In this paper, we explore the application of semidefinite programming to the realm of quantum codes, specifically focusing on codeword stabilized (CWS) codes with entanglement assistance. Notably, we utilize the isotropic subgroup of the CWS group and the set of word operators of a CWS-type quantum code to derive an upper bound on the minimum distance. Furthermore, this characterization can be incorporated into the associated distance enumerators, enabling us to construct semidefinite constraints that lead to SDP bounds on the minimum distance or size of CWS-type quantum codes. We illustrate several instances where SDP bounds outperform LP bounds, and there are even cases where LP fails to yield meaningful results, while SDP consistently provides tight and relevant bounds. Finally, we also provide interpretations of the Shor-Laflamme weight enumerators and shadow enumerators for codeword stabilized codes, enhancing our understanding of quantum codes.

翻译：本文探索了半定规划在量子码领域的应用，重点关注具有纠缠辅助的码字稳定子（CWS）量子码。值得注意的是，我们利用CWS群的各向同性子群以及CWS型量子码的字算子集合，导出了最小距离的上界。进一步地，这一特性可融入相关的距离计数器中，从而能够构造半定约束条件，进而得到CWS型量子码最小距离或码字大小的半定规划（SDP）界。我们通过多个实例展示了SDP界优于线性规划（LP）界的情况，甚至存在LP无法给出有意义结果而SDP始终提供紧致且相关界的情形。最后，我们还提供了码字稳定子码的Shor-Laflamme权重计数器和影子计数器的解释，这加深了我们对量子码的理解。