Covariant Mappings for the Description of Measurement, Dissipation and Decoherence in Quantum Mechanics

Abstract

The general formalism of quantum mechanics for the description of statistical experiments is briefly reviewed, introducing in particular position and momentum observables as POVM characterized by their covariance properties with respect to the isochronous Galilei group. Mappings describing state transformations both as a consequence of measurement and of dynamical evolution for a closed or open system are considered with respect to the general constraints they have to obey and their covariance properties with respect to symmetry groups. In particular different master equations are analyzed in view of the related symmetry group, recalling the general structure of mappings covariant under the same group. This is done for damped harmonic oscillator, two-level system and quantum Brownian motion. Special attention is devoted to the general structure of translation-covariant master equations. Within this framework a recently obtained quantum counterpart of the classical linear Boltzmann equation is considered, as well as a general theoretical framework for the description of different decoherence experiments, pointing to a connection between different possible behaviours in the description of decoherence and the characteristic functions of classical Lévy processes.

Fundamental limits to frequency estimation: a comprehensive microscopic perspective

“…Moreover, stemming from the microscopic picture, it allows to take into account the effect of the dissipative dynamics being dependent on the parameter being sensed-which we demonstrate to significantly improve the attainable sensing precision at a single-probe level. Last but not least, it gives a clear interpretation of the phase-covariance assumption [38][39][40], which forces the noise terms to commute with the parameterencoding Hamiltonian, as it is then naturally guaranteed by the secular approximation within which one discards fast oscillating terms in the master equation [41]. Hence, by considering the model yielding non-secular dynamics induced by the baths with Ohmic spectral densities, we are able to explicitly show that it is the Zeno limit (see references [42,43]) that dictates the asymptotic precision scaling also when the phasecovariance is broken.…”

“…Importantly, all the dynamics for which such limitation was proven are characterized by the fact that the action of the noise commutes with the unitary encoding of the parameter. In other terms, the dynamics of the probes, besides being independent and identical, is phase-covariant (PC) [38][39][40], which means that at any time t the quantum channel Λ ω 0 (t) can be decomposed into the unitary encoding term and a noise term, and these two commute. More precisely, the dynamics of a twolevel system is said to be PC, if for any rotation by an angle φ,…”

Non-Markovian dynamics under time-translation symmetry

“…We would like to point out, that time-translation symmetry and the form of the corresponding master equations was mainly addressed in the literature for the case of strictly Markovian dynamics [82][83][84][85]. An exception is the case of the time-translation symmetry for a qubit (mostly called phase covariance in this context), for which a large number of publications have appeared recently, motivated by the phase estimation problems [86][87][88][89][90].…”

Phase covariant qubit dynamics and divisibility

“…for all g ∈ G and all density operators . Covariance implies some particular structure on the dynamical map Φ(t) [27] and, in the special case of the Markov semigroup, on the generator L [25,26,28].…”