a density operator $\Delta$ belongs to $\DD(\FFF)$ if and only if $\Tr (\gamma_\Delta) < \infty$. Note that $\DD(\FFF)$ contains all the Slater determinant states $|\Phi\rangle\!\langle \Phi |$ where $|\Phi\rangle$ is a Slater determinant wave function (\[Slater determinant in Fock space\]) in $\FFF$.

Free states {#Free states}
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In this section we shall define and discuss the kind of many-fermion states we consider uncorrelated.

We restrict our attention to states that are represented by density operators on a fermion Fock space, and which have finite average particle number. Among such states, we consider Slater determinant states to be uncorrelated, as well as any state that can be represented as a “substate" or restriction of a Slater determinant state. A state can be represented as a substate of a Slater determinant state if and only if it is “gauge-invariant quasi-free" [@AlickiFannes] and its $1$-pdm has finite rank. We want limits of free states to be free, too. The smallest class of states containing all substates of Slater determinant states and limits of such is the class of “free" states, which we define as follows:

\[free def\] A density operator $\Gamma$ on a fermion Fock space is “free" if it represents a gauge-invariant 