MacWilliams Identities for Intrinsic Quantum Codes

[2604.16023] MacWilliams Identities for Intrinsic Quantum Codes Skip to main content Search Submit Donate Log in Quantum Physics arXiv:2604.16023 (quant-ph) [Submitted on 17 Apr 2026 ( v1 ), last revised 26 Aug 2026 (this version, v2)] Title: MacWilliams Identities for Intrinsic Quantum Codes Authors: Eric Kubischta , Ian Teixeira View PDF HTML (experimental) Abstract: This paper develops a MacWilliams-type enumerator theory and semidefinite programming bounds for intrinsic quantum codes. An intrinsic code is a subspace $\mathcal{C}\subset V$ of a finite-dimensional unitary representation $V$ of a group $G$, with errors organized into the isotypic components of $\mathscr{L}(V)$ under conjugation. The Knill-Laflamme conditions persist under every $G$-equivariant realization of $V$, aallowing intrinsic bounds to apply across diverse physical encodings. Without a group action, we first construct projector and twirl quadratic enumerators for an arbitrary orthogonal decomposition of $\mathscr{L}(V)$, analogous respectively to the $A$ and $B$ enumerators of Shor-Laflamme theory. These enumerators satisfy positivity, normalization, and Knill-Laflamme inequalities, but the two superoperator families need not span the same space and therefore admit no canonical duality. A group action places their generalized versions in the same equivariant intertwiner algebra, and the change of basis between them defines the intrinsic MacWilliams transform. When the conjugation representation has multiplicities, this algebra is noncommutative and the enumerators are matrix-valued; positivity and detection are encoded by Loewner-order constraints, making the natural programming bound a semidefinite program. The multiplicity-free specialization recovers the known scalar MacWilliams transform and linear-programming framework. For symmetric powers, intrinsic depth equals ordinary qudit distance, yielding extremality certificates for permutation-invariant qudit codes. Finally, for the $27$-dimensional $\mathrm{SU}(3)$ representation $V=(2,2)$, an exact dual certificate proves the sharp bound $K\leq5$ for codes detecting the full adjoint isotypic component. An explicit $5$-dimensional code attains the bound and, under the canonical embedding, gives a four-qutrit distance-$2$ code. Comments: 33 pages. Major Revision: reorganized to emphasize the general matrix-valued MacWilliams theory; clarified relation to prior multiplicity-free work; added the Shor-Laflamme specialization; and expanded the non-multiplicity-free $\mathrm{SU}(3)$ example with a sharp $K\leq 5$ semidefinite bound, exact dual certificate, and four-qutrit interpretation Subjects: Quantum Physics (quant-ph) ; Information Theory (cs.IT) Cite as: arXiv:2604.16023 [quant-ph] (or arXiv:2604.16023v2 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2604.16023 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Eric Kubischta [ view email ] [v1] Fri, 17 Apr 2026 12:53:17 UTC (111 KB) [v2] Wed, 26 Aug 2026 20:48:50 UTC (122 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: quant-ph < prev | next > new | recent | 2026-04 Change to browse by: cs cs.IT math math.IT References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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