the three terms in equation (\[martingale\]) converge to $S( \Delta \| \Gamma_{\Delta} )$, $S( \Gamma_\Delta \| \Gamma )$, and $S( \Delta \| \Gamma )$ as $n \longrightarrow \infty$. This establishes (\[superprop-equation\]) even when $S(\Delta )$ is infinite.

Appendices {#Appendices}
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The following nine appendices dilate on the technical background necessary for a thorough understanding of this review, and include a couple of deferred proofs. The appendices are titled:

\[Relative entropy for density operators\] Relative entropy for density operators

\[Fermion Fock spaces\] Fermion Fock spaces

\[Many-fermion states\] Many-fermion states

\[Substates of many-fermion states\] Substates of many-fermion states

\[1-particle density matrices\] $1$-particle density matrices

\[Gauge-invariant quasi-free states\] Gauge-invariant quasi-free states

\[Free states appendix\] Free states

\[Proof of Proposition 1\] Proof of Proposition \[free at last\]

\[Proof of Lemma 1\] Proof of Lemma \[good lemma\]

Relative entropy for density operators {#Relative entropy for density operators}
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The general definition of relative entropy for normal states on von Neumann algebras requires some modular theory [@OhyaPetz]. However, for density operators on a Hilbert space $\mathcal{X}$, which represent normal states on the von Neumann algebra of bounded operators on $\mathcal{X}$, a more elementary definition of relative entropy is available.

Let $A$ and $B$ denote density operators 