If information is conserved and the universe is finite, it's also cyclic, and the entropy law is false.
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Furthermore, technically the observable universe (any system of interest to humans is within this subsystem) is a subsystem of the universe, due to the Hubble horizon. That makes the observable universe an open quantum system, not closed, and its entropy increases over time.
As you know, what happens beyond the Hubble universe can’t possibly affect us, so no, the observable universe is a closed system, and yet it changes spontaneously and its entropy doesn’t increase. So the second law is false.
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Entropy is not a local quantity, it depends on the spectrum of the quantum field density matrix. There is an interaction Hamiltonian at the boundary, causing the effective dynamics of the intra-Hubble-radius state to increase in entropy.
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Trying to give you a chance to just accept that your original tweet is not quite right. I would stop digging at this point. I can do this all day... have a Master's in quantum cosmology, quantum thermodynamics, and quantum field theory.
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No, the density matrix of the universe doesn’t depend on anything outside it, or the outside would be observable.
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Observable universe is locally coupled to the outside system as quantum field theory is local and translationally invariant, same theory everywhere. There is a spatial gradient term in the Hamiltonian which couples the inside and outside.
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