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Is adiabatic quantum Hamiltonian of variable dimension possible? This is very hypothetical and I am afraid may not have enough merit to belong to this forum. I would still like to elaborate. Here is an example.

Is adiabatic quantum evolution possible on a system which contains four spins when the evolution starts, throws away (remove) two spins during the process and finally is left with two spins by the time the evolution finishes?

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A Hamiltonian which does that is certainly not allowed, as the corresponding evolution isn't unitary. If you want particles leaking out of your system you need a more generalized framework of open quantum system, which is used for modeling systems coupled to baths under rather general circumstances and use Kraus operator representation. A lot of work was done in this field in the last several decades and is often used in the context of noisy quantum computation. See, for instance, these lecture notes by John Preskill

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