estimate (that is an underestimate) of the mutual information. Based on this quantity, we have an entanglement indicator that is the analogue of Eq. (\[eqn:KL\_MI\_link\]), namely, $$\begin{aligned}
{\varepsilon_{\small\textsc{bd}}}({\theta_{\textsc{a}}},{\theta_{\textsc{b}}})=& D_{\textsc{b}}[w(X_{{\theta_{\textsc{a}}}},{\theta_{\textsc{a}}};X_{{\theta_{\textsc{b}}}},{\theta_{\textsc{b}}})\!:\! w_{\textsc{a}}(X_{{\theta_{\textsc{a}}}},\theta) w_{\textsc{b}}(X_{{\theta_{\textsc{b}}}},{\theta_{\textsc{b}}})].
\label{eqn:epsBD}\end{aligned}$$

The dependence on ${\theta_{\textsc{a}}}$ and ${\theta_{\textsc{b}}}$ of each of the foregoing entanglement indicators $\varepsilon$ is removed by averaging over a representative set of values of those variables. We denote the corresponding averaged value by $\xi$. In the context of bipartite CV models, we have shown in earlier work [@sharmila; @sharmila2] that averaging ${\varepsilon_{\small\textsc{tei}}}({\theta_{\textsc{a}}},{\theta_{\textsc{b}}})$ over $25$ different values of $({\theta_{\textsc{a}}},{\theta_{\textsc{b}}})$ selected at equal intervals in the range $[0,\pi)$ yields a reliable entanglement indicator $\xi_{\textsc{tei}}$. A similar averaging of each of the quantities ${\varepsilon_{\small\textsc{ipr}}}, {\varepsilon_{\small\textsc{pcc}}}$ and ${\varepsilon_{\small\textsc{bd}}}$ yields $\xi_{\textsc{ipr}}, \xi_{\textsc{pcc}}$ and $\xi_{\textsc{bd}}$, respectively.

Next, we turn to hybrid systems of field-atom interactions. For a two-level atom with ground state $\ket{g}$ and excited state $\ket{e}$, the quorum of observables is  [@thew] $$\begin{aligned}
\label{eqn:atomops}
\nonumber \sigma_{x}=
\textstyle{\frac{1}{2}} (\ket{e}\bra{g}&+\ket{g}\bra{e}), 
\;\sigma_{y}=\textstyle{\frac{1}{2}} i(\ket{g}\bra{e}-\ket{e}\bra{g}), \\
& \sigma_{z}=\textstyle{\frac{1}{2}} (\ket{e}\bra{e}-\ket{g}\bra{g}), \end{aligned}$$ where $\sigma_{i}$ is a Pauli matrix. Let $\sigma_{z}\ket{m}=m\ket{m}$. Then $U(\vartheta,\varphi) \ket{m} =  \ket{\vartheta,\varphi,m}$, where $U(\vartheta,\varphi)$ is a general SU(2) transformation parametrized by $(\vartheta, \varphi)$. Denoting $(\vartheta, \varphi)$ by the unit vector ${\mathbf{n}}$, the qubit tomogram is given 