Positivity and nonadditivity of quantum capacities using generalized erasure channels

The positivity and nonadditivity of the one-letter quantum capacity (maximum coherent information) $Q^{(1)}$ is studied for two simple examples of complementary quantum channel pairs $(B,C)$. They are produced by a process, we call it gluing, for combining two or more channels to form a composite. (We discuss various other forms of gluing, some of which may be of interest for applications outside those considered in this paper.) An amplitude-damping qubit channel with damping probability $0leq p leq 1$ glued to a perfect channel is an example of what we call a generalized erasure channel characterized by an erasure probability $lambda$ along with $p$. A second example, using a phase-damping rather than amplitude-damping qubit channel, results in the dephrasure channel of Ledtizky et al. [Phys. Rev. Lett. 121, 160501 (2018)]. In both cases we find the global maximum and minimum of the entropy bias or coherent information, which determine $Q^{(1)}(B_g)$ and $Q^{(1)}(C_g)$, respectively, and the ranges in the $(p,lambda)$ parameter space where these capacities are positive or zero, confirming previous results for the dephrasure channel. The nonadditivity of $Q^{(1)}(B_g)$ for two channels in parallel occurs in a well defined region of the $(p,lambda)$ plane for the amplitude-damping case, whereas for the dephrasure case we extend previous results to additional values of $p$ and $lambda$ at which nonadditivity occurs. For both cases, $Q^{(1)}(C_g)$ shows a peculiar behavior: When $p=0$, $C_g$ is an erasure channel with erasure probability $1-lambda$, so $Q^{(1)}(C_g)$ is zero for $lambda leq 1/2$. However, for any $p>0$, no matter how small, $Q^{(1)}(C_g)$ is positive, though it may be extremely small, for all $lambda >0$. Despite the simplicity of these models we still lack an intuitive understanding of the nonadditivity of $Q^{(1)}(B_g)$ and the positivity of $Q^{(1)}(C_g)$.