Universal random codes: capacity regions of the compound quantum multiple-access channel with one classical and one quantum sender

Abstract

We consider the compound memoryless quantum multiple-access channel (QMAC) with two sending terminals. In this model, the transmission is governed by the memoryless extensions of a completely positive and trace preserving map which can be any element of a prescribed set of possible maps. We study a communication scenario, where one of the senders aims for transmission of classical messages, while the other sender sends quantum information. Combining powerful universal random coding results for classical and quantum information transmission over point-to-point channels, we establish universal codes for the mentioned two-sender task. Conversely, we prove that the two-dimensional rate region achievable with these codes is optimal. In consequence, we obtain a multi-letter characterization of the capacity region of each compound QMAC for the considered transmission task.

Appendices

Auxiliary results

For the convenience of the reader, some auxiliary standard results used in the text are collected.

Lemma 3

(Gentle measurement lemma [35]) Let $\rho \in \mathcal {S}(\mathcal {K})$, $E \in \mathcal {L}(\mathcal {H})$, $0 \le E \le \mathbb {1}_{\mathcal {H}}$. It holds

$$\begin{aligned} \Vert \sqrt{E}\rho \sqrt{E} - \rho \Vert _1 \ \le \ 3 \cdot \sqrt{1 - \mathrm {tr}E \rho }. \end{aligned}$$

Lemma 4

[39] Let $\Psi , \rho , \sigma \in \mathcal {S}(\mathcal {K})$ be states, where $\Psi$ is pure. Then

$$\begin{aligned} F(\Psi , \rho ) \ \ge \ F(\Psi , \sigma ) - \tfrac{1}{2}\Vert \rho - \sigma \Vert _1 \end{aligned}$$

Lemma 5

[39] Let $\Psi \in \mathcal {S}(\mathcal {K}_A), \rho \in \mathcal {S}(\mathcal {K}_B), \sigma \in \mathcal {S}(\mathcal {K}_{A} \otimes \mathcal {K}_B)$ Then

$$\begin{aligned} F(\Psi \otimes \rho , \sigma ) \ \ge \ 1 - \Vert \rho - \sigma _B\Vert _1 - 3\left( 1 - F(\Psi , \sigma _A)\right) . \end{aligned}$$

Lemma 6

[6, 37] Let $\rho , \sigma \in \mathcal {S}(\mathcal {K}_{A} \otimes \mathcal {K}_B)$, $\Vert \rho - \sigma \Vert _1 \le \epsilon \le 1$. Then

$$\begin{aligned} |I(A\rangle B, \rho ) - I(A \rangle B, \sigma )| \ \le \ 6 \epsilon \log \dim \mathcal {K}_A + (2 + 4 \epsilon ) h(2 \epsilon / 1 + 2 \epsilon ), \end{aligned}$$

(57)

where $h(x) = - x \log x - (1-x) \log (1-x)$, $s \in (0,1)$ is the binary Shannon entropy.

Convexity of the capacity formula

In this appendix, we show that the functional expressions in Theorem 3 do not need further convexification. Following the arguments given in [39] for the case of a perfectly known QMAC, we show that for given $\mathfrak {M}\subset \mathcal {C}(\mathcal {H}_A \otimes \mathcal {H}_B, \mathcal {H}_C)$, the set

$$\begin{aligned} \hat{C}_1(\mathfrak {M}) := \bigcup _{l=1}^\infty \bigcup _{p,V,\Psi } \bigcap _{\mathcal {M}\in \mathfrak {M}} \hat{C}^{(1)}(\mathfrak {M}^{\otimes l}, p, V, \Psi ), \end{aligned}$$

(58)

is convex, i.e. we show

Lemma 7

Let $\mathfrak {M}\in \mathcal {C}(\mathcal {H}_A \otimes \mathcal {H}_B, \mathcal {H}_C)$. It holds

$$\begin{aligned} \mathrm {conv}(\hat{C}_1(\mathfrak {M})) = \hat{C}_1(\mathfrak {M}). \end{aligned}$$

(59)

Proof

We have to show that for each $\lambda \in (0,1)$, and any two rate pairs $(R_1^{(i)},R_2^{(i)}) \in \hat{C}_1(\mathfrak {M})$, $i = 1,2$, the their convex combination $(\overline{R}_1, \overline{R}_2)$ with $\overline{R}_j^{(i)} = \lambda R_j^{(1)} + (1- \lambda ) R_j^{(2)}$ for $i = 1,2$ is also a member of $\hat{C}_1(\mathfrak {M})$. Assume

$$\begin{aligned} (R_1^{(i)}, R_2^{(i)}) \in \bigcap _{\mathcal {M}\in \mathfrak {M}} \hat{C}^{(1)}(\mathcal {M}^{\otimes l_i}, p_i, V_i, \Psi _i) \end{aligned}$$

(60)

for some $l_i, p_i, V_i, \Psi _i$, i.e. with effective states

$$\begin{aligned} \omega _i(\mathcal {M}) := \omega (\mathcal {M}^{\otimes l_i}, p_i, V_i, \Psi _i)&(i \in {1,2}, \mathcal {M}\in \mathfrak {M}) \end{aligned}$$

(61)

according to (3), the equations

$$\begin{aligned} l_i \cdot R_1^{(i)}&\le \underset{\mathcal {M}\in \mathfrak {M}}{\inf } \ I(X_i; C^{l_i}, \omega _i) \ \text {and} \nonumber \\ l_i \cdot R_2^{(i)}&\le \underset{\mathcal {M}\in \mathfrak {M}}{\inf } \ I(B^{l_i}\rangle C^{l_iX_i}, \omega _i) \end{aligned}$$

(62)

are fulfilled. Fix $\delta > 0$, and let $k,n \in \mathbb {N}$, $0< k < n$ such that

$$\begin{aligned} |\lambda - \tfrac{k}{n}| \ < \frac{\delta }{R_1^{(1)} + R_2^{(1)} + R_1^{(2)} + R_2^{(2)}}. \end{aligned}$$

(63)

Set $t_1 := k l_2$, $t_2 := (n-k) l_1$. With

$$\begin{aligned} \tilde{\omega }(\mathcal {M}) \ := \ \omega _1(\mathcal {M})^{\otimes t_1} \otimes \omega _2(\mathcal {M})^{\otimes t_2}, \end{aligned}$$

(64)

which is unitarily equivalent to

$$\begin{aligned} \omega (\mathcal {M}^{\otimes nl_1l_2}, p_1^{t_2}\otimes p_2^{t_2}, V_1^{\otimes t_1} \otimes V_2^{\otimes t_2}, \Psi _1^{\otimes t_1} \otimes \Psi _2^{\otimes t_2}), \end{aligned}$$

(65)

we have

$$\begin{aligned} I(X_1^{t_1}X_2^{t_2}; C^{l_1l_2n}, \tilde{\omega }(\mathcal {M}))&\ = \ I(X_1^{t_1}X_2^{t_2}; C^{l_1l_2n}, \omega _1(\mathcal {M})^{\otimes t_1} \otimes \omega _2(\mathcal {M})^{\otimes t_2}) \end{aligned}$$

(66)

$$\begin{aligned}&\ = \ I(X_1^{t_1}; C^{l_1t_1}, \omega _1(\mathcal {M})^{\otimes t_1}) + I(X_2^{t_2}; C^{l_2t_2}, \omega _2(\mathcal {M})^{\otimes t_2}) \end{aligned}$$

(67)

$$\begin{aligned}&\ = \ t_1 \cdot I(X_1; C^{l_1}, \omega _1(\mathcal {M})) + t_2 \cdot I(X_2; C^{l_2}, \omega _2(\mathcal {M})) \end{aligned}$$

(68)

$$\begin{aligned}&\ \ge t_1 l_1 R_1^{(1)} + t_2 l_2 R_1^{(2)} \end{aligned}$$

(69)

$$\begin{aligned}&\ \ge k l_1 l_2 R_1^{(1)} + (n-k) l_1 l_2 R_1^{(2)}. \end{aligned}$$

(70)

where the second and third equalities above are by additivity of the quantum mutual information evaluated on product states, and the inequality is by (62). Consequently, we have

$$\begin{aligned} \frac{1}{l_1 l_2 n} \underset{\mathcal {M}\in \mathfrak {M}}{\inf } I(X_1^{t_1}X_2^{t_2}; C^{l_1l_2n}, \tilde{\omega }(\mathcal {M}))&\ \ge \ \frac{k}{n} R_1^{(1)} + (1 - \frac{k}{n} R_1^{(2)} \end{aligned}$$

(71)

$$\begin{aligned}&\ \ge \ \lambda R_1^{(1)} + (1-\lambda ) R_1^{(2)} - \delta . \end{aligned}$$

(72)

In a similar manner, also the inequality

$$\begin{aligned} \frac{1}{l_1 l_2 n} \underset{\mathcal {M}\in \mathfrak {M}}{\inf } I(B^{n l_1 l_2} \rangle C^{n l_1 l_2}X_1^{t_1}X_2^{t_2}, \tilde{\omega }(\mathcal {M}))&\ \ge \ \lambda R_2^{(1)} + (1-\lambda ) R_2^{(2)} - \delta \end{aligned}$$

(73)

is verified. By (72) and (73),

$$\begin{aligned}&(\overline{R}_1, \overline{R}_2) \ \in \ \left( \bigcap _{\mathcal {M}\in \mathfrak {M}} \frac{1}{l_1l_2} \hat{C}^{(1)}(\mathcal {M}^{l_1l_2n}, p_1^{t_1}\otimes p_2^{t_2}, V_1^{\otimes t_1} \otimes V_2^{\otimes t_2}, \Psi _1^{\otimes t_1} \otimes \Psi _2^{\otimes t_2})\right) _\delta \nonumber \\&\qquad \qquad \subset \ \hat{C}_1(\mathfrak {M})_\delta . \end{aligned}$$

(74)

Since $\delta > 0$ can be chosen arbitrarily, the claim of the lemma follows. $\square$