Can one incorporate quantum field theory into quantum information theory? My understanding is that quantum information is primarily formulated in terms of non-relativistic quantum mechanics language. And it can deal pretty well with, for example, elastic scattering in quantum mechanics, and deal with the related quantum decoherence, for example in this paper
However, is there a formulation in which quantum field theory is incorporated into the quantum information theory, such that I can use it to deal with other situation, for example, absorption? The problem with absorption is that, before the scattering and after the scattering, the Hilbert space are not the same, and "to trace out the environment" is not really well-defined in this sense.
 A: It's true that quantum information theory is usually talked about in non-relativistic language, but quantum field theory (QFT) already fits perfectly well into the framework of quantum information. People compute entanglement entropy and perform partial traces all the time, particularly in research related to the Black Hole Information Paradox. In fact, many important thought experiments involve performing quantum computation on Hawking radiation (excitations of quantum fields like photons emitted from the black hole); that's about as quantum-information as you can get! See this answer for some examples and relevant links.
As for difficulties of tracing out systems in quantum field theory, these operations are perfectly well-defined in QFT (up to usually not-too-important subtleties regarding short-distance spatial cut-offs, i.e. how sharply do you define your boundary when tracing out spatial regions). Importantly, in all of quantum mechanics (including QFT), your Hilbert space never changes unless you explicitly trace out a portion of your system, in which case you're simply restricting your attention to a subsystem. It's part of the mathematical structure of quantum mechanics, and I would say even the axioms, that time evolution maps a state vector to another state vector in the same Hilbert space. In your example of scattering, the Hilbert space consists of ingoing modes, outgoing modes, and the scatterer. The only thing than changes after scattering is which modes are occupied.
The links in the above post cover a lot of ground, and range from arXiv preprints to more accessible pieces like Quanta articles. Here are a few more examples that you might find helpful depending on what you're looking for:

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*Lectures on entanglement entropy in field theory and holography - from the abstract: 'a brief introduction to the relevant aspects of quantum information theory is included.'


*This Stack Exchange answer gives an example of decoherence in the context of atom-photon interactions.


*Came across a textbook chapter called Decoherence in Quantum Field Theory that might be of interest.
A: So if you're not looking into the stars, but at quantum computers, then probably you want a condensed-matter take on QFT and fortunately for you that already exists and is pervasive in the field—open up a condensed matter textbook to the TOC and look for chapters with phrases like “Fock space” or “second quantization.” Quantum optics texts also do similar stuff but I have to confess I get a little overwhelmed with jargon like the difference between “heterodyne” and “homodyne” when I peek into that world.
See e.g. Nazarov’s Advanced Quantum Mechanics: a practical guide or numerous other works.
The basic idea is that for example you might have an system that can be in an excited state, or it can decay emitting a photon. You put it in a mirrored box. Fock space says we can take your $|\text g\rangle,|\text e\rangle$ ground and excited states of the atom and form outer products with the number of photons in some particle-in-a-box mode, $|0\rangle,|1\rangle,\dots$. Some Hamiltonian term looks like $|\text g1\rangle\langle e0|+|\text g2\rangle\langle e1|+\dots+\text{H.c.},$ say, which captures this notion that you can excite a photon while decaying the molecule. This has a nice expression in terms of the CAPs (Creation and Annihilation oPeratorS) of the theory, a fermionic annihilator $c=|\text g\rangle\langle \text e|$ and a bosonic one $b$, as  $c^\dagger b+b^\dagger c.$ Maybe then we introduce a wall of the box held on with a cantilever, if this wall absorbs a photon it’ll become a phonon in the cantilever’s vibrational mode, say. Stuff like that can happen. Then maybe you move all of this into an interaction picture and the CAPs get $e^{-i\omega t}$ time dependences.
So you'll notice that we're doing quantum fields, and you can see we can have annihilation of particles, but the Hamiltonian is central rather than the Lagrangian. And that's just because we don't particularly care about relativity here, so we can afford to use the simpler formalism. You’ll also notice that Fock space has everything you need in order to trace out a density matrix to get a reduced density matrix. So for our molecule in a box the $|g\rangle\langle \text g|$ term is just $$\rho_{\text{gg}}=\langle \text g0|\rho|\text g0\rangle+\langle \text g1|\rho|\text g1\rangle+\dots.$$For a textbook which uses these sorts of things to show that classic Lindbladian evolution of a non-isolated quantum state see e.g. Wiseman & Milburn’s Quantum Measurement and Control. That will give a feel of the sorts of approximations you are also likely to see and so forth.
