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Simultaneous Decoding of Classical Coset Codes over $3-$User Quantum Broadcast Channel: New Achievable Rate Regions

Fatma Gouiaa, Arun Padakandla

TL;DR

The paper addresses the problem of characterizing inner bounds for reliable bit transmission over a general $3$-user classical-quantum broadcast channel ($3$-CQBC). It introduces a two-stage coding strategy that combines partitioned coset codes with Sen's tilting, smoothing and augmentation (TSA) to enable simultaneous decoding of both unstructured and algebraically structured codewords. The first stage (Step I) derives an inner bound $oldsymbol{ ilde{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{ackslash}}}}}}$ by decoding bivariate interference components using a TSA-based simultaneous decoder; the second stage (Step II) enlarges the rate region to $oldsymbol{US}$ by augmenting with unstructured IID Marton-type codes, yielding a bound that subsumes all prior inner bounds. The main contributions are the explicit inner bounds (Theorems analogous to Thm.3CQBCStepIInnerBound and Thm.Step2) that integrate coset-code structure with TSA in a quantum network setting, and the demonstration that jointly decoding bivariate and univariate interference yields strictly larger capacity regions for the $3$-CQBC. This advances coding strategies for quantum broadcast channels and informs practical strategies for quantum networks where interference management is crucial.

Abstract

Combining the technique of employing coset codes for communicating over a quantum broadcast channel and the recent discovery of \textit{tilting, smoothing and augmentation} by Sen to perform simultaneous decoding over network quantum channels, we derive new inner bounds to the capacity region of a $3-$user classical quantum broadcast channel that subsumes all known.

Simultaneous Decoding of Classical Coset Codes over $3-$User Quantum Broadcast Channel: New Achievable Rate Regions

TL;DR

The paper addresses the problem of characterizing inner bounds for reliable bit transmission over a general -user classical-quantum broadcast channel (-CQBC). It introduces a two-stage coding strategy that combines partitioned coset codes with Sen's tilting, smoothing and augmentation (TSA) to enable simultaneous decoding of both unstructured and algebraically structured codewords. The first stage (Step I) derives an inner bound by decoding bivariate interference components using a TSA-based simultaneous decoder; the second stage (Step II) enlarges the rate region to by augmenting with unstructured IID Marton-type codes, yielding a bound that subsumes all prior inner bounds. The main contributions are the explicit inner bounds (Theorems analogous to Thm.3CQBCStepIInnerBound and Thm.Step2) that integrate coset-code structure with TSA in a quantum network setting, and the demonstration that jointly decoding bivariate and univariate interference yields strictly larger capacity regions for the -CQBC. This advances coding strategies for quantum broadcast channels and informs practical strategies for quantum networks where interference management is crucial.

Abstract

Combining the technique of employing coset codes for communicating over a quantum broadcast channel and the recent discovery of \textit{tilting, smoothing and augmentation} by Sen to perform simultaneous decoding over network quantum channels, we derive new inner bounds to the capacity region of a user classical quantum broadcast channel that subsumes all known.

Paper Structure

This paper contains 12 sections, 2 theorems, 52 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Role of Coset Codes in Quantum Broadcast Channels
  4. Need for simultaneous decoding of coset codes
  5. Evolving Structure of a General Coding Strategy
  6. Simultaneous Decoding of Coset Codes
  7. Step I : Simultaneous decoding of Bivariate Components
  8. Step II: Enlarging $\alpha_{S}$ via Unstructured IID codes to $\alpha_{US}$
  9. Verification of \ref{['Eqn:ClosenessOfTiltedState']}
  10. Upper bound on first two terms of \ref{['Eqn:TheFinalThreeTermsForEff5TxMAC']}
  11. Verification of Standard Inequalities
  12. Verification of \ref{['seperation of the error analysis of the three users']}

Key Result

Theorem 1

Let $\hat{\alpha}_{S} \in [0,\infty)^{4}$ be the set of all rate-cost quadruples $(R_{1},R_{2},R_{3},\tau) \in [0,\infty)^{4}$ for which there exists (i) finite sets $\mathcal{V}_{j}: j \in [3]$, (ii) finite fields $\mathcal{U}_{ij}=\mathcal{F}_{\upsilon_{j}}$ for each $ij \in \Xi\mathrel{\ensuresta for all $A \subseteq \left\{12,13,21,23,31,32\right\}, B \subseteq \left\{ 1,2,3 \right\}, C \subse

Figures (6)

  • Figure 1: Three independent messages to be communicated over a $3-$CQBC.
  • Figure 2: Depiction of all RVs in the full blown coding strategy. In Sec. \ref{['SubSec:StepICodingTheorem']} (Step I) only RVs in the gre dashed box are non-trivial, with the rest trivial.
  • Figure 3: The $9$ codebooks on the left are used by Tx. The $3$ rightmost cosets are obtained by adding corresp. cosets. Rx $k$ decodes into $4$ codes in row $k$.
  • Figure 4: $9$ codes employed in the proof of Thm. \ref{['Thm:3CQBCStepIInnerBound']}. 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 auxiliary finite sets $\mathcal{V}_{j}$ 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 $U_{ij}\oplus U_{kj}$.
  • Figure 5: Communication over the effective CQMAC : The five RVs or codebooks on the left form only a sub-collection of RVs or codes handled by the encoder.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Theorem 1
  • Remark 1
  • proof
  • Remark 2
  • Theorem 2