Analytical calculation formulas for capacities of classical and classical-quantum channels

Abstract

We derive an analytical calculation formula for the channel capacity of a classical channel without any iteration while its existing algorithms require iterations and the number of iteration depends on the required precision level. Hence, our formula is its first analytical formula without any iteration. We apply the obtained formula to examples and see how the obtained formula works in these examples. Then, we extend it to the channel capacity of a classical-quantum (cq-) channel. Many existing studies proposed algorithms for a cq-channel and all of them require iterations. Our extended analytical algorithm have also no iteration and output the exactly optimum values.

Figures (5)

• Figure 1: Graphs of functions $g_1,g_2,$ and $g_3$ with logarithmic scale. Red dashed curve expresses $g_1$. Blue dashed curve expresses $g_2$. Purple solid curve expresses $g_3$. Green solid line expresses $1$. Red dashed curve $g_1$ and Blue dashed curve $g_2$ across Green solid line at $0.3588$ and $0.4286$, respectively.
• Figure 2: Graph of the function $\widehat{Q}_{1,X}$. Black solid curve expresses $\widehat{Q}_{1,X}(2)$. Red dashed curve expresses $\widehat{Q}_{1,X}(3)$. Green solid line expresses $\widehat{Q}_{1,X}(4)$. The values $\widehat{Q}_{1,X}(3)$ and $\widehat{Q}_{1,X}(4)$ are always positive. The value $\widehat{Q}_{1,X}(2)$ is positive only when $\epsilon \le 0.3972$.
• Figure 3: Graphs of functions $C_1,C_3,C_4,C_{*}$ and $C_{**}$. Black solid curve expresses $C_1$. Blue dashed curve expresses $C_3$. Red dashed curve expresses $C_4$. Green solid line expresses $C_*$. Purple solid curve expresses $C_{**}$. Its enlarged view is given as Fig. \ref{['Capacity-en']}.
• Figure 4: Enlarged view of graphs of functions $C_1,C_3,C_4,C_{*}$ and $C_{**}$. The explanations for 5 curves are the same as Fig. \ref{['Capacity-c']}. 4 curves $C_1$, $C_3$, $C_4$, and $C_{**}$ intersect at $0.3972$. In particular, $C_1$ touches $C_{**}$ at $0.3972$$C_3$ and $C_4$ touch $C_{*}$ at $0.3588$ and $0.4286$, respectively. That is, the inequalities $C_1\ge C_{**}$ and $C_3,C_4 \ge C_{*}$ hold always.
• Figure 5: Relation among various distributions appearing in Step 1 of the proof of Theorem \ref{['T-1']}. This figure shows the topological relation among the distributions $Q_{Y,*}=Q_{Y,t_*}, Q_{Y,0}$, $Q_{Y,t_0}$, the exponential family ${\cal E}_0$, and the mixture family ${\cal M}_{x_0}$.

Theorems & Definitions (18)

• Lemma 1
• Lemma 2
• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Lemma 3
• Theorem 4
• proof
• Theorem 5