MacWilliams Identities for Intrinsic Quantum Codes

arXiv:2604.16023v2 Announce Type: replace 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.

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.

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

Quantum Physics (quant-ph); Information Theory (cs.IT)

arXiv:2604.16023 [quant-ph]

(or arXiv:2604.16023v2 [quant-ph] for this version)

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MacWilliams Identities for Intrinsic Quantum Codes

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