B) \ = \   -  \Tr ( A \log B ) - S(A)$$ if $S(A) < \infty$, and that $$\label{entropy inequality}
  S( A ) \ \le\  -\Tr(A \log B)$$ even if $S(A)$ is infinite. When $ S(A) = \infty$ formula (\[relative entropy\]) cannot be used and $S( A \| B )$ may still be finite.

Fermion Fock spaces {#Fermion Fock spaces}
-------------------

Let $\HH$ be a Hilbert space, the $1$-particle Hilbert space. Unit vectors in $\HH$ are called “orbitals."

The fermion Fock space over $\HH$, which we denote $\FFF(\HH)$, is the Hilbert space direct sum of alternating tensor powers of the $1$-particle Hilbert space $\HH$. That is, $$\label{Fock space}
   \FFF(\HH) \equals \CCC \oplus \HH \oplus \wedge^2 \HH \oplus \cdots \oplus \wedge^m \HH \oplus \cdots \cdots$$ where $\wedge^m \HH $ denotes the $m^{th}$ exterior (alternating tensorial) power of $\HH$. The first component of $\FFF(\HH)$ contains a distinguished unit vector $|\Omega \rangle$ called the “vacuum vector."

The $m$-particle Hilbert space $\wedge^m \HH$ is the completion of the span of all tensor products $h_1 \wedge h_2 \wedge \ldots \wedge h_m$, where $h_1,\ldots,h_m$ are any $m>0$ vectors in $\HH$. The tensors $h_1
\wedge h_2 \wedge \ldots \wedge h_m$ are formally multilinear in $h_1,\ldots,h_m$ and 