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Constructions of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes

Xiujing Zheng, Liqi Wang, Shixin Zhu

TL;DR

By leveraging generalized Reed-Solomon codes and the Hermitian entanglement-assisted quantum error-correcting framework, the paper constructs three families of entanglement-assisted quantum MDS codes (EAQMDS) with explicit length formulas and parameter ranges. The approach uses the hull structure of GRS codes and the rank of $HH^\dagger$ to determine the required ebits, achieving parameters that meet the EA-quantum Singleton bound $n-k+c \ge 2(d-1)$. Notably, several lengths are sums of divisors of $q^2-1$ and some are not divisors of $q^2-1$, expanding the catalog of EAQMDS codes with larger minimum distances than previously known. These constructions broaden the landscape of feasible EAQMDS codes for quantum communication with pre-shared entanglement.

Abstract

By generalizing the stabilizer quantum error-correcting codes, entanglement-assisted quantum error-correcting (EAQEC) codes were introduced, which could be derived from any classical linear codes via the relaxation of self-orthogonality conditions with the aid of pre-shared entanglement between the sender and the receiver. In this paper, three classes of entanglement-assisted quantum error-correcting maximum-distance-separable (EAQMDS) codes are constructed through generalized Reed-Solomon codes. Under our constructions, the minimum distances of our EAQMDS codes are much larger than those of the known EAQMDS codes of the same lengths that consume the same number of ebits. Furthermore, some of the lengths of the EAQMDS codes are not divisors of $q^2-1$, which are completely new and unlike all those known lengths existed before.

Constructions of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes

TL;DR

By leveraging generalized Reed-Solomon codes and the Hermitian entanglement-assisted quantum error-correcting framework, the paper constructs three families of entanglement-assisted quantum MDS codes (EAQMDS) with explicit length formulas and parameter ranges. The approach uses the hull structure of GRS codes and the rank of to determine the required ebits, achieving parameters that meet the EA-quantum Singleton bound . Notably, several lengths are sums of divisors of and some are not divisors of , expanding the catalog of EAQMDS codes with larger minimum distances than previously known. These constructions broaden the landscape of feasible EAQMDS codes for quantum communication with pre-shared entanglement.

Abstract

By generalizing the stabilizer quantum error-correcting codes, entanglement-assisted quantum error-correcting (EAQEC) codes were introduced, which could be derived from any classical linear codes via the relaxation of self-orthogonality conditions with the aid of pre-shared entanglement between the sender and the receiver. In this paper, three classes of entanglement-assisted quantum error-correcting maximum-distance-separable (EAQMDS) codes are constructed through generalized Reed-Solomon codes. Under our constructions, the minimum distances of our EAQMDS codes are much larger than those of the known EAQMDS codes of the same lengths that consume the same number of ebits. Furthermore, some of the lengths of the EAQMDS codes are not divisors of , which are completely new and unlike all those known lengths existed before.
Paper Structure (8 sections, 17 theorems, 76 equations, 4 tables)

This paper contains 8 sections, 17 theorems, 76 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. New EAQMDS codes from GRS codes
  4. New EAQMDS codes of length $n=\frac{b({q^2}-1)}{a}+\frac{{q^2}-1}{a}$ with $a\vert (q+1)$
  5. The case $a+b\equiv 1\ ({\rm mod}\ 2)$
  6. The case $a+b\equiv 0\ ({\rm mod}\ 2)$
  7. New EAQMDS codes of length $n=\frac{b({q^2}-1)}{a}$ with $a\vert (q-1)$
  8. Conclusion

Key Result

Theorem 1.1

(AAHM22LA18BDH06GHW22)(EA-quantum Singleton bound) For any $[[n,k,d;c]]_q$ EAQEC code with $d \le \frac{n+2}{2}$ must satisfy where $0\le c \le n-1$. Furthermore, it is indeed the quantum Singleton bound if $c=0$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 29 more