Projecting Density Matrix into Edge States Subspace

I’m reading and trying to understand the paper by Diehl, S., Rico, E., Baranov, M. et al. "Topology by dissipation in atomic quantum wires". Nature Phys 7, 971–977 (2011). https://doi.org/10.1038/nphys2106, (also available at https://arxiv.org/abs/1105.5947)

I have faced some difficulties, though, mainly with notation.

The article starts off by studying a dissipative N-sites Kitaev chain through a Master Equation (2), then proceeds to show the Density Matrix’s evolution is governed by the evolution of the 2Nx2N Correlation Matrix, $Gamma_{ab}(t)=frac{i}{2}langle[c_a,c_b]rho(t)rangle$, where we see a 2×2 subspace is a constant of motion (5). This subspace of the Correlation Matrix is referred to as the "edge mode subspace" and corresponds to the fermionic operator $ã_N = frac{i}{2}(c_{2N} + c_{1})$, which does not appear in the Hamiltonian nor in the Lindblad Operators.

My main doubts regard the equation (27) in the Appendix (arXiv version), section "Properties of the finite size system", "Dissipative isolation of the subspace" subsection:

begin{equation}
partial_t begin{pmatrix} rho_{pp}&rho_{pq}\rho_{qp}&rho_{qq} end{pmatrix}
= sum_j begin{pmatrix} 0&-frac{1}{2}rho_{pq}J^dagger_{j,qq}J_{j,qq}\-frac{1}{2}J^dagger_{j,qq}J_{j,qq}rho_{qp}&L_{j,qq}[rho_{qq}] end{pmatrix},
end{equation}

where

begin{equation}
L_{j,qq}[rho_{qq}] = J_{j,qq}rho_{qq}J_{j,qq}^dagger – frac{1}{2} {J_{j,qq}^dagger J_{j,qq}, rho_{qq}}
end{equation}
and there were introduced "projectors on the edge (zero mode) and bulk subspaces, p and q = 1−p, respectively".

My questions are:

1. What is the dimensionality of this matrix? Is it $2^Ntimes2^N$ as the Fock space for N fermions or is it smaller, for a fixed particle number? What about $rho_{pp}$? Is it $2times2$?

2. What exactly are these projectors? By the looks of the equation I would say they are

begin{equation}
p = vert 0rangle langle 0vert + vert ã_N^daggerrangle langle ã_N^daggervert, quad q = {1!!1}_{2^N} – p.
end{equation}
Is this correct?

From my point of view, a more meaningfull projection would be tracing:

begin{equation}
rho_{pp} = Tr_{bulk}(rho) = sum_i ({1!!1}_2 otimes langle i vert) rho ({1!!1}_2 otimes vert i rangle),
end{equation}
but then I have no idea what is meant by $rho_{pq}$ and other off-diagonal terms. This leads to my final question:

3. What is meant by $J_{i,pq} = 0$? I believe it is $pJ_iq=0$, but using the first projectors above, this equation does not seem to hold for the case analyzed in the main text, i.e. the operator $ã_1$ (which is one of the $J_i$‘s) connects $langle ã_N^dagger vert$ (inside the subspace generated by p) with $vert ã_1^dagger ã_N^dagger rangle$ (outside that subspace).

---

Update:

By delving deeper into the paper I understood that the density matrices being considered have the so-called normal Gaussian form

begin{equation}
rho = : Pi_{ineq j} [{1!!1} – frac{i}{2}hat{c}_iG_{ij}hat{c}_j] :,
end{equation}

apart from a normalization factor. The operation $:f:$ is the normal-ordering operation (https://en.wikipedia.org/wiki/Normal_order) and $G(t)$ is a matrix related to the correlation matrix:

begin{equation}
Gamma = itanh(iG/2),
end{equation}

such that $G$ "stores" all the information of $rho$. I’m thinking that the projectors $p, q$ might be related to this, somehow, but I still haven’t figured out how.