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Semigrupos cuánticos de Markov: Pasado, presente y futuro

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Agregado Echeverry, Julián Andrés
Abstract (Spanish)
Los semigrupos cuánticos de Markov (SCM) son una extensión no conmutativa de los semigrupos de Markov definidos en probabilidad clásica. Ellos representan una evolución sin memoria de un sistema microscopico acorde a las leyes de la física cuántica y a la estructura de los sistemas cuánticos abiertos. Esto significa que la dinámica reducida del sistema principal es descrita por un espacio de Hilbert separable complejo 𝔥 por medio de un semigrupo 𝒯=(𝒯t)t≥0, el cual actúa sobre una subálgebra de von Neumann 𝔐 del álgebra 𝔓(𝔥) de todos los operadores lineales acotados definidos en 𝔥. Por simplicidad, algunas veces asumiremos que 𝔐=𝔓(𝔥). El semigrupo 𝓣 corresponde al cuadro de Heisenberg en el sentido que dado cualquier observable x, 𝓣t(x) describe su evolución en el tiempo t. De esta forma, dada una matriz de densidad p, su dinámica (cuadro de Schrödinger) es dada por el semigrupo predual 𝓣*t(ρ) , donde tr(ρ𝓣t(x))=tr(𝓣*t(ρ)x), tr(⋅) denota la operación traza. En este trabajo ofrecemos una exposición de varios resultados básicos sobre SCM. Además discutimos aplicaciones de SCM en teoría de la información cuántica y computación cuántica.
Abstract (English)
Quantum Markov semigroups (SCM) are a non-commutative extension of the Markov semigroups defined in classical probability. They represent an evolution without memory of a microscopic system according to the laws of quantum physics and the structure of open quantum systems. This means that the reduced dynamics of the main system is described by a complex separable Hilbert space 𝔥 by means of a semigroup 𝓣=(𝓣t)t≥0, acting on a von Neumann algebra 𝔓(𝔥) of the linear operators defined on 𝔥. For simplicity, we will sometimes assume that 𝔐=𝔓(𝔥). The semigroup 𝓣 corresponds to the Heisenberg picture in the sense that given any observable x, 𝓣t(x) describes its evolution at time t. Thus, given a density matrix p, its dynamics (Schrödinger's picure) is given by the predual semigroup 𝓣*t(ρ), where tr(ρ𝓣t(x))=tr(𝓣*t(ρ)x), tr(⋅) denote trace of a matrix. In this paper we offer an exposition of several basic results on SCM. We also discuss SCM applications in quantum information theory and quantum computing.
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10 páginas
URI: https://repositorio.escuelaing.edu.co/handle/001/1682
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AC - GIMATH: Grupo de investigación en Matemáticas de la Escuela Colombiana de Ingeniería
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