Transition behavior in the capacity of correlated-noisy channels in arbitrary dimensions

Core claim

The paper's central claim is that for the correlated channel $E(\rho)=(1-\mu)(\Phi\otimes\Phi^*)(\rho)+\mu\Phi_c(\rho)$, with $\Phi(\rho)=\sum_\alpha p_\alpha U_\alpha\rho U_\alpha^\dagger$ and $\Phi_c(\rho)=\sum_\alpha p_\alpha (U_\alpha\otimes U_\alpha^*)\rho(U_\alpha\otimes U_\alpha^*)^\dagger$, the optimal input states exhibit a forced transition. If the intersection of the phase-eigenspaces $I_\alpha=\{|\psi\rangle: U_\alpha^{-1}U_{\alpha_0}|\psi\rangle=e^{ic_\alpha}|\psi\rangle\}$ is empty, then at $\mu=1$ no separable input achieves zero output entropy, while the maximally entangled state $|\psi\rangle=d^{-1/2}\sum_i |i,i\rangle$ passes through undisturbed and gives zero entropy. Since at $\mu=0$ the channel is a product channel whose optimal input is assumed separable, a critical $\mu$ must exist at which the optimal input changes. For Kraus operators that commute modulo a phase and form an irreducible group representation, the two-use mutual information is maximized exactly by the output-entropy-minimizing states, so the transition is a genuine capacity effect. In the symmetric generalized Pauli channels studied, the transition is numerically sharp and always between complete separability and maximal entanglement.