Consider a Gaussian operator$^{1}$ $M=Ce^{i\sum_{n,m=1}^N a^\dagger_n L_{nm} a_m}\equiv e^{ia^\dagger La}$, with $a_n^\dagger,a_n$ fermionic creation and annihilation operators, $C\in\mathbb{C}$, and $L$ a complex $N\times N$ matrix. I am not assuming that $L$ is Hermitian.
A set of these Gaussian operators, $M_p=C_pe^{ia^\dagger L_p a}$, $p=1,2,\ldots P$, represents the Kraus operators of a channel that maps a Gaussian state$^{2}$ onto a mixture of Gaussian states: $\rho\mapsto \sum_{p} M_p\rho M_p^\dagger$. The mapping is trace preserving if $\sum_{p} M_p^\dagger M_p$ equals the identity, hence if
$$\sum_{p=1}^P |C_p|^2e^{-ia^\dagger L_p^\dagger a}e^{ia^\dagger L_p a}=I.$$
Is it possible to write this sum rule directly as a restriction on $C_p$, $L_p$, without involving fermion operators?
$^{1}$ Projectors are included as a limit, for example, $a_1^\dagger a_1=\lim_{u\rightarrow\infty}e^{-u}e^{ua_1^\dagger a_1}$.
$^{2}$ Unlike in the bosonic case, any fermionic Gaussian state can be decomposed as a convex sum of pure Gaussian states. The converse does not hold, so the convex-Gaussian channel is not a Gaussian channel (it does not map Gaussian states onto Gaussian states). I thank Norbert Schuch for pointing this out to me.
