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Vicent Gimeno

Publications and source records attributed to Vicent Gimeno.

At least 19 recordsLinked to original sources
arXiv2023

Multiple photon Hamiltonian in linear quantum optical networks

We give an alternative derivation for the explicit formula of the effective Hamiltonian describing the evolution of the quantum state of any number of photons entering a linear optics multiport. The description is based on the effective Hamiltonian of the optical system for a single photon and comes from relating the evolution in the Lie group that describes the unitary evolution matrices in the Hilbert space of the photon states to the evolution in the Lie algebra of the Hamiltonians for one and multiple photons. We give a few examples of how a group theory approach can shed light on some properties of devices with two input ports.

quant-ph↗
arXiv2023

A method to determine which quantum operations can be realized with linear optics with a constructive implementation recipe

The evolution of quantum light through linear optical devices can be described by the scattering matrix $S$ of the system. For linear optical systems with $m$ possible modes, the evolution of $n$ input photons is given by a unitary matrix $U=φ_{m,M}(S)$ given by a known homomorphism, $φ_{m,M}$, which depends on the size of the resulting Hilbert space of the possible photon states, $M$. We present a method to decide whether a given unitary evolution $U$ for $n$ photons in $m$ modes can be achieved with linear optics or not and the inverse transformation $φ_{m,M}^{-1}$ when the transformation can be implemented. Together with previous results, the method can be used to find a simple optical system which implements any quantum operation within the reach of linear optics. The results come from studying the adjoint map bewtween the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.

quant-ph↗
arXiv2023

QOptCraft: A Python package for the design and study of linear optical quantum systems

The manipulation of the quantum states of light in linear optical systems has multiple applications in quantum optics and quantum computation. The package QOptCraft gives a collection of methods to solve some of the most usual problems when designing quantum experiments with linear interferometers. The methods include functions that compute the quantum evolution matrix for n photons from the classical description of the system and inverse methods that, for any desired quantum evolution, will either give the complete description of the experimental system that realizes that unitary evolution or, when this is impossible, the complete description of the linear system which approximates the desired unitary with a locally minimal error. The functions in the package include implementations of different known decompositions that translate the classical scattering matrix of a linear system into a list of beam splitters and phase shifters and methods to compute the effective Hamiltonian that describes the quantum evolution of states with n photons. The package is completed with routines for useful tasks like generating random linear optical systems and computing matrix logarithms. The routines are chosen to avoid usual numerical problems when dealing with the unitary matrices that appear in the description of linear systems.

quant-ph↗
arXiv2021

Quantum collision finding for homomorphic hash functions

Hash functions are a basic cryptographic primitive. Certain hash functions try to prove security against collision and preimage attacks by reductions to known hard problems. These hash functions usually have some additional properties that allow for that reduction. Hash functions which are additive or multiplicative are vulnerable to a quantum attack using the hidden subgroup problem algorithm for quantum computers. Using a quantum oracle to the hash, we can reconstruct the kernel of the hash function, which is enough to find collisions and second preimages. When the hash functions are additive with respect to the group operation in an Abelian group, there is always an efficient implementation of this attack. We present concrete attack examples to provable hash functions, including a preimage attack to $\oplus$-linear hash functions and for certain multiplicative homomorphic hash schemes.

cs.CR↗
arXiv2021

First Eigenvalue of the Laplacian of a Geodesic Ball and Area-Based Symmetrization of its Metric Tensor

Given a Riemmanian manifold, we provide a new method to compute a sharp upper bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of radius less than the injectivity radius of the manifold. This upper bound is obtained by transforming the metric tensor into a rotationally symmetric metric tensor that preserves the area of the geodesic spheres. The provided upper bound can be computed using only the area function of the geodesic spheres contained in the geodesic ball and it is sharp in the sense that the first eigenvalue of geodesic ball coincides with our upper bound if and only if the mean curvature pointed inward of each geodesic sphere is a radial function.

math.DG↗
arXiv2020

Optimal approximation to unitary quantum operators with linear optics

Linear optical systems acting on photon number states produce many interesting evolutions, but cannot give all the allowed quantum operations on the input state. Using Toponogov's theorem from differential geometry, we propose an iterative method that, for any arbitrary quantum operator $U$ acting on $n$ photons in $m$ modes, returns an operator $\widetilde{U}$ which can be implemented with linear optics. The approximation method is locally optimal and converges. The resulting operator $\widetilde{U}$ can be translated into an experimental optical setup using previous results.

quant-ph↗
arXiv2018

Generalized partially linear models on Riemannian manifolds

The generalized partially linear models on Riemannian manifolds are introduced. These models, like ordinary generalized linear models, are a generalization of partially linear models on Riemannian manifolds that allow for response variables with error distribution models other than a normal distribution. Partially linear models are particularly useful when some of the covariates of the model are elements of a Riemannian manifold, because the curvature of these spaces makes it difficult to define parametric models. The model was developed to address an interesting application, the prediction of children's garment fit based on 3D scanning of their body. For this reason, we focus on logistic and ordinal models and on the important and difficult case where the Riemannian manifold is the three-dimensional case of Kendall's shape space. An experimental study with a well-known 3D database is carried out to check the goodness of the procedure. Finally it is applied to a 3D database obtained from an anthropometric survey of the Spanish child population. A comparative study with related techniques is carried out.

stat.ME↗
arXiv2018

Parabolicity, Brownian escape rate and properness of self-similar solutions of the direct and inverse Mean Curvature Flow

We study some potential theoretic properties of homothetic solitons $Σ^n$ of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in $\mathbb{R}^{n+m}$, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons $Σ^n$ with $n>2$ are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the Mean Exit Time function for the Brownian motion defined on $Σ$ as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of \cite{CaoLi}.

math.DG↗
arXiv2017

Mean Curvature of Hypersurfaces in Killing Submersions with Bounded Shadow

Given a complete hypersurface isometrically immersed in an ambient manifold, in this paper we provide a lower bound for the norm of the mean curvature vector field of the immersion assuming that: 1) The ambient manifold admits a Killing submersion with unit-length Killing vector field. 2)The projection of the image of the immersion is bounded in the base manifold. 3)The hypersurface is stochastically complete, or the immersion is proper.

math.DG↗
arXiv2016

Upper Bounds for the Poincaré Recurrence Time in Quantum Mixed States

In this paper by using geometric techniques, we provide upper bounds for the Poincaré recurrence time of a quantum mixed state with discrete spectrum of energies. In the case of discrete but finite spectrum we obtain two type of upper bounds; one of them depends on the uncertainty in the energy, and the other depends only on the (finite) number of states. In the case of discrete but non-finite spectrum we obtain in the same way two upper bounds defining the number of relevant states according to an statistical measurement. These bounds correspond to two different situations in the quantum recurrence process. The first bound is a recurrence time estimation purely quantum, while the other bound that is related with the number of relevant states survives in the classical limit.

quant-ph↗
arXiv2016

Dirichlet spectrum and Green function

In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/λ_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower estimates for the series $\sum λ_{i}^{-2}(Ω)$ where $Ω$ is an extrinsic ball of a proper minimal surface of $\mathbb{R}^{3}$. In the second part we show that the first eigenvalue of bounded domains is given by iteration of the Green operator and taking the limit, $λ_{1}(Ω)=\lim_{k\to \infty} \Vert G^k(f)\Vert_{2}/\Vert G^{k+1}(f)\Vert_{2}$ for any function $f>0$. In the third part we obtain explicitly the $L^{1}(Ω, μ)$-momentum spectrum of a bounded domain $Ω$ in terms of its Green operator. In particular, we obtain the first eigenvalue of a weighted bounded domain in terms of the $L^{1}(Ω, μ)$-momentum spectrum, extending the work of Hurtado-Markvorsen-Palmer on the first eigenvalue of rotationally invariant balls.

math.DG↗
arXiv2015

Asymptotically extrinsic tamed submanifolds

We study, from the extrinsic point of view, the structure at infinity of open submanifolds isometrically immersed in the real space forms of constant sectional curvature $κ\leq 0$. We shall use the decay of the second fundamental form of the the so-called tamed immersions to obtain a description at infinity of the submanifold in the line of the structural results in the papers Internat. Math. Res. Notices 1994, no. 9, authored by R. E. Greene, P. Petersen and S. Zhou and Math. Ann. 2001, 321 (4), authored by A. Petrunin and W. Tuschmann. We shall obtain too an estimation from below of the number of its ends in terms of the volume growth of a special class of extrinsic domains, the extrinsic balls.

math.DG↗
arXiv2015

Conformal type of ends of revolution in space forms of constant sectional curvature

In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the $3$-dimensional Euclidean space or in the $3$-dimensional sphere is parabolic. In the case of ends of revolution in the hyperbolic $3$-dimensional space, we find sufficient conditions to attain parabolicity for complete ends of revolution using their relative position to the complete flat surfaces of revolution.

math.DG↗
arXiv2015

Mean curvature, volume and properness of isometric immersions

We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact $2$-Riemannian manifold $M$ with non-positive Gaussian curvature and finite topology, using the study of the focal points of the transverse Jacobi fields to a geodesic ray in $M$ . In particular, we have explored the relation between the minimal focal distance of a geodesic ray and the total curvature of an end containing that geodesic ray.

math.DG↗
arXiv2015

On the total curvature and extrinsic area growth of surfaces with tamed second fundamental form

In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In the particular case of a surface of nonnegative curvature, we prove that the surface is diffeomorphic to the Euclidean plane if the surface has tamed second fundamental form, and that the surface is isometric to the Euclidean plane if the surface has strongly tamed second fundamental form. In the last part of the paper we characterize the fundamental tone of any submanifold of tamed second fundamental form immersed in an ambient space with a pole and quadratic decay of the radial sectional curvatures.

math.DG↗
arXiv2014

Ends, fundamental tones, and capacities of minimal submanifolds via extrinsic comparison theory

We study the volume of extrinsic balls and the capacity of extrinsic annuli in minimal submanifolds which are properly immersed with controlled radial sectional curvatures into an ambient manifold with a pole. The key results are concerned with the comparison of those volumes and capacities with the corresponding entities in a rotationally symmetric model manifold. Using the asymptotic behavior of the volumes and capacities we then obtain upper bounds for the number of ends as well as estimates for the fundamental tone of the submanifolds in question.

math.DG↗