltonian cycle. We base our analysis on a special case of the spectral preorder introduced in [FCLP20a] and we use the magnetic potential as a spectral control parameter.

math.CO↗

arXiv2020

Spectral preorder and perturbations of discrete weighted graphs

In this article, we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential. The first preorder is expressed through the existence of a graph homomorphism respecting the magnetic potential and fulfilling certain inequalities for the weights. The second preorder refers to the spectrum of the associated Laplacian of the magnetic weighted graph. These relations give a quantitative control of the effect of elementary and composite perturbations of the graph (deleting edges, contracting vertices, etc.) on the spectrum of the corresponding Laplacians, generalising interlacing of eigenvalues. We give several applications of the preorders: we show how to classify graphs according to these preorders and we prove the stability of certain eigenvalues in graphs with a maximal d-clique. Moreover, we show the monotonicity of the eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic Cheeger constants with respect to the geometric preorder. Finally, we prove a refined procedure to detect spectral gaps in the spectrum of an infinite covering graph.

math.CO↗

arXiv2020

Amenability and paradoxicality in semigroups and C*-algebras

We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner's type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence. In the context of inverse semigroups $S$ we give a characterization of invariant measures on $S$ (in the sense of Day) in terms of two notions: $domain$ $measurability$ and $localization$. Given a unital representation of $S$ in terms of partial bijections on some set $X$ we define a natural generalization of the uniform Roe algebra of a group, which we denote by $\mathcal{R}_X$. We show that the following notions are then equivalent: (1) $X$ is domain measurable; (2) $X$ is not paradoxical; (3) $X$ satisfies the domain Følner condition; (4) there is an algebraically amenable dense *-subalgebra of $\mathcal{R}_X$; (5) $\mathcal{R}_X$ has an amenable trace; (6) $\mathcal{R}_X$ is not properly infinite and (7) $[0]\not=[1]$ in the $K_0$-group of $\mathcal{R}_X$. We also show that any tracial state on $\mathcal{R}_X$ is amenable. Moreover, taking into account the localization condition, we give several C*-algebraic characterizations of the amenability of $X$. Finally, we show that for a certain class of inverse semigroups, the quasidiagonality of $C_r^*\left(X\right)$ implies the amenability of $X$. The converse implication is false.

math.OA↗

arXiv2019

Covering graphs, magnetic spectral gaps and applications to polymers and nanoribbons

In this article, we analyze the spectrum of discrete magnetic Laplacians (DML) on an infinite covering graph $\widetilde{G} \rightarrow G=\widetilde{G} /Γ$ with (Abelian) lattice group $Γ$ and periodic magnetic potential $\widetildeβ$. We give sufficient conditions for the existence of spectral gaps in the spectrum of the DML and study how these depend on $\widetildeβ$. The magnetic potential may be interpreted as a control parameter for the spectral bands and gaps. We apply these results to describe the spectral band/gap structure of polymers (polyacetylene) and of nanoribbons in the presence of a constant magnetic field.

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arXiv2018

Notions of Infinity in Quantum Physics

In this article we will review some notions of infiniteness that appear in Hilbert space operators and operator algebras. These include proper infiniteness, Murray von Neumann's classification into type I and type III factors and the class of F{/o} lner C*-algebras that capture some aspects of amenability. We will also mention how these notions reappear in the description of certain mathematical aspects of quantum mechanics, quantum field theory and the theory of superselection sectors. We also show that the algebra of the canonical anti-commutation relations (CAR-algebra) is in the class of F{/o} lner C*-algebras.

math-ph↗

arXiv2018

Amenability of coarse spaces and K-algebras

In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over commutative fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.

math.RA↗

arXiv2018

Amenability and uniform Roe algebras

Amenability for groups can be extended to metric spaces, algebras over commutative fields and $C^*$-algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra $C_u^*(X)$ over a metric space $(X,d)$ with bounded geometry. In particular, we show that the following conditions are equivalent: (1) $(X,d)$ is amenable; (2) the translation algebra generating $C_u^*(X)$ is algebraically amenable (3) $C_u^*(X)$ has a tracial state; (4) $C_u^*(X)$ is not properly infinite; (5) $[1]_0\neq [0]_0$ in the $K_0$-group $K_0(C_u^*(X))$; (6) $C_u^*(X)$ does not contain the Leavitt algebra as a unital $*$-subalgebra; (7) $C_u^*(X)$ is a Følner $C^*$-algebra in the sense that it admits a net of unital completely positive maps into matrices which is asymptotically multiplicative in the normalized trace norm. We also show that every possible tracial state of the uniform Roe algebra $C_u^*(X)$ is amenable.

math.OA↗

arXiv2018

Spectral gaps and discrete magnetic Laplacians

The aim of this article is to give a simple geometric condition that guarantees the existence of spectral gaps of the discrete Laplacian on periodic graphs. For proving this, we analyse the discrete magnetic Laplacian (DML) on the finite quotient and interpret the vector potential as a Floquet parameter. We develop a procedure of virtualising edges and vertices that produces matrices whose eigenvalues (written in ascending order and counting multiplicities) specify the bracketing intervals where the spectrum of the Laplacian is localised. We prove Higuchi-Shirai's conjecture for Z-periodic trees and apply our technique in several examples like the polypropylene or the polyacetylene to show the existence spectral gaps.

math.CO↗

arXiv2014

On Self-adjoint extensions and symmetries in Quantum Mechanics

Given a unitary representation of a Lie group $G$ on a Hilbert space $\mathcal{H}$, we develop the theory of $G$-invariant self-adjoint extensions of symmetric operators both using von Neumann's theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the $G$-invariant unbounded operator. We also prove a $G$-invariant version of the representation theorem for quadratic forms. The previous results are applied to the study of $G$-invariant self-adjoint extensions of the Laplace-Beltrami operator on a smooth Riemannian manifold with boundary on which the group $G$ acts. These extensions are labeled by admissible unitaries $U$ acting on the $L^2$-space at the boundary and having spectral gap at $-1$. It is shown that if the unitary representation $V$ of the symmetry group $G$ is traceable, then the self-adjoint extension of the Laplace-Beltrami operator determined by $U$ is $G$-invariant if $U$ and $V$ commute at the boundary. Various significant examples are discussed at the end.

math-ph↗

arXiv2013

Amenable traces and Følner C*-Algebras

In the present article we review an approximation procedure for amenable traces on unital and separable C*-algebras acting on a Hilbert space in terms of Følner sequences of non-zero finite rank projections. We apply this method to improve spectral approximation results due to Arveson and Bédos. We also present an abstract characterization in terms of unital completely positive maps of unital separable C*-algebras admitting a non-degenerate representation which has a Følner sequence or, equivalently, an amenable trace. This is analogous to Voiculescu's abstract characterization of quasidiagonal C*-algebras. We define Følner C*-algebras as those unital separable C*-algebras that satisfy these equivalent conditions. Finally we also mention some permanence properties related to these algebras.

math.OA↗