Can decoherence account for suppression of interference in all basis? Let the system formed by particle(microscopic or macroscopic)- environment coupling be described by $|\phi><\phi| $ . In decoherence approach, to retrieve the classical properties, one focuses on $Tr_a (|\phi><\phi|)$. Where the subscript 'a' denotes a particular basis spanning the Hilbert space of the  environment. The amplitudes associated with the states in this reduced density matrix are then interpreted as classical probabilities. I have a single questions with this approach in relation to the idea of retrieving classical dynamics.


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*As for the particle is concerned, this consideration is with reference to only one of the observable associated with said particle. i.e. In the state $|\Phi>$, the entanglement involves the eigenstates of an observable of the environment on the one hand and the eigenstates of an observable of the particle on the other hand, and when one says that the particle's quantum coherence is approximately lost, it is only with reference to the eigenstates of the latter observable characterizing the particle.
But for the claim 'classical physics has been retrieved' to be validated, one would expect a similar loss of coherence for all the physical quantities associated with the said particle.
Which means, not just it's position(say) must be shown to apparently become classical, but also it's momentum, internal energy and other classical physical quantities that are associated with that particle.
Is such a calculation possible within the decoherence approach? If so, how does one proceed? And does it work in the broadest generality possible(That is for all environment-system interaction and for all initial states)?

 A: As an example of simultaneous approximate decoherence with respect to the bases associated with two non-commuting observables, consider position and momentum.
Decoherence tends to diagonalize the density matrix in the position basis because interactions are local in that basis. However, the kinetic terms compete with this, so decoherence in the position basis is never perfect. Over time, these two effects tend to continually balance each other, so that positions and momenta both have approximately well-defined time-dependent values (for macroscopic purposes). For momenta, this occurs simply because the momentum observable is (proportional to) the time-derivative of the position observable, so if positions are continually being approximately localized, then so are the momenta. 
Once positions and momenta both have approximately well-defined values, the energy does, too, because the energy (at least in one broad class of models) is expressed in terms of position and momentum observables.
One way to approach the calculations is to start with estimates of the decoherence rate and asymptotic coherence length in the position basis, as in 


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*Joos and Zeh (1985), "The emergence of classical properties through interaction with the environment," Zeitschrift für Physik B 59: 223-243, https://link.springer.com/article/10.1007/BF01725541

*Tegmark (1993), "Apparent wave function collapse caused by scattering", Found. Phys. Lett., 6: 571-590, https://arxiv.org/abs/gr-qc/9310032
(Tegmark's paper is actually critiquing a modification of quantum theory, but calculations using ordinary quantum theory are included for comparison.) Once we have those quantities, we can compare them to the rate of dispersion due to the kinetic term and make estimates about where the balance lies. That's only a back-of-the-envelope approach, but it has the virtue of building intuition.
Does it work for all environment-system interactions and for all initial states?   Probably not, but I suppose it depends on what you mean by "all." In any quantum system involving various "objects" (particles, fields, or whatever) interacting with each other, those objects will tend to become entangled. However, with a complex pattern of arbitrary not-necessarily-local interactions in the Hamiltonian, it's not clear to me that any given list of observables would consistently approach well-defined values for all initial states. I don't know of any "theorems" with that level of generality, though. Maybe some interesting results in that department are just waiting to be discovered. However, if we restrict attention to conventional quantum field theory in which the Hamiltonian is local in space, then approximate decoherence in the position basis (and, via time-dependence, the momentum basis) will probably tend to dominate. 
That assumes, of course, that we're talking about a theory with one or more characteristic spatial scales built into it, as in QED or QCD.  If we consider a conformal field theory instead, as in the CFT side of the AdS/CFT correspondence, then my argument fails. That's good, because otherwise the AdS/CFT correspondence wouldn't be viable as a theory of quantum gravity in the higher-dimensional AdS spacetime. For that reason, a study of decoherence in CFT would probably be very interesting.
