Qubit channels that achieve capacity with two states

This paper considers a class of qubit channels for which three states are always sufficient to achieve the Holevo capacity. For these channels, it is known that there are cases where two orthogonal states are sufficient, two nonorthogonal states are required, or three states are necessary. Here a systematic theory is given which provides criteria to distinguish cases where two states are sufficient, and determine whether these two states should be orthogonal or nonorthogonal. In addition, we prove a theorem on the form of the optimal ensemble when three states are required, and present efficient methods of calculating the Holevo capacity.

Figure 1

The values of $A$ (solid line) and ${\lambda }_m^2-{\lambda }_3^2$ (dashed line) as a function of ${\lambda }_m$ for ${\lambda }_3=t=1∕2$. The shaded region shows the region of values of $A$ such that the optimal ensemble may require three states. Results for ${\lambda }_m>1∕\sqrt{2}$ are not shown, because the maps for ${\lambda }_m>1∕\sqrt{2}$ are not CPTP.

Figure 2

The difference between the two-state capacity and the three-state capacity vs the value of $A$. Random samples are shown as gray points, and the numerically obtained upper bound is shown as the solid line. The cross and plus are examples from Ref. 6. The cross is for ${\lambda }_1={\lambda }_2=0.6$ and ${\lambda }_3=t=0.5$, and the plus is for ${\lambda }_1=t=0.5$ and ${\lambda }_2={\lambda }_3=0.435$.