Environment induced decoherence for Markovian evolutions

Abstract

We study environmental decoherence for a quantum Markov semigroup T acting on\ud an arbitrary von Neumann algebra M. In particular, we analyze the relationships\ud between the decomposition of the algebra induced by decoherence and a sort of\ud “isometric-sweeping decomposition” for the space of states. Moreover, when the\ud semigroup has a faithful normal invariant state, we embed the algebra M in its\ud completion \hat{M} with respect to the scalar product induced by the faithful state,\ud and we compare the decomposition induced by decoherence with some other kind\ud of isometric-sweeping decompositions of M and \hat{M}

On Period, Cycles and Fixed Points of a Quantum Channel

“…The previous theorem allows us to prove that this is true whenever the channel has an invariant faithful density; moreover, in the same case, we can deduce there is environmental decoherence choosing the decomposition M 1 = N and M 2 = M s . This last consideration is an almost direct consequence stated for instance in [14,Proposition 31]. Due to the previous remark, these conclusions hold also for the continuous time case, so for instance, it can generalize many of the results concerning EID for quantum dynamical semigroups as treated in [14] (see in particular Section IV).…”

“…Some aspects of this rigidity were already known and were object of interest in many papers in the last years, related to different problems: e.g. the structure of the invariant states and irreducible decompositions [8,13], decoherence free algebra and environmental decoherence [14,17], the notion of sufficiency in quantum statistics [30,36,34], periodicity and ergodic properties [12]. In finite dimension, the structure of the channel and its spectrum, cycles and multiplicative properties were investigated in [43,44].…”

“…For instance, in [21] a characterization of fixed points and of the DFA was found in terms of the Lindblad form of the generator of the quantum dynamical semigroup. The DFA also appears in [14] or [29], linked with environmental decoherence and other forms of decompositions of the algebra.…”

“…The fact that M 1 coincides with N and can be the image of a normal conditional expectation is in general an interesting but not clear point as far as we know ( [14]). The previous theorem allows us to prove that this is true whenever the channel has an invariant faithful density; moreover, in the same case, we can deduce there is environmental decoherence choosing the decomposition M 1 = N and M 2 = M s .…”

“…When there exists an invariant normal faithful state, the cyclic properties of such channels are investigated passing through the decoherence free algebra and the fixed points domain. Both these spaces are proved to be images of a normal conditional expectation so that their consequent atomic structure are analyzed in order to give a better description of the action of the channel and, for instance, of its Kraus form and invariant densities.Point (ii) already appeared in [14] (see also references therein) and was the original representation of the decoherence free algebra used in [9] when introducing environmental decoherence.The following results are well known.Proposition 4. Assume that there is a faithful normal invariant state for Φ.…”