Theorem: Exact marginal matrices and diagnostics

In the declared occupation bases,

\rho_R=\mathcal N_\eta(\rho_M)
 &=
 \begin{pmatrix}
 1-p+pt^2&0&0&\eta z\\
 0&p\eta t&0&0\\
 0&0&p\eta t&0\\
 \eta z^*&0&0&p\eta^2
 \end{pmatrix},
 \label{eq:rhoR}\\
 \rho_E=\mathcal C_\eta(\rho_M)
 &=
 \begin{pmatrix}
 1-p+p\eta^2&0&0&t z\\
 0&p\eta t&0&0\\
 0&0&p\eta t&0\\
 t z^*&0&0&pt^2
 \end{pmatrix}.
 \label{eq:rhoE}

For

on the respective subsystems,

The distinguished occupation-basis coherences are

The accessible purity is

The residual purity follows by interchanging $\eta$ and $t$.

Proof. Insert (eq:input) into the Kraus expressions. Only $A_{00}$ contributes the accessible coherence. For the complement, the only off-diagonal residual matrix units are $00,11$ and their adjoints, with coefficient $tz$. This gives (eq:rhoR) and (eq:rhoE).

Tracing either marginal against its number operator gives

The coherence formulas are their displayed entries. For a Hermitian matrix, $\mathrm{Tr}{\rho }^2={\sum }_{i,j}|{\rho }_{ij}{|}^2$. Applying this identity to (eq:rhoR) gives (eq:purity).