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Achievable Rate Regions for Multi-terminal Quantum Channels via Coset Codes

Fatma Gouiaa, Arun Padakandla

TL;DR

This work develops new inner bounds for the classical-quantum capacity regions of two 3-user quantum networks: the 3-user classical-quantum interference channel (3-CQIC) and the 3-user classical-quantum broadcast channel (3-CQBC). The authors fuse structured coset codes, notably nested coset codes, with Sen’s tilting, smoothing, and augmentation (TSA) to enable true simultaneous decoding of multiple codebooks, a key to achieving improved throughput in multi-terminal quantum channels. The approach unfolds in two stages for both networks: (i) Step I establishes an initial inner bound (denoted $\alpha_S$ for CQIC and $\alpha_S$ for CQBC) based on simultaneous decoding of bi-variate coset components; (ii) Step II enlarges these regions to $\alpha_{US}$ by incorporating unstructured IID (Han-Kobayashi/Marton-style) components for univariate interference, yielding inner bounds that subsume prior results. The framework demonstrates that coset-code structures, coupled with joint decoding of bivariate interference, can strictly outperform traditional unstructured coding in quantum networks and provides a unified methodology applicable to both interference and broadcast scenarios, with explicit rate-region characterizations grounded in quantum information quantities and tilted-state analysis.

Abstract

We undertake a Shannon theoretic study of the problem of communicating classical information over (i) a $3-$user quantum interference channel (QIC) and (ii) a $3-$user quantum broadcast channel (QBC). Our focus is on characterizing inner bounds. In our previous work, we had demonstrated that coding strategies based on coset codes can yield strictly larger inner bounds. Adopting the powerful technique of \textit{tilting}, \textit{smoothing} and \textit{augmentation} discovered by Sen recently, and combining with our coset code strategy we derive a new inner bound to the classical-quantum capacity region of both the $3-$user QIC and $3-$user QBC. The derived inner bound subsumes all current known bounds.

Achievable Rate Regions for Multi-terminal Quantum Channels via Coset Codes

TL;DR

This work develops new inner bounds for the classical-quantum capacity regions of two 3-user quantum networks: the 3-user classical-quantum interference channel (3-CQIC) and the 3-user classical-quantum broadcast channel (3-CQBC). The authors fuse structured coset codes, notably nested coset codes, with Sen’s tilting, smoothing, and augmentation (TSA) to enable true simultaneous decoding of multiple codebooks, a key to achieving improved throughput in multi-terminal quantum channels. The approach unfolds in two stages for both networks: (i) Step I establishes an initial inner bound (denoted for CQIC and for CQBC) based on simultaneous decoding of bi-variate coset components; (ii) Step II enlarges these regions to by incorporating unstructured IID (Han-Kobayashi/Marton-style) components for univariate interference, yielding inner bounds that subsume prior results. The framework demonstrates that coset-code structures, coupled with joint decoding of bivariate interference, can strictly outperform traditional unstructured coding in quantum networks and provides a unified methodology applicable to both interference and broadcast scenarios, with explicit rate-region characterizations grounded in quantum information quantities and tilted-state analysis.

Abstract

We undertake a Shannon theoretic study of the problem of communicating classical information over (i) a user quantum interference channel (QIC) and (ii) a user quantum broadcast channel (QBC). Our focus is on characterizing inner bounds. In our previous work, we had demonstrated that coding strategies based on coset codes can yield strictly larger inner bounds. Adopting the powerful technique of \textit{tilting}, \textit{smoothing} and \textit{augmentation} discovered by Sen recently, and combining with our coset code strategy we derive a new inner bound to the classical-quantum capacity region of both the user QIC and user QBC. The derived inner bound subsumes all current known bounds.

Paper Structure

This paper contains 20 sections, 5 theorems, 82 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Inner Bounds to the Capacity Region of a $3-$CQIC
  3. Preliminaries and Problem Statement for $3-$CQIC
  4. Need for simultaneous decoding of coset codes over $3-$CQIC
  5. Specifications of an efficient coding strategy for the $3-$CQIC
  6. Simultaneous Decoding of Coset Codes over the $3-$CQIC
  7. Step I : Simultaneous Decoding of Bi-variate Components over the $3-$CQIC
  8. Step II: Enlarging $\alpha_{S}$ via Unstructured IID Codes to $\alpha_{US}$
  9. Inner Bounds to the Capacity Region of a $3-$CQBC
  10. Preliminaries and Problem Statement for $3-$CQBC
  11. Role of Coset Codes in Quantum Broadcast Channels
  12. Simultaneous decoding of coset codes over the $3-$CQBC
  13. Evolving Structure of a General Coding Strategy
  14. Simultaneous Decoding of Coset Codes over the $3-$CQBC
  15. Step I : Simultaneous decoding of Bivariate Components over a $3-$CQBC
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\hat{\alpha}_{S} \in [0,\infty)^{6}$ be the set of all rate-cost vectors $(\underline{R},\underline{\tau})$ for which there exists for all $j \in [3]$, (i) finite fields $\mathcal{U}_{ji}=\mathcal{F}_{q_{i}},\mathcal{U}_{jk}=\mathcal{F}_{q_{k}}$, (ii) PMF $p_{\underline{\underline{U}} ~\!\!\und holds for every $j \in [3]$, $A_{j} \subseteq \{ji,jk\}$, where $U_{j}^{\oplus}= U_{ij}\oplus U_{kj

Figures (12)

  • Figure 1: Three independent messages to be communicated over a $3-$CQBC.
  • Figure 2: Three independent messages to be communicated over a $3-$CQIC.
  • Figure 3: Depiction of all RVs in the full blown coding strategy. In Sec. \ref{['SubSec:StageICodeStrategy3CQIC']} (Step I) only RVs in the gre dashed box are non-trivial, with the rest trivial.
  • Figure 4: The box with 3 codes depicts the codes at Tx $j$. Rx $j$ decodes into these 4 codes.
  • Figure 5: $9$ codes employed in the proof of Thm. \ref{['Thm:3CQICStageIRateRegion']}. The $U_{ji} : ji \in \llbracket 3 \rrbracket$ are coset codes built over finite fields. Codes with the same color are built over the same finite field and the smaller of the two is a sub-coset of the larger. The black codes are built over the finite sets $\mathcal{X}_{j}$ using the conventional IID random code structure. Row $j$ depicts the codes of Tx, Rx $j$. Rx $j$, in addition to decoding into codes depicted in row $j$ also decodes $U_{ij}\oplus U_{kj}$.
  • ...and 7 more figures

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Example 2
  • Theorem 1
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • ...and 10 more