Abstract

AGREDO E., Julián A. Quantum Markov semigroups (QMS): past, present and future panorama. Orinoquia [online]. 2017, vol.21, suppl.1, pp.20-29. ISSN 0121-3709.  https://doi.org/10.22579/20112629.427.

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=(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(≥), where tr(P𝓣 t (x))= tr (𝓣 *t (P)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.

Keywords : Quantum computation; quantum Markov semigroup; information theory..

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