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Entanglement-Assisted Codes Outside the Stabilizer Framework

Jaszmine DeFranco, Andrew Nemec

TL;DR

It is shown how entanglement-assisted codes can be constructed from arbitrary quantum codes by associating them with quantum codes for erasure channels, and gives examples of permutation-invariant and XP-stabilizer entanglement-assisted codes, the first outside of the stabilizer and codeword-stabilized frameworks.

Abstract

We show how entanglement-assisted codes can be constructed from arbitrary quantum codes by associating them with quantum codes for erasure channels. If a subset of physical qubits is correctable for an erasure error, then it naturally forms the receiver's share of a bipartite state that can be used for entanglement-assisted communications, both in the noiseless and noisy ebit error models. In the case of degenerate codes, we show that the receiver's share of the bipartite state can sometimes be compressed, at the cost of potentially reduced error-correction ability in the noisy ebit error model. We also give examples of permutation-invariant and XP-stabilizer entanglement-assisted codes, the first outside of the stabilizer and codeword-stabilized frameworks.

Entanglement-Assisted Codes Outside the Stabilizer Framework

TL;DR

It is shown how entanglement-assisted codes can be constructed from arbitrary quantum codes by associating them with quantum codes for erasure channels, and gives examples of permutation-invariant and XP-stabilizer entanglement-assisted codes, the first outside of the stabilizer and codeword-stabilized frameworks.

Abstract

We show how entanglement-assisted codes can be constructed from arbitrary quantum codes by associating them with quantum codes for erasure channels. If a subset of physical qubits is correctable for an erasure error, then it naturally forms the receiver's share of a bipartite state that can be used for entanglement-assisted communications, both in the noiseless and noisy ebit error models. In the case of degenerate codes, we show that the receiver's share of the bipartite state can sometimes be compressed, at the cost of potentially reduced error-correction ability in the noisy ebit error model. We also give examples of permutation-invariant and XP-stabilizer entanglement-assisted codes, the first outside of the stabilizer and codeword-stabilized frameworks.
Paper Structure (8 sections, 12 theorems, 39 equations)

This paper contains 8 sections, 12 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum Error Correction
  4. Erasure and Replacer Channels
  5. Entanglement-Assisted Quantum Error Correction
  6. Entanglement-Assisted Subspace Codes
  7. Receiver Share Compression via Degeneracy
  8. Conclusion

Key Result

Theorem 1

Let $\mathcal{C}\subseteq\mathcal{H}$ be a quantum code with projector $P$. Then $\mathcal{C}$ corrects the set of errors $\mathcal{E}\subseteq\mathcal{L}\!\left(\mathcal{H}\right)$ if and only if for all $E_i,E_j\in\mathcal{E}$, where $\lambda_{ij}\in\mathbb{C}$.

Theorems & Definitions (25)

  • Theorem 1: Error Correction Condition for Quantum Subspace Codes
  • Theorem 2: CHKNN2025
  • Corollary 3
  • proof
  • Example 4
  • Theorem 5: GHW2022
  • Remark 6
  • Theorem 7: Error Correction Condition for Noiseless Ebits
  • Theorem 8: Error Correction Condition for Noisy Ebits
  • Theorem 9
  • ...and 15 more