The Quantum Multiple-Access Channel with Cribbing Encoders

TL;DR

The work studies quantum multiple-access channels with cribbing encoders, where Encoder 2 measures a cribbing environment entangled with Encoder 1, making perfect cribbing forbidden by the no-cloning principle. It develops (i) achievable regions for strictly-causal, causal, and non-causal cribbing, (ii) a regularized capacity characterization for robust cribbing, and (iii) a partial decode-forward scheme for strictly-causal non-robust cribbing, drawing an analogy to relay channels. The results extend to classical-quantum MACs, with a complete capacity result for noiseless cribbing and a cutset bound for noisy cribbing, and include a bosonic-channel example showcasing cribbing gains. A partial decode-forward approach broadens the toolbox for non-robust cribbing, and the work clarifies how cribbing interacts with quantum entanglement and environmental measurements in multi-user quantum networks, with potential applications to robust 6G-like quantum communication architectures.

Abstract

Communication over a quantum multiple-access channel (MAC) with cribbing encoders is considered, whereby Transmitter 2 performs a measurement on a system that is entangled with Transmitter 1. Based on the no-cloning theorem, perfect cribbing is impossible. This leads to the introduction of a MAC model with noisy cribbing. In the causal and non-causal cribbing scenarios, Transmitter 2 performs the measurement before the input of Transmitter 1 is sent through the channel. Hence, Transmitter 2's cribbing may inflict a "state collapse" for Transmitter 1. Achievable regions are derived for each setting. Furthermore, a regularized capacity characterization is established for robust cribbing, i.e. when the cribbing system contains all the information of the channel input. Building on the analogy between the noisy cribbing model and the relay channel, a partial decode-forward region is derived for a quantum MAC with non-robust cribbing. For the classical-quantum MAC with cribbing encoders, the capacity region is determined with perfect cribbing of the classical input, and a cutset region is derived for noisy cribbing. In the special case of a classical-quantum MAC with a deterministic cribbing channel, the inner and outer bounds coincide.

Figures (4)

• Figure 1: The quantum multiple-access channel $\mathcal{M}_{A_1 A_2\rightarrow B}$ with cribbing at Encoder 2.
• Figure 2: The beam splitter relation of the single-mode bosonic MAC. The left beam splitter corresponds to the cribbing channel $\mathcal{L}_{A_1\to A_1' E}$, while the right beam splitter describes the communication channel $\mathcal{N}_{A_1' A_2\to B}$. Alice 1 encodes the message $m_1$ by a coherent state, and sends her transmission through $n$ uses of the cribbing channel $\mathcal{L}_{A_1\to A_1' E}$. The output $E^n$ is the cribbing system which will be measured by the second transmitter in a strictly-causal manner. At time $i$, Alice 2 encodes a coherent state that depends on her message $m_2$ and on the previous cribbing measurement outcomes $z^{i-1}$, she sends the state, and then performs a measurement on $E^i$ to obtain a measurement outcome $z_i$ which she will use in the next time instance. The inputs $A_{1,i}'^n$ and $A_{2,i}$ are sent through the communication channel $\mathcal{N}_{A_1' A_2\to B}$, and Bob receives the output $B_i$. Bob performs a measurement on $B^n$ to obtain an estimate $(\hat{m}_1,\hat{m}_2)$ of the senders' messages.
• Figure 3: Achievable regions for the single-mode bosonic MAC. The decode forward achievable region with strictly-causal cribbing is the area below the thick blue line (see (\ref{['eq:inRscG']})). For comparison, the capacity region without cribbing $\mathcal{C}_{\text{none}}(\mathcal{N}\circ\mathcal{L})$ is depicted as the area below the red dashed line (see (\ref{['eq:inR0G']})). The transmission rate of Alice 1 can be significantly higher using cribbing.
• Figure 4: Partial decode-forward cribbing scheme. The block index $j\in [1:T]$ is indicated at the top. In the following rows, we have the corresponding elements: (1), (2) auxiliary sequences; (3) codewords of Alice 1; (4) cribbing estimates by Alice 2; (5) codewords of Alice 2; (6) estimated messages at the decoder. The arrows in the fourth row indicate that the Alice 2 measures and encodes forward with respect to the block index, while the arrows in the sixth row indicate that Bob decodes backwards.

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Definition 4
• Remark 2
• Remark 3
• Theorem 1: see Winter:01pSavov:12z
• Remark 4
• Lemma 2