&&= \bra{x,y} \tr_{_{N_M \setminus 1}}\{\tr_{_{N_I \setminus 1}} 
    \{\hat{\rho}\}\} \ket{x,y},\label{eq:rho2inter}\end{aligned}$$ where $N_{\sigma} \setminus n$ stands for integrating out $N_{\sigma}-n$ coordinates of component $\sigma$ and $\bar{\sigma} \neq \sigma$.

Phase separation: few body mixture {#sec:few_body}
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We start our analysis with a few-body system consisting of $N_M = 5$ majority particles with $N_I\in\{1,2\}$ impurities. First, we uncover the mechanism responsible for the phase separation on the mean-field level and subsequently we perform a comparison to the correlated many-body treatment by means of ML-X.

Mean-field approach: Basic mechanism of phase separation {#sec:few_body_MF}
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For the mean field description we choose a single species orbital $S=1$, yielding a non-entangled state $\ket{\Psi(t)} = \ket{\Psi^M(t)} \otimes \ket{\Psi^I(t)}$ on the species layer. On the particle layer a single SPF $s_{\sigma}=1$ is used for each component, meaning that particles of the same species are forced to condense into the same single-particle state $\varphi^{\sigma}(z,t)$ and $\ket{N_{\sigma}}$ is the only possible number state on the particle level. Thus, our ansatz is $\ket{\Psi(t)} = \ket{N_{M}(t)} \otimes \ket{N_{I}(t)}$ and only $\varphi^{\sigma}(z,t)$ are time-dependent. As a result of imaginary time propagation we end up with the ground state orbitals $\varphi_{_{MF}}^{\sigma}(z)$. The interpretation of the mean-field treatment is that each species feels in addition to 