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.
From: Ching-Yi Lai [view email]
[v1]
Mon, 1 Feb 2016 07:35:36 UTC (100 KB)
[v2]
Fri, 19 Aug 2016 06:29:42 UTC (125 KB)
[v3]
Wed, 17 May 2017 15:41:03 UTC (128 KB)
[v4]
Thu, 3 Sep 2026 15:18:46 UTC (128 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。