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arXiv:2604.16023 · IEEE Transactions on Information Theory

MacWilliams Identities for Intrinsic Quantum Codes

Eric Kubischta, Ian Teixeira

quant-phcs.ITSubmitted Apr 17, 2026• Announced Apr 20, 2026• 0 citations• Revised Aug 26, 2026 (v2)
v1 · Apr 17, 2026v2 · Aug 26, 2026
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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

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This paper develops a MacWilliams-type enumerator theory and semidefinite programming bounds for intrinsic quantum codes. An intrinsic code is a subspace C⊂V\mathcal{C}\subset VC⊂V of a finite-dimensional unitary representation VVV of a group GGG, with errors organized into the isotypic components of L(V)\mathscr{L}(V)L(V) under conjugation. The Knill-Laflamme conditions persist under every GGG-equivariant realization of VVV, 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 L(V)\mathscr{L}(V)L(V), analogous respectively to the AAA and BBB 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 272727-dimensional SU(3)\mathrm{SU}(3)SU(3) representation V=(2,2)V=(2,2)V=(2,2), an exact dual certificate proves the sharp bound K≤5K\leq5K≤5 for codes detecting the full adjoint isotypic component. An explicit 555-dimensional code attains the bound and, under the canonical embedding, gives a four-qutrit distance-222 code.
We introduce an intrinsic formulation of quantum error correction based on representation theory, in which error-protection structure is encoded directly in a unitary group representation, rather than being tied to a particular embedding into a larger Hilbert space. In this framework, error models are classified according to the isotypic decomposition of the conjugation action on the operator algebra. Our main result, the \emph{Schur bootstrap}, shows that if an intrinsic code satisfies the Knill--Laflamme conditions on a given symmetry sector, then the same error-protection relations hold for every extrinsic realization obtained from a group-equivariant isometric embedding into a larger Hilbert space. Thus a single intrinsic verification certifies the corresponding symmetry-resolved error-correction conditions across an entire family of physical realizations. We further introduce an intrinsic notion of distance, called depth, defined via adjoint order. For standard multi-qudit systems this coincides with conventional code distance, while for more general representations it refines the usual weight-based notion. We also prove an intrinsic Eastin--Knill theorem: any intrinsic code of depth at least two has a discrete logical symmetry group, with the obstruction to continuous covariant gates arising from the representation-theoretic structure of the adjoint action. We illustrate the framework with several examples, including a minimal SU(2)\mathrm{SU}(2)SU(2) construction that unifies permutation-invariant qubit codes and bosonic codes, and higher-dimensional constructions exhibiting transversal Clifford symmetries and realizations beyond qubit systems.
As emerging quantum architectures evolve into heterogeneous networks combining different physical substrates, such as qubits for logic and higher-dimensional qudits for robust communication, the traditional scalar metrics of quantum error correction become insufficient. To address this, we introduce a mathematical framework based on dimension multisets to characterize quantum error-correcting codes (QECC) and absolutely maximally entangled (AME) states in mixed-dimensional Hilbert spaces. By replacing scalar weights with multisets, we accurately capture the exact physical composition of error supports across these diverse systems. Our central result is the mixed-dimensional quantum MacWilliams identity, which establishes the formal algebraic relationship between Shor-Laflamme enumerators and unitary weight enumerators. From this foundation, we deduce the mixed-dimensional shadow identity and derive rigorous, generalized constraints on code parameters, explicitly formulating the mixed-dimensional quantum Hamming, Singleton and Scott bounds, and developing a linear program to systematically evaluate code viability. For the Singleton bound, a tighter bound that has no homogeneous analogue is derived for pure mixed-dimensional codes. Finally, we deploy this enumerator machinery to thoroughly analyze AME states, utilizing shadow inequalities to constrain their existence and introducing a combinatorial grid method for the explicit construction of mixed-dimensional tripartite AME states.
We derive an explicit formula for the intrinsic MacWilliams transform for permutation-invariant qudit codes. Such codes naturally live in symmetric power representations, where the relevant error sectors are determined by the irreducible decomposition of the conjugation action on the associated operator space. Using the multiplicity-free structure of this decomposition and the corresponding intertwiner algebra, we identify the intrinsic MacWilliams matrix with a finite Racah transform. The entries are given by a terminating hypergeometric series, and the rows of the matrix are Racah orthogonal polynomials with parameters determined explicitly by the block length and local dimension. Computing the spectrum of the degree-one twirl reveals that this spectrum lies on an affine quadratic lattice. Then we derive a tridiagonal multiplication rule from the representation theory of the adjoint sector. As consequences, we obtain closed-form orthogonality, detailed-balance, and involutivity identities for the transform. The resulting formula supplies an explicit MacWilliams matrix for computing linear programming bounds on permutation-invariant qudit codes.
We construct the three-point continuous-variable (CV) quantum MacWilliams identity, extending the two-point framework of Burchards, and give its closed-form integral kernel. Its configuration space carries a symplectic invariant with no classical counterpart, which encodes the GKP quantization condition and a three-point sign phase. Using the identity, we derive the semidefinite-programming bounds it supports on the dimension of CV quantum error-correcting codes, and we prove, in two collapse theorems, that the three-point apparatus does not improve on the two-point bound. For GKP lattice codes the three-point optimum equals the Burchards two-point linear-programming optimum identically. This is an exact determination of the lattice three-point optimum, so the E8E_8E8​ and Leech magic functions saturate it rather than beat it. For general bosonic codes a completely-positive reformulation bypasses the positivity obstruction that rules out the natural factored-form constructions; the phase-sign condition together with Choi positivity then force the three-point term to vanish. We certify this collapse for radial Choi forms on the first eight Laguerre levels at one mode, and leave the full trace-class cone open. Both collapses have a single cause with no classical analogue, the code projector: it orients the bound correctly but also removes the full positivity that powers the classical three-point improvement.
A unified Fourier framework for quantum weight enumerators of prime-power-dimensional qudits is developed in this paper. We introduce the notions of Weyl power spectrum and Weyl-conjugation autocorrelation for the Weyl-operator quantities nderlying the Shor--Laflamme weight enumerators. Representing these quantities as formal vectors, we show that they form a symplectic Fourier transform pair, from which the quantum MacWilliams identity follows by applying the Hamming-weight functional. This framework extends to a family of twisted Fourier transforms, including the Rains shadow transform. We show that identical-copy Bell sampling corresponds to a twisted autocorrelation related to the shadow enumerator through Weyl translation and Hamming projection. We further prove that all twisted Fourier transforms are Weyl translations of either the ordinary MacWilliams transform or the Rains shadow transform. This two-class reduction implies that the twists yield no additional independent coefficientwise positivity constraints under compatible projections. Nevertheless, the framework leads to refined MacWilliams and shadow identities that retain more Fourier information than ordinary Hamming weight enumerators. We employ these refined identities in linear programming, together with simplex mapping and shadow-coset counting, to exclude surviving candidate weight distributions. These techniques yield tighter bounds for binary and qutrit quantum codes, including nonexistence results for specific code parameters.
The MacWilliams extension theorem fails for module alphabets with non-cyclic socle, and the label alphabet of qudit stabilizer codes, $\F_{q^2}$ over $\F_q$, is such an alphabet. Quantum error correction, however, only ever sees \emph{self-orthogonal} additive codes, and whether that rigidity rescues the theorem---equivalently, whether every weight-preserving isomorphism of stabilizer groups is implemented by local Cliffords and a qudit permutation---was asked by Gluesing-Luerssen and Pllaha and answered negatively by Pllaha for particular qubit codes. We develop the negative answer systematically and at the smallest possible scales. For every prime power qqq we construct a pair of [[q+1,q−1]]q[[q+1,q-1]]_q[[q+1,q−1]]q​ stabilizer codes and a weight-preserving isomorphism between them extending to no monomial transformation; the codespaces are inequivalent even under arbitrary local unitaries combined with permutations, though they share Shor--Laflamme enumerators. Self-orthogonality is automatic here, by two elementary lemmas which also show that Dyshko's threshold-length counterexamples were already self-orthogonal, unremarked. For qubits we prove by exhaustive search that length 333 is minimal and the counterexample essentially unique. Dropping the ``idle qudit'' invariant that detects these, we find the minimal full-support lengths: 444 for a non-extendable isometry, 555 for a weight-isometric pair that is not monomially equivalent, realized by explicit [[5,2]][[5,2]][[5,2]] codes; at length 666 all nontrivial stabilizer elements can have weight ≥4\ge 4≥4. Whether these codespaces are locally unitarily equivalent is posed as an open problem, connecting the extension problem to the LU--LC circle of questions.