Types of quantum field thories
This is statement is to be tacitly understood as applying to dynamics taking place on an a-priori fixed spacetime (a fixed “background”). If however one takes also gravity into account as part of the “system”, then the statement of the second law becomes much more subtle.
The generalized second law of theoremodynamics is an attempt to generalize the naive second law to general relativity. A central idea is that the Bekenstein-Hawking entropy of horizons (such as of black holes) has to be added to the naive entropy of the matter in the “universe”.
A detailed discussion is in
Aron Wall, Ten proofs of the generalized second law (arXiv:0901.3865)
Aron Wall, A proof of the generalized second law for rapidly-evolving Rindler horizons (arXiv:1007.1493)
Aron Wall, A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices (arXiv:1105.3445)
The first paper critiques previous work. The second proves the generalized 2nd law in a semiclassical framework on a background spacetime that has both boost and null translation symmetries. The argument makes heavy use of the concept of relative entropy. The third paper generalizes the second one to a much wider class of spacetimes, assuming some axioms about the behavior of quantum fields on these spacetimes.
To show that entropy increases, a crucial step in Wall’s proof is to use the fact that the relative entropy obeys
S(\rho'|\sigma') \leq S(\rho|\sigma)
where the states and are obtained by restricting the states and to a subalgebra of the original algebra of observables.