Positivity and nonadditivity of quantum capacities using generalized erasure channels

Abstract

We consider various forms of a process, which we call {\em gluing}, for combining two or more complementary quantum channel pairs $(\mathcal{B},\mathcal{C})$ to form a composite. One type of gluing combines a perfect channel with a second channel to produce a \emph{generalized erasure channel} pair $({\mathcal{B}}_g,{\mathcal{C}}_g)$. We consider two cases in which the second channel is (i) an amplitude-damping, or (ii) a phase-damping qubit channel; (ii) is the \emph{dephrasure channel} of Leditzky et al. For both (i) and (ii), $({\mathcal{B}}_g,{\mathcal{C}}_g)$ depends on the damping parameter $0\le p\le 1$ and a parameter $0\le \lambda \le 1$ that characterizes the gluing process. In both cases we study $Q^{(1)}({\mathcal{B}}_g)$ and $Q^{(1)}({\mathcal{C}}_g)$, where $Q^{(1)}$ is the channel coherent information, and determine the regions in the $(p,\lambda )$ plane where each is zero or positive, confirming previous results for (ii). A somewhat surprising result for which we lack any intuitive explanation is that $Q^{(1)}({\mathcal{C}}_g)$ is zero for $\lambda \le 1/2$ when $p=0$, but is strictly positive (though perhaps extremely small) for all values of $\lambda >0$ when $p$ is positive by even the smallest amount. In addition we study the nonadditivity of $Q^{(1)}({\mathcal{B}}_g)$ for two identical channels in parallel. It occurs in a well-defined region of the $(p,\lambda )$ plane in case (i). In case (ii) we have extended previous results for the dephrasure channel without, however, identifying the full range of $(p,\lambda )$ values where nonadditivity occurs. Again, an intuitive explanation is lacking.

13 pages (two-column), 6 figures, v2 matches published version

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