MacWilliams Identities for Intrinsic Quantum Codes

• URL: http://arxiv.org/abs/2604.16023v1
• Date: Fri, 17 Apr 2026 12:53:17 GMT
• Title: MacWilliams Identities for Intrinsic Quantum Codes
• Abstract summary: An intrinsic quantum code is a subspace of a representation $V$ of a group $G$.<n>Errors are organized by the decomposition of the conjugation representation on $mathcalL(V)$ into isotypic subspaces.
• Score: 4.297070083645049
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop an intrinsic enumerator framework for quantum error correction in unitary representations of symmetry groups. An intrinsic quantum code is a subspace of a representation $V$ of a group $G$, and errors are organized by the decomposition of the conjugation representation on $\mathcal{L}(V)$ into isotypic subspaces. Associated with any orthogonal decomposition of $\mathcal{L}(V)$ we introduce two families of quadratic enumerators, called projector and twirl enumerators, which satisfy positivity, normalization, and Knill--Laflamme type inequalities. When the conjugation representation is multiplicity--free, these enumerators are related by a linear transform that we interpret as an intrinsic MacWilliams identity. For $G=\mathrm{SU}(2)$, we compute this transform explicitly in terms of Wigner $6j$-symbols. Applied to symmetric-power representations, this gives linear programming bounds for permutation-invariant qubit and qudit codes, including extremality results for the four-qubit, seven-qubit, and three-qutrit examples treated here. We also develop the general equivariant theory in the presence of multiplicities, where the enumerators become matrix-valued, the MacWilliams transform becomes block unitary, and the resulting feasibility problem becomes semidefinite; we illustrate this theory in a first non-multiplicity-free $\mathrm{SU}(3)$ example.

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