Capacities of the covariant Pauli channel

We study four well-known capacities of a two-parameter family of qubit Pauli channels. These are the channels that are covariant under the $\text{SO}\left(2\right)$ group and contain the depolarizing channel as a special case. We find exact expressions for the classical capacity and entanglement-assisted capacities, and analytically determine the regions where the quantum capacity of the channel vanishes. We then use a flag extension to find upper bound for the quantum capacity and private capacity of these channels in the entire region of parameter space and also obtain the lower bound for the quantum capacity by calculating the single-shot quantum capacity numerically. In conjunction with previous results on depolarizing channels, our result is one step forward for determining the capacities of the full Pauli channel.

Figure 1

Spaces of the input and output systems ($S,S^{\prime }$) and environments ($E,E^{\prime }$) and the channels transforming their states ($\mathrm{\Lambda },{\mathrm{\Lambda }}^c$) [Eqs. (6) and (7)].

Figure 2

Two different regions for the classical capacity (36), (37). In each region, the classical capacity is given $C_{cl}=1-h\left(\xi \right)$ where $h\left(\xi \right)$ is the binary entropy function.

Figure 3

Zero-capacity regions: the entanglement-breaking (EB) and antidegradable (AD) regions are shown in the parameters space. The curved dotted line indicates the solution of inequality (49).

Figure 4

Two different regions for the best upper bound. In one region the best upper bound is given by $A$ as in Eq. (65) or (66) and in the other region, the classical capacity $C_{cl}\left(\mathrm{\Lambda }\right)=1-h\left(\xi \right)$, with $\xi =\frac{1+p_0-p_3}{2}$ is the best upper bound. The curved dotted boundaries depicts the solution of Eq. (76).

Figure 5

In all the figures, the upper curves show the best upper bound Eq. (75) and the lower curves show the best lower bounds Eq. (79) of quantum capacity vs. $p_0+p_3$ for fixed values of $\epsilon :=\frac{p_0-p_3}{p_0+p_3}$.