Quantum Markov semigroups (QMS): past, present and future panorama
DOI:
https://doi.org/10.22579/20112629.427Keywords:
Quantum computation, quantum Markov semigroup, information theoryAbstract
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 ???? by means of a semigroup ????=(????t)t≥0, acting on a von Neumann algebra ????(????) of the linear operators defined on ????. For simplicity, we will sometimes assume that ????=????(????). The semigroup ???? corresponds to the Heisenberg picture in the sense that given any observable x, ????t(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(ρ), where tr(ρ????t(x))=tr(????*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.Downloads
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