Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators

TL;DR

This paper develops linear programming bounds for entanglement-assisted quantum error-correcting codes, including nonadditive codes, by introducing split weight enumerators and deriving new algebraic bounds, leading to improved limitations on code parameters.

Contribution

It introduces split weight enumerators for quantum codes, derives MacWilliams identities, and establishes new linear programming bounds that improve existing bounds for entanglement-assisted quantum codes.

Findings

First LP bound improves Hamming bound for large relative distance.

No [[27,15,5]] or [[28,14,6]] quantum stabilizer codes exist.

Bounds on minimum distance of maximal-entanglement codes up to length 20 are tightened.

Abstract

Linear programming approaches have been applied to derive upper bounds on the size of classical codes and quantum codes. In this paper, we derive similar results for general quantum codes with entanglement assistance, including nonadditive codes, by considering a type of split weight enumerators. After deriving the MacWilliams identities for these split weight enumerators, we are able to prove algebraic linear programming bounds, such as the Singleton bound, the Hamming bound, and the first linear programming bound. In particular, we show that the first linear programming bound improves the Hamming bound when the relative distance is sufficiently large. On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of small quantum codes. In particular, we show that there is no [[27,15,5]] or [[28,14,6]] quantum stabilizer code. We also discuss the existence of some entanglement-assisted quantum stabilizer codes with maximal entanglement. As a result, the upper and lower bounds on the minimum distance of maximal-entanglement quantum stabilizer codes with length up to 20 are significantly improved.