Quantum Relay Channels

TL;DR

This work extends quantum Shannon theory to fully quantum relay channels by deriving three complementary lower bounds: partial decode-forward, measure-forward, and assist-forward. Each bound corresponds to a distinct relaying strategy, with PDF allowing the relay to decode part of the message, MF using measurement compression, and AF exploiting entanglement-assisted transmission across an orthogonal-receiver setup. The authors establish a single-letter capacity formula for Hadamard relay channels under the PDF scheme and provide bounds for depolarizing and other channels, thereby connecting quantum relay performance to environment-assisted and entanglement-assisted paradigms. The results advance understanding of quantum network cooperation and have potential implications for quantum repeaters and long-distance quantum communication. Overall, the paper presents a unified framework and actionable bounds that generalize classical relay results to the fully quantum regime, with concrete examples illustrating the gains from quantum relaying.

Abstract

Communication over a fully quantum relay channel is considered. We establish three bounds based on different coding strategies, i.e., partial decode-forward, measure-forward, and assist-forward. Using the partial-decode forward strategy, the relay decodes part of the information, while the other part is decoded without the relay's help. The result by Savov et al. (2012) for a classical-quantum relay channel is obtained as a special case. Based on our partial-decode forward bound, the capacity is determined for Hadamard relay channels. In the measure-forward coding scheme, the relay performs a sequence of measurements and then sends a compressed representation of the measurement outcome to the destination receiver. The measure-forward strategy can be viewed as a generalization of the classical compress-forward bound. At last, we consider quantum relay channels with orthogonal receiver components. The assist-forward bound is based on a new approach, whereby the transmitter sends the message to the relay and simultaneously generates entanglement assistance between the relay and the destination receiver. Subsequently, the relay can transmit the message to the destination receiver with rate-limited entanglement assistance.

Figures (6)

• Figure 1: A three-terminal relay network.
• Figure 2: Coding for a fully quantum relay channel $\mathcal{N}_{A D \rightarrow B E}$. The quantum systems of Alice, Bob, and the relay are marked in red, blue, and brown, respectively.
• Figure 3: A diagram of the quantum relay channel: The transmitters at the sender and the relay are labeled as $A$ and $D$, and the receivers at the destination and the relay as $B$ and $E$, respectively.
• Figure 4: A graphical representation for the noiseless relay channel in Example \ref{['Example:Trivial_AF']}. Alice can transmit three qubits to the relay, from $A_0$ to $E$, and a single qubit to Bob, from $A_1$ to $B_1$. The relay can send a single qubit to Bob, from $D$ to $B_2$.
• Figure 5: Partial decode-forward strategy. The block index $j\in [1:T]$ is indicated at the top. In the following rows, we have the corresponding elements: (1) auxiliary sequences; (2) codewords of Alice; (3) relay estimates; (4) relay codewords; (5), (6) estimated messages at the destination receiver. The arrows in the third row indicate that the relay measures and encodes forward with respect to the block index, while the arrows in the fifth row indicate that Bob decodes backwards.

Theorems & Definitions (22)

• Definition 1: Degraded relay channel
• Definition 2
• Definition 3
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 1
• Corollary 2: see SavovWildeVu:12c
• Remark 5