Y-channel are upper bounded by $$\begin{aligned}
\label{CS1}
&\hspace{-0.3cm}R_{jk}+R_{jl}\leq\min\left\{I(X_j;Y_r|X_k,X_l,X_r),\right.\nonumber\\
&\hspace{4.5cm}\left.I(X_r;Y_k,Y_l|X_k,X_l)\right\}\\
\label{CS2}
&\hspace{-0.3cm}R_{jl}+R_{kl}\leq\min\left\{I(X_j,X_k;Y_r|X_l,X_r),I(X_r;Y_l|X_l)\right\}\end{aligned}$$ for all distinct $j,k,l\in\{1,2,3\}$, where $(X_1,X_2,X_3,X_r)$ is a zero-mean Gaussian random vector with joint distribution $p(x_1,x_2,x_3,x_r)$, such that $\mathbb{E}[X_j^2]\leq P$ and $\mathbb{E}[X_r^2]\leq P_r$.

The first bound (\[CS1\]) in Theorem \[CutSetBounds\] is obtained by considering the cuts $\mathcal{T}=\{U_j\}$ and $\mathcal{T}=\{U_j,R\}$, respectively for the first and second arguments of the $\min$ operation. These cuts are shown for the case of $j=1$ in Figure \[Cut\] labeled as cut 1 and cut 2 respectively. The last bound (\[CS2\]) in Theorem \[CutSetBounds\] is obtained by considering the complementary cuts. Namely, the first and the second arguments of the $\min$ operation are obtained by considering $\mathcal{T}=\{U_j,R\}^c$ and $\mathcal{T}=\{U_j\}^c$ respectively. The following bounds are obtained as a corollary from Theorem \[CutSetBounds\].

![A cut in the Y-channel. Cut 1 splits the set $\mathcal{S}=\{U_1,U_2,U_3,R\}$ into $\mathcal{T}=\{U_1\}$ and $\mathcal{T}^c$. This can be used to obtain a bound on $R_{12}+R_{13}$ if we consider information flow from $\mathcal{T}$ to $\mathcal{T}^c$, and on $R_{21}+R_{31}$ if we consider information flow from $\mathcal{T}^c$ to $\mathcal{T}$. Similarly, using cut 2 we can obtain one more bound on both $R_{12}+R_{13}$ and $R_{21}+R_{31}$.[]{data-label="Cut"}](Y_Channel_Cut.eps){width="0.8\columnwidth"}

\[CSG\] The achievable rates in the Y-channel must satisfy $$\begin{aligned}
\label{CSG1}
R_{jk}+R_{jl}&\leq C\left(\min\left\{h_j^2P,h_k^2P_r+h_l^2P_r\right\}\right)\\
\label{CSG2}
