Three-Receiver Quantum Broadcast Channels: Classical Communication with Quantum Non-unique Decoding

TL;DR

This investigation includes a comprehensive study of pinching across tensor product spaces, presenting the findings as the asymptotic counterpart to the one-shot codes constructed by this work by employing quantum non-unique decoding.

Abstract

In network communication, it is common in broadcasting scenarios for there to exist a hierarchy among receivers based on information they decode due, for example, to different physical conditions or premium subscriptions. This hierarchy may result in varied information quality, such as higher-quality video for certain receivers. This is modeled mathematically as a degraded message set, indicating a hierarchy between messages to be decoded by different receivers, where the default quality corresponds to a common message intended for all receivers, a higher quality is represented by a message for a smaller subset of receivers, and so forth. We extend these considerations to quantum communication, exploring three-receiver quantum broadcast channels with two- and three-degraded message sets. Our technical tool involves employing quantum non-unique decoding, a technique we develop by utilizing the simultaneous pinching method. We construct one-shot codes for various scenarios and find achievable rate regions relying on various quantum Rényi mutual information error exponents. Our investigation includes a comprehensive study of pinching across tensor product spaces, presenting our findings as the asymptotic counterpart to our one-shot codes. By employing the non-unique decoding, we also establish a simpler proof to Marton's inner bound for two-receiver quantum broadcast channels without the need for more involved techniques. Additionally, we derive no-go results and demonstrate their tightness in special cases.

Figures (3)

• Figure 1: Two-receiver quantum broadcast channel with private and common messages. A common message $M_0$ is intended for both receivers, while two private messages $M_1$ and $M_2$ are intended for receiver $B_1$ and $B_2$, respectively (here and throughout private does not imply confidentiality, but rather individualized). $M_{0b_i}$ denotes the estimate made by receiver $B_i$ about the message $M_0$. Similarly, $M_{1b_1}$ denotes the estimate made by receiver $B_1$ about the the message $M_1$ and $M_{2b_2}$ denotes the estimate made by receiver $B_2$ about the the message $M_2$
• Figure 2: three-receiver multilevel quantum broadcast channel with two-degraded message set. A common message $M_0$ is intended for all receivers and another message $M_1$ is intended solely for receiver $B_1$. $M_{0b_i}$ denotes the estimate made by receiver $B_i$ about the message $M_0$. Similarly, $M_{1b_1}$ denotes the estimate made by receiver $B_1$ about the the message $M_1$. Here, receiver $B_2$ is degradable with respect to the receiver $B_1$, i.e. there exists a cptp map $\mathcal{M}^{B_1\to B_2}$ such that $\rho_x^{B_2}=\mathcal{M}(\rho_x^{B_1})$ for any $x\in\mathcal{X}$.
• Figure 3: General three-receiver quantum broadcast channel with three-degraded message set. There are three independent messages $(M_0,M_1,M_2)$ such that $M_0$ is a common message intended for all receivers, $M_1$ is intended for receivers $B_1$ and $B_2$ and $M_2$ is intended solely for receiver $B_1$. $\hat{M}_{jb_i}$ represents the estimate made by receiver $B_i$ for the message $j=0,1,2$.

Theorems & Definitions (33)

• Lemma 1: Hayashi-Nagaoka inequality 1207373
• Proposition 2
• Definition 1
• Theorem 3
• Remark 1
• Remark 2
• Remark 3
• Definition 2
• Theorem 4
• Theorem 5