If information is conserved and the universe is finite, it's also cyclic, and the entropy law is false.
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No, if a subsystem is isolated, then the subsystem's free Hamiltonian is the total Hamiltonian. Time evolution is governed by the Liouville equation, the Hamiltonian dynamics are unitary, the eigenvalues of the density matrix are conserved, and hence so is its Von Neumann entropy
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The second law: dS≥0 is not violated when entropy is conserved. If the system is coupled to an environment the LVN equation becomes the Lindblad equation, which does not conserve entropy in general, and typically increases the entropy of the subsystem.
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Building quantum computers is literally the science of how to minimize this increase of entropy (maintain unitarity). The theory of open quantum systems is very well tested and well understood. If we could violate the second law, that'd be great. Sadly, it's not the case.
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