Numerical analysis of capacities for two-qubit unitary operations

We present numerical results on the capacities of two-qubit unitary operations for performing communication and creating entanglement. The capacities for communication considered are based upon the increase in Holevo information of an ensemble. Our results indicate that the capacity may be accurately estimated using ensemble sizes and ancilla dimensions of 4. In addition, the calculated values of these capacities were close to, and in some cases equal to, the similarly defined entangling capacities; this result indicates connections between these capacities.

Figure 1

Capacities with zero initial entanglement or Holevo information for the operation $U_1$. The values of $E_U$ are shown as the solid line, and the values of ${\chi }_U^{\prime }$, ${\chi }_U^{(2,1)}$, ${\chi }_U^{(4,4)}$ are shown as the circles, plusses and squares, respectively (these symbols overlap and are not separately visible).

Figure 2

Capacities with zero initial entanglement or Holevo information for the operation $U_2$. The values of $E_U$ are shown as the solid line, and the values of ${\chi }_U^{\prime }$, ${\chi }_U^{(4,2)}$, ${\chi }_U^{(4,3)}$, and ${\chi }_U^{(4,4)}$ are shown as the circles, plusses, crosses, and squares, respectively.

Figure 3

Capacities with zero initial entanglement or Holevo information for the operation $U_3$. The values of $E_U$ are shown as the solid line, and the values of ${\chi }_U^{\prime }$, ${\chi }_U^{(4,2)}$, ${\chi }_U^{(4,3)}$, and ${\chi }_U^{(4,4)}$ are shown as the circles, plusses, crosses, and squares, respectively.

Figure 4

Capacities with arbitrary initial states or ensembles for the operation $U_1$. The values of $\mathrm{\Delta }{E}_U^{\left(1\right)}$ and $\mathrm{\Delta }{E}_U^{\left(2\right)}$ are shown as the solid line, and the values of $\mathrm{\Delta }{\chi }_U^{(2,1)}$ and $\mathrm{\Delta }{\chi }_U^{(2,2)}$ are shown as the circles and plusses, respectively.

Figure 5

The differences between $\mathrm{\Delta }{\chi }_U^{*}$ and $\mathrm{\Delta }{\chi }_U^{(2,1)}$ for the operation $U_1$. The values of $\mathrm{\Delta }{\chi }_U^{(2,2)}-\mathrm{\Delta }{\chi }_U^{(2,1)}$ and $\mathrm{\Delta }{\chi }_U^{(2,3)}-\mathrm{\Delta }{\chi }_U^{(2,1)}$ are shown as the plusses and crosses, respectively.

Figure 6

The differences between $\mathrm{\Delta }{\chi }_U^{*}$ and $\mathrm{\Delta }{\chi }_U^{(2,2)}$ for the operation $U_1$. The values of $\mathrm{\Delta }{\chi }_U^{(2,3)}-\mathrm{\Delta }{\chi }_U^{(2,2)}$, $\mathrm{\Delta }{\chi }_U^{(2,4)}-\mathrm{\Delta }{\chi }_U^{(2,2)}$, and $\mathrm{\Delta }{\chi }_U^{(2,5)}-\mathrm{\Delta }{\chi }_U^{(2,2)}$ are shown as the crosses, squares, and asterisks, respectively.

Figure 7

Capacities with arbitrary initial states or ensembles for the operation $U_2$. The values of $\mathrm{\Delta }{E}_U^{\left(1\right)}$ and $\mathrm{\Delta }{E}_U^{\left(2\right)}$ are shown as the dotted and solid lines, respectively, and the values of $\mathrm{\Delta }{\chi }_U^{(4,1)}$, $\mathrm{\Delta }{\chi }_U^{(4,2)}$, and $\mathrm{\Delta }{\chi }_U^{(4,3)}$ are shown as the circles, plusses, and crosses, respectively.

Figure 8

The differences between $\mathrm{\Delta }{\chi }_U^{*}$ and $\mathrm{\Delta }{\chi }_U^{(4,2)}$ for the operation $U_2$. The values of $\mathrm{\Delta }{\chi }_U^{(4,3)}-\mathrm{\Delta }{\chi }_U^{(4,2)}$, $\mathrm{\Delta }{\chi }_U^{(4,4)}-\mathrm{\Delta }{\chi }_U^{(4,2)}$, and $\mathrm{\Delta }{\chi }_U^{(4,5)}-\mathrm{\Delta }{\chi }_U^{(4,2)}$ are shown as the crosses, squares, and asterisks, respectively.

Figure 9

Capacities with arbitrary initial states or ensembles for the operation $U_3$. The values of $\mathrm{\Delta }{E}_U^{\left(2\right)}$ are shown as the solid line, and the values of $\mathrm{\Delta }{\chi }_U^{(4,2)}$, $\mathrm{\Delta }{\chi }_U^{(4,3)}$, and $\mathrm{\Delta }{\chi }_U^{(4,4)}$ are shown as the plusses, crosses, and squares, respectively.

Figure 10

The differences between $\mathrm{\Delta }{\chi }_U^{*}$ and $\mathrm{\Delta }{\chi }_U^{\left(4,2\right)}$ for the operation $U_3$. The values of $\mathrm{\Delta }{\chi }_U^{\left(4,3\right)}-\mathrm{\Delta }{\chi }_U^{\left(4,2\right)}$, $\mathrm{\Delta }{\chi }_U^{\left(4,4\right)}-\mathrm{\Delta }{\chi }_U^{\left(4,2\right)}$, and $\mathrm{\Delta }{\chi }_U^{\left(4,5\right)}-\mathrm{\Delta }{\chi }_U^{\left(4,2\right)}$ are shown as the crosses, squares, and asterisks, respectively.