Absorption capacity of separable noise: Bell-mixing thresholds on entanglement and teleportation

Abstract

We study Bell-mixing lines $\rho_\lambda=\lambda\Phi^+ +(1-\lambda)\sigma$, where $\Phi^+$ is a fixed Bell reference and $\sigma$ is a separable two-qubit noise state. Along this line there are two operational crossings: the state becomes entangled, and it reaches quantum teleportation advantage over classical strategies. We package these crossings as capacities of the noise state. The entanglement absorption capacity $C_{\rm abs}(\sigma)$ is the largest amount of Bell reference that $\sigma$ can absorb while the partial transpose remains positive. The fidelity absorption capacity $C_F(\sigma)$ is the largest amount of Bell reference that $\sigma$ can absorb while keeping the maximal teleportation fidelity at or below the classical bound $2/3$. The thresholds corresponding to the two crossing points are obtained from the same M\"obius map, $\lambda_* = C_{\rm abs}/(1+C_{\rm abs})$ and $\lambda_F = C_F/(1+C_F)$. We derive closed-form capacities and thresholds for product noise states and separable complex $X$ noise states. For product noise, $C_{\rm abs}$ depends only on local marginal purities, while $C_F$ also depends on orientation relative to the maximally entangled reference. For $X$ noise states, both capacities are explicit in all four Bell frames. We also study three extensions: arbitrary pure-state references, the evolution of $X$ noise states and their capacities under local amplitude-damping and dephasing channels, and decomposition certificates that give lower bounds on the capacities, hence on the thresholds, for general separable noise.