The New SI and the Fundamental Constants of Nature

The launch in 2019 of the new international system of units is an opportunity to highlight the key role that the fundamental laws of physics and chemistry play in our lives and in all the processes of basic research, industry and commerce. The main objective of these notes is to present the new SI in an accessible way for a wide audience. After reviewing the fundamental constants of nature and its universal laws, the new definitions of SI units are presented using, as a unifying principle, the discrete nature of energy, matter and information in these universal laws. The new SI system is here to stay: although the experimental realizations may change due to technological improvements, the definitions will remain unaffected. Quantum metrology is expected to be one of the driving forces to achieve new quantum technologies of the second generation. ----- La puesta en marcha en 2019 del nuevo sistema internacional de unidades es una oportunidad para resaltar el papel fundamental que las leyes fundamentales de la F\'{\i}sica y la Qu\'{\i}mica juegan en nuestra vida y en todos los procesos de la investigaci\'on fundamental, la industria y el comercio. El principal objetivo de estas notas es presentar el nuevo SI de forma accesible para una audiencia amplia. Tras repasar las constantes fundamentales de la naturaleza y sus leyes universales, se presentan las nuevas definiciones de las unidades SI utilizando como principio unificador la naturaleza discreta de la energ\'{\i}a, la materia y la informaci\'on en esas leyes universales. El nuevo sistema SI tiene vocaci\'on de futuro: aunque las realizaciones experimentales cambien por mejoras tecnol\'gicas, las definiciones permanecer\'an inalteradas. La Metrolog\'{\i}a cu\'antica est\'a llamada a ser uno de las fuerzas motrices para conseguir nuevas tecnolog\'{\i}as cu\'anticas de segunda generaci\'on.Comment: Revtex file, color figures. English version y en espa\~no

Topological Heat Transport and Symmetry-Protected Boson Currents

The study of non-equilibrium properties in topological systems is of practical and fundamental importance. Here, we analyze the stationary properties of a two-dimensional bosonic Hofstadter lattice coupled to two thermal baths in the quantum open-system formalism. Novel phenomena appear like chiral edge heat currents that are the out-of-equilibrium counterparts of the zero-temperature edge currents. They support a new concept of dissipative symmetry-protection, where a set of discrete symmetries protects topological heat currents, differing from the symmetry-protection devised in closed systems and zero-temperature. Remarkably, one of these currents flows opposite to the decreasing external temperature gradient. As the starting point, we consider the case of a single external reservoir already showing prominent results like thermal erasure effects and topological thermal currents. Our results are experimentally accessible with platforms like photonics systems and optical lattices.Comment: RevTeX4 file, color figure

The Renormalization Group Method and Quantum Groups: the postman always rings twice

We review some of our recent results concerning the relationship between the Real-Space Renormalization Group method and Quantum Groups. We show this relation by applying real-space RG methods to study two quantum group invariant Hamiltonians, that of the XXZ model and the Ising model in a transverse field (ITF) defined in an open chain with appropriate boundary terms. The quantum group symmetry is preserved under the RG transformation except for the appearence of a quantum group anomalous term which vanishes in the classical case. This is called {\em the quantum group anomaly}. We derive the new qRG equations for the XXZ model and show that the RG-flow diagram obtained in this fashion exhibits the correct line of critical points that the exact model has. In the ITF model the qRG-flow equations coincide with the tensor product decomposition of cyclic irreps of $SU_q(2)$ with $q^4=1$.Comment: LATEX file, 21 pages, no figures. To appear in "From Field Theory to Quantum Groups", World Scientific. Proceedings to honor J.Lukierski in his 60th birthda

Analytic Formulations of the Density Matrix Renormalization Group

We present two new analytic formulations of the Density Matrix Renormalization Group Method. In these formulations we combine the block renormalization group (BRG) procedure with Variational and Fokker-Planck methods. The BRG method is used to reduce the lattice size while the latter are used to construct approximate target states to compute the block density matrix. We apply our DMRG methods to the Ising Model in a transverse field (ITF model) and compute several of its critical properties which are then compared with the old BRG results.Comment: LATEX file, 25 pages, 8 figures available upon reques

Insertion Sort is O(n log n)

Traditional Insertion Sort runs in O(n^2) time because each insertion takes O(n) time. When people run Insertion Sort in the physical world, they leave gaps between items to accelerate insertions. Gaps help in computers as well. This paper shows that Gapped Insertion Sort has insertion times of O(log n) with high probability, yielding a total running time of O(n log n) with high probability.Comment: 6 pages, Latex. In Proceedings of the Third International Conference on Fun With Algorithms, FUN 200

Systematic Analysis of Majorization in Quantum Algorithms

Motivated by the need to uncover some underlying mathematical structure of optimal quantum computation, we carry out a systematic analysis of a wide variety of quantum algorithms from the majorization theory point of view. We conclude that step-by-step majorization is found in the known instances of fast and efficient algorithms, namely in the quantum Fourier transform, in Grover's algorithm, in the hidden affine function problem, in searching by quantum adiabatic evolution and in deterministic quantum walks in continuous time solving a classically hard problem. On the other hand, the optimal quantum algorithm for parity determination, which does not provide any computational speed-up, does not show step-by-step majorization. Lack of both speed-up and step-by-step majorization is also a feature of the adiabatic quantum algorithm solving the 2-SAT ``ring of agrees'' problem. Furthermore, the quantum algorithm for the hidden affine function problem does not make use of any entanglement while it does obey majorization. All the above results give support to a step-by-step Majorization Principle necessary for optimal quantum computation.Comment: 15 pages, 14 figures, final versio