The definitions are as follows: $$\begin{aligned}
\label{eq:species_entropy}
    S_{vN} && = - \sum_{i=1}^{S} \lambda_i \ln{\lambda_i} \\
\label{eq:particle_entropy}
    S_{vN}^{\sigma} && = - \sum_{i=1}^{s_{\sigma}} m_i \ln{m_i} \; \;
    \text{with} \ \hat{\rho}_1^{\sigma}
    =\sum_{i=1}^{s_{\sigma}} m_i \ket{m_i}\bra{m_i}\end{aligned}$$ where $\lambda_i$ are expansion coefficients from , $m_i$ natural populations satisfying $\sum_{i=1}^{s_{\sigma}}m_i=1$ and $\ket{m_i}$ natural orbitals of the spectrally decomposed one-body density operator $\hat{\rho}_1^{\sigma}$. The entanglement entropy is bounded by the equal distribution of orbitals $S_{vN} \leq \ln(S)$, whereas for two dominantly occupied orbitals we expect $S_{vN} \leq \ln(2)\approx0.7$. If $S_{vN}=0$, then there is no entanglement between the species and the wave-function is a simple product state on the species layer. Similarly, fragmentation entropy $S_{vN}^{\sigma}=0$ means that all particles occupy the same SPF and the species is thus condensed. For parameter values where this is fulfilled a mean-field treatment is well justified. However, in Fig. \[fig:entanglement.eps\] we recognize that for stronger couplings $g_{_{MI}}$ this is not the case. Particularly, in the vicinity of the critical region $a_I/a_M \approx \sqrt[7]{N_I/N_M}$ at positive $g_{_{MI}}$, identified in the previous section as highly competitive, the entanglement entropy $S_{vN}$ is very pronounced (Fig. \[fig:entanglement.eps\] first row). The fragmentation entropy of the majority species $S_{vN}^M$ is comparatively weaker and slightly shifted towards a smaller length 