Dephrasure Channel and Superadditivity of Coherent Information

The quantum capacity of a quantum channel captures its capability for noiseless quantum communication. It lies at the heart of quantum information theory. Unfortunately, our poor understanding of nonadditivity of coherent information makes it hard to understand the quantum capacity of all but very special channels. In this Letter, we consider the dephrasure channel, which is the concatenation of a dephasing channel and an erasure channel. This very simple channel displays remarkably rich and exotic properties: we find nonadditivity of coherent information at the two-letter level, a substantial gap between the threshold for zero quantum capacity and zero single-letter coherent information, a big gap between single-letter coherent and private information, and positive quantum capacity for all complementary channels. Its clean form simplifies the evaluation of coherent information substantially and, as such, we hope that the dephrasure channel will provide a much-needed laboratory for the testing of new ideas about nonadditivity.

Figure 1

Heat map of the quantity $\underset{\lambda }{\max }\frac{1}{2}{I}_c({\rho }_2,{\mathcal{N}}_{p,q}^{\otimes 2})-I_c({\mathcal{N}}_{p,q})$. The repetition code ${\rho }_2$ is defined in Eq. (16). The functions $g(p)$ (green) and $j(p)$ (red) are defined in Eqs. (12) and (15), respectively. The function $k(p)$ (orange) defined in Eq. (10) bounds the region $\mathcal{A}$ of antidegradability of ${\mathcal{N}}_{p,q}$ defined in Eq. (9).

Figure 2

Plot of the coherent information $I_c(·,{\mathcal{N}}_{p,3p}^{\otimes n})/n$ in the interval $p\in [0.107,0.118]$ for the repetition code ${\rho }_n$ for $n=1,\dots ,5$ defined in Eq. (16) (solid lines), the generalized $Z$-diagonal code ${\theta }_4$ defined in Eq. (36) in Ref. [10] for $k=4$ (dashed line), and the nondiagonal code ${\chi }_3$ defined in Eq. (37) in Ref. [10] (dashed-dotted lines). The zero line is plotted as a dashed gray line for reference. The inset plot shows the repetition codes ${\rho }_n$ in the interval $p\in [0.118,0.1202]$, showing that repetition codes increase the threshold of the dephrasure channel.

Figure 3

Plot of the optimal single-letter coherent information $I_c({\mathcal{N}}_{p,3p})$ for $p\in [0.08,0.125]$ (blue), and a lower bound to the single-letter private information, $\underset{\lambda }{\max }{I}_p({\mathfrak{E}}_{\lambda },{\mathcal{N}}_{p,3p})$ (red), where the private code ${\mathfrak{E}}_{\lambda }$ is defined in Eq. (19).