Figure 1

Phase boundaries for dense coding with a four-by-four entangled state using unitary encoding. The ${\lambda }_1$ axis lies perpendicular to the page. Regions of $N$ unitary messages are labeled by the appropriate integer (except for $N=11,13,14$, for which there is no space, but the reader can use his or her imagination to fill in). There is no region of only $N=15=d^2-1$ unitary messages [5], and the “maximally entangled” point (MES) at the far left of the plot where ${\lambda }_n=1/d=0.25\forall n$ allows $N=d^2=16$. All boundaries to the right of regions with $N\ge 2d$ have significant portions lying at ${\lambda }_0=d/N$. Notice also that the boundary to the right of the region with $N=2d+1=9$ messages bends at the lower part of the plot (these data points, which in this view appear to lie within the $N=9$ region, are actually part of the boundary separating that region from the region of $N=8$) toward the triangular face corresponding to ${\lambda }_1={\lambda }_0$ (the left-most two lines, meeting at MES, form two sides of this triangle), and that to the right of the $N=3d+1=13$ and $N=14$ regions bend at the top toward the line corresponding to ${\lambda }_2={\lambda }_1={\lambda }_0$ (the line in the upper-left of the figure extending upward from MES).Reuse & Permissions