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Steane enlargement of Entanglement-Assisted Quantum Error-Correcting Codes

Carlos Galindo, Fernando Hernando, Ryutaroh Matsumoto

TL;DR

This work extends Steane enlargement to entanglement-assisted quantum error-correcting codes in the Euclidean setting, deriving parameter formulas for enlarged codes and analyzing how enlargement interacts with entanglement. The authors present a general construction from a CSS-type EAQECC built from a code $C=igra B_rig rangleigoplusigra B_tig rangle$ by using an invertible $t imes t$ matrix $A$ with no $ ield_q$-eigenvalues to form an enlarged code $ ilde{D}_A$, yielding $[[n, n-2r-t+c', d'; c']]_q$ with explicit lower bounds for $d'$ and a computable $c'$. They treat two orthogonality regimes, showing how $c'$ and the distance bounds behave when $igra B_tig rangle eq 0$ is contained in or orthogonal to the dual, and they illustrate parameter gains through concrete BCH-code constructions. The BCH-focused results exploit subfield-subcodes and cyclotomic cosets to produce families of enlarged EAQECCs with often minimal entanglement (frequently $c=1$) and explicit distance bounds, complemented by several new code tables in binary, quaternary, and nonbinary settings. Overall, the paper broadens the toolkit for constructing larger EAQECCs with controlled entanglement resources, leveraging the structure of BCH codes for practical quantum communication applications.

Abstract

We introduce a Steane-like enlargement procedure for entanglement-assisted quantum error-correcting codes (EAQECCs) obtained by considering Euclidean inner product. We give formulae for the parameters of these enlarged codes and apply our results to explicitly compute the parameters of enlarged EAQECCs coming from some BCH codes.

Steane enlargement of Entanglement-Assisted Quantum Error-Correcting Codes

TL;DR

This work extends Steane enlargement to entanglement-assisted quantum error-correcting codes in the Euclidean setting, deriving parameter formulas for enlarged codes and analyzing how enlargement interacts with entanglement. The authors present a general construction from a CSS-type EAQECC built from a code by using an invertible matrix with no -eigenvalues to form an enlarged code , yielding with explicit lower bounds for and a computable . They treat two orthogonality regimes, showing how and the distance bounds behave when is contained in or orthogonal to the dual, and they illustrate parameter gains through concrete BCH-code constructions. The BCH-focused results exploit subfield-subcodes and cyclotomic cosets to produce families of enlarged EAQECCs with often minimal entanglement (frequently ) and explicit distance bounds, complemented by several new code tables in binary, quaternary, and nonbinary settings. Overall, the paper broadens the toolkit for constructing larger EAQECCs with controlled entanglement resources, leveraging the structure of BCH codes for practical quantum communication applications.

Abstract

We introduce a Steane-like enlargement procedure for entanglement-assisted quantum error-correcting codes (EAQECCs) obtained by considering Euclidean inner product. We give formulae for the parameters of these enlarged codes and apply our results to explicitly compute the parameters of enlarged EAQECCs coming from some BCH codes.
Paper Structure (7 sections, 19 theorems, 49 equations, 3 tables)

This paper contains 7 sections, 19 theorems, 49 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Steane enlargement of EAQECCs
  3. The case when $\langle B_t \rangle$ and $C$ are Euclidean orthogonal
  4. The case when $\langle B_r \rangle$ and $\langle B_t \rangle$ are Euclidean orthogonal
  5. Steane enlargement of EAQECCs given by BCH codes
  6. Results
  7. Examples

Key Result

Theorem 1

Consider two $q$-ary linear codes $C_i$, $i=1,2$, with respective parameters $[n,k_i,d_i]_q$, and assume that $C_2^{\perp_e} \subseteq C_1$. Then, there exists a stabilizer quantum code with parameters $[[n, k_1+ k_2 -n, d]]_q$ with minimum distance

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • Proposition 8
  • ...and 24 more