Spin alignment phenomenon for partially erasing channels

Let $\mathcal N_0^{A_0\to B_0B_1}$ and $\mathcal N_1^{A_1\to B_0B_1}$ be the partially erasing channels defined by fixed states $\sigma_0\forin\fomathcal D(B_0)$ and $\sigma_1\forin\fomathcal D(B_1)$, with $A_isimeq B_i$ for $i=0,1$. For $n\forin\fomathbb N$ and $vec x\forin\fo\ry{0,1}^n$, let $\mathcal N_{vec x}$ be their product channel and let $\rho_{vec x}\forin\fomathcal D(\bigotimes_{i=1}^n A_{x_i}^{(i)})$. Fix a probability distribution ${p_{vec x}}_{vec x\forin\fo\ry{0,1}^n}$. Spin alignment phenomenon. The minimization

$$auto$$

of the von Neumann entropy has an optimal choice of input states satisfying

$$\rho_{vec x}=\bigotimes_{i=1}^n \tau_{x_i}^{(i)},$$

where $\tau_0^{(i)}=|\psi_{\max}\ranglelanglepsi_{\max}|_{A_0^{(i)}}$ and $\tau_1^{(i)}=|\phi_{\max}\ranglelanglephi_{\max}|_{A_1^{(i)}}$, with $\psi_{\max}$ and $\phi_{\max}$ any unit vectors in the maximal-eigenspaces of $\sigma_0$ and $\sigma_1$, respectively. Equivalently, each freely chosen spin is aligned with a maximal eigenvector of the corresponding fixed state. This phenomenon extends the spin-alignment result cited in the source and identifies a product-form entropy minimizer for every fixed distribution of channel labels; the extent to which it supports the paper's broader one-way distillable-entanglement claims is the surrounding motivation.