The factorization of the process tensor is essentially equivalent to the validity of quantum regression 8. Moreover, it is closely related to the recent approach to quantum Markovianity proposed in 17, 18, 19 in which the Markovianity of the corresponding process is characterized in terms of the so-called quantum process tensor of the system. The approach of 3, 4 adopted in this paper is closely related to the mathematical formulation of quantum Markov stochastic processes proposed in 13, 14, 15 (cf. One should always take into account this ambiguity when dealing with concepts of (non–)Markovianity in open quantum systems (cf. An exhaustive discussion of all concepts of Markovianity in quantum mechanics and the hierarchical relations between them is reported in 8.
#Boson x mod full#
Other approaches take into account the full unitary dynamics of the system and the environment in order to account for the unraveling of information backflow via external interventions or measurements on the system: among many others, the validity of the factorization (or Born) approximation-roughly speaking, the idea that the environment, not “feeling” the action of the system, evolves independently of the presence of the latter-or the quantum regression formula. Some of them solely involve the properties of the reduced dynamics: for instance, the completely positive divisibility 6 (and, as a particular case, the semigroup property) and the monotonicity of the distinguishability between arbitrary states 7. Diverse mathematical properties, each focusing on particular aspects of the problem under investigation, have been-more or less formally-put forward as definitions of quantum Markovianity. In this regard, it is worth recalling that no universally accepted definition of quantum Markovianity exists. Importantly, both definitions are strictly dependent on the choice of the measurement basis. Markovian, if the Markov property ( 3) holds.Ĭlassical, if the consistency condition ( 2) (or, in the Markov case, Eq. ( 6)) holds. In this framework, a (classical) stochastic process corresponds to a collection of functions \(\\), we shall denote as quantum process any sequence of sharp measurements in said basis on the system defining the corresponding family of joint probability distributions as in Eq. Kolmogorov’s pioneering book 1, the modern theory of stochastic processes has been developed in a measure-theoretical axiomatic framework which proves to be powerful and flexible enough for the description of diverse classical physical phenomena-Brownian motion being a paradigmatic example-as well as having applications in other natural and social sciences 2. Long before the dawn of quantum mechanics, randomness has always occupied a central role in the description of natural phenomena.
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