Classical Feedback in a Quantum Network

TL;DR

The paper investigates how classical feedback, implemented through measurement on the receiver's quantum output, can boost communication rates over a quantum multiple-access channel. It develops a coding framework that extends classical block Markov schemes to the quantum setting, employing a three-layer superposition code and backward decoding, and leverages the quantum multiparty packing lemma to establish achievability results. Two main inner bounds are derived: the Quantum Cover–Leung region and the Partial Decode region, with the latter strictly enlarging the former in general. The qubit SWAP channel serves as a key example where feedback yields a strictly larger achievable region than the no-feedback case. These results reveal that classical feedback can meaningfully enhance quantum multi-user networks, despite the intrinsic no-cloning limitation, and lay groundwork for exploring feedback vs. entanglement-assisted resources in hybrid classical-quantum systems.

Abstract

Communication over a quantum multiple access channel (MAC) is considered with classical feedback. Since the no-cloning prohibits universal copying of arbitrary quantum states, classical feedback is generated through measurement. An achievable rate region is derived using partial information decoding at each transmitter. Our region generalizes both the classical Cover-Leung region and the generalized feedback region. As an example, we show that the qubit SWAP channel can benefit from feedback.

Figures (7)

• Figure 1: Quantum MAC with classical feedback.
• Figure 2: Achievable regions for the qubit SWAP channel. The capacity region without feedback is the area within the dashed black line. An achievable rate region with classical feedback is indicated by the solid blue line.
• Figure 3: MAC Bayesian multiplex network.
• Figure 4: Bayesian multiplex network $\mathcal{B}_{\text{cl-fb}}$ generating the codebook $C$ for the quantum MAC with classical feedback, illustrating the Cover–Leung scheme with $T = 2$ blocks.
• Figure 5: The block index $t\in [1:T]$ is indicated at the top. In the following rows, we have the corresponding elements: (1) codeword of Alice 1; (2) Alice 1 estimates; (3) codeword of Alice 2; (4) estimated messages at Bob. The arrows in the second row indicate that Alice 1 estimates and encodes forward with respect to the block index, while the arrows in the fourth row indicate that Bob decodes backwards.

Theorems & Definitions (16)

• Definition 1: Code with classical feedback
• Remark 1: Operational description
• Remark 2: Communication without feedback
• Remark 3: Perfect feedback and generalized feedback
• Definition 2: Achievable rate pair
• Definition 3: Capacity region
• Theorem 1: Quantum Cover-Leung bound
• Remark 4: Full Decode Limitation
• Example 1
• Theorem 2