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

Semidefinite programming bounds on the size of entanglement-assisted codeword stabilized quantum codes