increasing sequence of finite-rank projectors $P_n$ on $\HH$ that converges strongly to the identity, and let $\HH_n$ denote the range of $P_n$. The finite-dimensional von Neumann algebras $\BB(\FFF(\HH_n))$ can be embedded into $\BB(\FFF(\HH))$ as subalgebras, which we denote here by $\BB_n$. Let $\Delta_n$, $(\Gamma_{\Delta})_n$, and $\Gamma_n$ denote the density operators on $\FFF(\HH_n)$ that represent the restrictions of $\Delta$, $\Gamma_\Delta$, and $\Gamma$ to the corresponding substates delimited by $\HH_n$ (as defined in Section \[Substates of many-fermion states\]). The density operator $(\Gamma_{\Delta})_n$ is free because it is a substate of a free state (see Section \[Free states appendix\]) and it has the same $1$-pdm as $\Delta_n$, whence $(\Gamma_{\Delta})_n = \Gamma_{\Delta_n}$. Since $\Delta_n$ is a density operator on a finite-dimensional space, it has finite von Neumann entropy, and therefore $$\label{martingale}
     S(\Delta_n\|\Gamma_{\Delta_n}) \ + \ S(\Gamma_{\Delta_n}\| \Gamma_n ) \ = \ S(\Delta_n \|\Gamma_n  )$$ by (\[superprop-equation\]), as proven above. The norm closure of $\bigcup \BB_n$ is equal to the CAR algebra [@AlickiFannes Theorem 6.6] in its Fock representation as a subalgebra of $\BB(\FFF(\HH))$. The bi-commutant of $\bigcup \BB_n$, which equals that of its closure, is therefore all of $\BB(\FFF(\HH))$. The “filtration" $(\BB_n)_{n=1}^\infty$ thus satisfies the hypothesis of Cor. 5.12(iv) of Ref. [@OhyaPetz], and therefore 