What is the status on quantum information (processing, communication...) in infinite-dimensional Hilbert spaces?

In canonical references for learning quantum information, such as Nielsen’s Quantum Computation and Quantum Information and Preskill’s lecture notes, the focus seems to be on systems with finite-dimensional Hilbert spaces. This is evident given the overall interested in the most basic quantum system: a qubit.

However, nature is largely comprised of systems with infinite-dimension Hilbert spaces (e.g. QFT), with finite-dimension systems consisting of only parts of these systems (e.g. the polarization of a photon) and engineered ones, used for our practical and operational purposes.

It is understandable, from this operational point of view, to consider finite-dimensional systems, and to engineer systems with this feature. It seems also reasonable that to consider the ultimate physical consequences of engineered systems only a finite number of states are considered (i.e. even though harmonic oscillators have an infinite tower of states, those with arbitrarily large energy are not accessible, if not non-physical; if not even this, at least other systems will interact with only a finite interval of states, considering UV and IR cutoffs).

It is also interesting, nonetheless, to consider the consequences of how informations "flows" in infinite-dimensional Hilbert spaces, especially those with a continuum of states. Such as in the references alluded to above, these topics do not seem to be very much expounded upon.

From research I have made, at most we have the literature on continuous variable quantum information, which is still strict as it seems to focus intensively on Gaussian states, or the papers on relativistic quantum information, where the quantum fields are often traced out very soon. The topics which I am thinking about include the whole machinery of superoperators, quantum channels, metrology measures, etc.

1. Is there a full-blown literature on these topics that I am missing? That is, are my premises, including the physical ones, even reasonable?
2. If not, what are the difficulties that impede extensive research and results in this direction?
3. Physically, what is there to gain from considering infinite-dimensional Hilbert spaces, if anything?
4. Tangentially, I could imagine a connection between this machinery and that of algebraic quantum field theory, is there something in this direction too?

I would like to emphasise that the physical part of this question is also of interest. What I mean by this is, for instance, the physical considerations going in the choice of finite-dimensional systems for quantum information processing ("to what extent are they really choices and not our incapacity?" would be a question).