Mixed-Dimensional Quantum MacWilliams Identity

• The mixed-dimensional quantum MacWilliams identity generalizes classical MacWilliams duality by using refined weight enumerators based on dimension multisets and mixed alphabets.
• It connects paired enumerators (Shor–Laflamme, unitary, and shadow) through explicit linear transform relations tailored to heterogeneous Hilbert spaces.
• This framework underpins new quantum bounds, AME state analysis, and stabilizer code constructions by integrating algebraic and quantum error-decomposition methods.

Searching arXiv for relevant papers on mixed-dimensional and quantum MacWilliams identities. The mixed-dimensional quantum MacWilliams identity is a quantum generalization of classical MacWilliams duality in which error supports are not summarized by a single scalar weight, but by a finer descriptor adapted to heterogeneous or mixed alphabets, mixed local dimensions, or mixed ring structures. In the literature surrounding additive codes over mixed alphabets, quantum stabilizer enumerators, and heterogeneous quantum architectures, the term refers to a family of transform relations connecting weight enumerators of a code and its dual, or of paired enumerator families such as Shor–Laflamme and unitary enumerators. The most explicit classical precursor in a mixed-alphabet setting is the MacWilliams theory for additive codes over $\mathbb{Z}_p\mathcal{R}\mathcal{S}$, where $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$ and $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$, together with Gray maps and CSS constructions yielding $p$-ary quantum codes (Debnath et al., 2022). A fully explicit quantum formulation for heterogeneous Hilbert spaces is given by the mixed-dimensional quantum MacWilliams identity based on dimension multisets, relating mixed-dimensional Shor–Laflamme and unitary weight enumerators and supporting Hamming, Singleton, Scott, and linear-programming bounds (González-Lociga et al., 28 Apr 2026).

1. Algebraic and quantum settings

The mixed-dimensional theme appears in two distinct but related settings. One is classical-algebraic: additive codes over direct products of rings such as $$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$, with $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$ and $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$, where the “mixed alphabet” is literally a direct product of symbols from $\mathbb{Z}_p$, $\mathcal{R}$, and $\mathcal{S}$, equipped with a common $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$0-module structure (Debnath et al., 2022). The other is genuinely quantum and heterogeneous: composite Hilbert spaces of the form

$\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$1

where the local dimensions $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$2 are not necessarily equal, so scalar support size is no longer an adequate proxy for error structure (González-Lociga et al., 28 Apr 2026).

In the mixed-alphabet ring setting, $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$3 and $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$4 are Frobenius rings, and $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$5 is itself Frobenius; this enables generating-character methods and MacWilliams transforms for complete, Hamming, symmetrized, and Lee weight enumerators (Debnath et al., 2022). In the heterogeneous quantum setting, the relevant refinement is not a ring-valued symbol alphabet but a multiset of subsystem dimensions. For a subsystem $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$6, its dimension multiset is

$\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$7

and this multiset determines both the scalar weight $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$8 and the dimensional weight $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$9 (González-Lociga et al., 28 Apr 2026). This replacement of scalar weight by a multiset is the defining technical move in the explicitly mixed-dimensional quantum theory.

A further, more representation-theoretic generalization replaces tensor-product weight altogether by error-sector decompositions in the conjugation representation on $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$0. In that framework, an intrinsic quantum code is a subspace $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$1, errors are organized by the isotypic decomposition of $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$2, and MacWilliams duality becomes a linear transform between projector and twirl enumerators (Kubischta et al., 17 Apr 2026). This suggests that “mixed-dimensional” can be understood either as heterogeneous tensor factors or, more generally, as a nonuniform representation-theoretic error architecture.

2. Mixed alphabets, Gray maps, and classical MacWilliams theory

The classical mixed-alphabet construction in (Debnath et al., 2022) begins with additive codes over $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$3. A $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$4-additive code of block length $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$5 is an $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$6-submodule of $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$7, and mixed constacyclic codes are realized as $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$8-submodules of

$\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$9

Under appropriate coprimality assumptions, generator polynomials factor through the three components, with generators of the form

$p$0

subject to divisibility chains in each coordinate ring (Debnath et al., 2022).

Gray maps are central because they transport mixed-alphabet codes into $p$1-linear codes while preserving distance and duality. Assuming $p$2 is a quadratic residue mod $p$3, one chooses $p$4 with $p$5, and defines

$p$6

These induce Gray maps

$p$7

that are isometries from Lee distance to Hamming distance and preserve orthogonality: $p$8 for the inner products defined on the mixed alphabet (Debnath et al., 2022).

This classical theory already exhibits the key ingredients later reused in the quantum setting: a nonuniform alphabet, a duality compatible with the weight structure, a Fourier- or character-based transform, and a mechanism translating mixed classical structure into uniform $p$9-ary stabilizer inputs via Gray images and CSS. The paper explicitly states that it does not write down a quantum MacWilliams formula, but that the combination of mixed-alphabet MacWilliams identities, Gray-map duality preservation, and CSS is “exactly the classical recipe used in the known quantum MacWilliams identities” (Debnath et al., 2022). This suggests that the mixed-alphabet construction is a precursor rather than a complete quantum theory.

3. Enumerators and transform identities

The central mathematical content of any MacWilliams theory is the existence of paired enumerators linked by a linear transform. In the mixed-alphabet Frobenius-ring setting, the complete weight enumerator of a $$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$0-additive code $$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$1 is defined by counting multiplicities of all $$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$2 symbols, and the MacWilliams identity takes the form

$$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$3

where $$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$4 is the character table derived from a generating character $$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$5 on the mixed ring (Debnath et al., 2022). Hamming, symmetrized, and Lee enumerators arise by specialization or variable-collapsing, yielding further MacWilliams identities, including

$$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$6

which has the same functional form as the classical linear-code identity, despite the underlying mixed alphabet and Lee metric (Debnath et al., 2022).

For quantum codes, the standard point of departure is the Shor–Laflamme formalism. In the qubit case, the standard quantum weight enumerators are

$$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$7

with

$$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$8

and the Shor–Laflamme identity is

$$\mathbb{Z}_p^q\times \mathcal{R}^r\times \mathcal{S}^s$$9

(Hu et al., 2018). The same paper refines this to double and complete weight enumerators tracking $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$0-weight, $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$1-weight, and full Pauli type counts, with corresponding MacWilliams transforms given by explicit linear substitutions (Hu et al., 2018). These refined enumerators show that quantum MacWilliams theory is not restricted to a single scalar weight and can accommodate split or typed statistics on error supports.

The fully explicit mixed-dimensional quantum formulation in (González-Lociga et al., 28 Apr 2026) defines mixed-dimensional Shor–Laflamme coefficients

$\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$2

$\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$3

indexed by dimension multisets $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$4, along with multiset unitary coefficients $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$5 defined by subsystem partial traces. Introducing variables $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$6 for each distinct local dimension $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$7, one forms multivariate enumerators $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$8, $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$9, $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$0, and $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$1. The main mixed-dimensional quantum MacWilliams identity is then

$\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$2

(González-Lociga et al., 28 Apr 2026). The dimension dependence enters componentwise, rather than through a single ambient $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$3, which is precisely what distinguishes the mixed-dimensional identity from its homogeneous predecessor.

4. Duality, shadows, and Krawtchouk structure

MacWilliams identities are inseparable from a duality theory. In the mixed-alphabet ring case, duals are defined by inner products that weight the $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$4-, $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$5-, and $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$6-components differently: $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$7 and constacyclic structure is preserved under duals by inversion of the multipliers (Debnath et al., 2022). The fact that $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$8 is Frobenius and admits a generating character is what makes the complete MacWilliams theory possible (Debnath et al., 2022).

In quantum coding, the analogous role is played by paired enumerators rather than by a literal linear dual code in every setting. In the qubit asymmetric theory, double and complete weight distributions satisfy coefficient-level Krawtchouk transforms: $\mathcal{S}=\mathbb{Z}_p[u]/\langle u^3\rangle$9

$\mathbb{Z}_p$0

with $\mathbb{Z}_p$1 the binary Krawtchouk polynomials (Hu et al., 2018). This coefficient-level transform is the quantum counterpart of Delsarte-style spectral duality.

The mixed-dimensional quantum theory again replaces a single Krawtchouk parameter by a product structure over distinct local dimensions. The linear relations between mixed-dimensional Shor–Laflamme, unitary, and shadow coefficients are expressed באמצעות generalized Krawtchouk polynomials indexed by the multiplicities $\mathbb{Z}_p$2 of each dimension $\mathbb{Z}_p$3 in the support multiset (González-Lociga et al., 28 Apr 2026). In particular, the shadow coefficients are linear combinations of the $\mathbb{Z}_p$4 with products of generalized Krawtchouk terms, and the shadow enumerator obeys

$\mathbb{Z}_p$5

(González-Lociga et al., 28 Apr 2026).

Shadow inequalities supply additional positivity constraints beyond MacWilliams duality itself. In the homogeneous AME analysis, generalized shadow inequalities together with the quantum MacWilliams identity yield nonexistence results for many AME parameters (Huber et al., 2017). In the mixed-dimensional setting, shadow coefficients remain non-negative and lead to a mixed-dimensional shadow identity and a linear-programming feasibility framework for code and AME existence (González-Lociga et al., 28 Apr 2026). This indicates that the mixed-dimensional quantum MacWilliams identity is not merely a formal transform: it is the algebraic backbone of a full enumerator-constrained optimization theory.

5. Relation to CSS constructions, stabilizer theory, and intrinsic generalizations

The ring-theoretic paper (Debnath et al., 2022) constructs quantum codes primarily from cyclic codes over $\mathbb{Z}_p$6, not directly from $\mathbb{Z}_p$7-codes, using Gray images and CSS. If $\mathbb{Z}_p$8 satisfies $\mathbb{Z}_p$9, then $\mathcal{R}$0 is a $\mathcal{R}$1-linear dual-containing code of length $\mathcal{R}$2, and CSS yields a $\mathcal{R}$3-ary stabilizer code with parameters $\mathcal{R}$4 in the self-orthogonal case (Debnath et al., 2022). The paper notes that a separable $\mathcal{R}$5-additive code is dual-containing iff each component is dual-containing, suggesting a route to hybrid constructions reflecting mixed alphabets (Debnath et al., 2022). This suggests that mixed-alphabet MacWilliams theory can feed into stabilizer theory through Gray-map images, even when the final quantum code lives over uniform $\mathcal{R}$6-level systems.

The modern mixed-dimensional quantum theory in (González-Lociga et al., 28 Apr 2026) differs conceptually. It does not pass through a uniform $\mathcal{R}$7-image, but works directly with heterogeneous physical systems $\mathcal{R}$8, generalized Pauli error bases, and dimension-multiset-indexed enumerators. Thus the final code itself is mixed-dimensional. The corresponding parameters are written informally as $\mathcal{R}$9, where $\mathcal{S}$0 is the dimensional minimum distance, defined as the smallest dimensional weight among undetectable errors (González-Lociga et al., 28 Apr 2026).

A broader conceptual extension appears in the intrinsic framework of (Kubischta et al., 17 Apr 2026). There, the code is a projector $\mathcal{S}$1 on an arbitrary representation space $\mathcal{S}$2, error “weights” are replaced by isotypic sectors $\mathcal{S}$3, and the MacWilliams transform becomes a unitary matrix $\mathcal{S}$4 or, with multiplicities, a block-unitary operator linking projector enumerators $\mathcal{S}$5 and twirl enumerators $\mathcal{S}$6: $\mathcal{S}$7 In multiplicity-free cases this yields linear-programming bounds; with multiplicities it becomes an SDP on matrix-valued enumerators (Kubischta et al., 17 Apr 2026). This suggests that the mixed-dimensional quantum MacWilliams identity can be interpreted as one member of a larger family of nonuniform quantum MacWilliams theories, all characterized by sectorwise transforms determined by symmetry and error decomposition.

6. Bounds, AME states, and significance

The principal utility of mixed-dimensional MacWilliams theory is the derivation of constraints on quantum-code and entanglement parameters. In the qubit asymmetric setting, refined MacWilliams identities support quantum Singleton, Hamming, and linear-programming bounds via Krawtchouk expansions and positivity arguments (Hu et al., 2018). In the mixed-dimensional setting of (González-Lociga et al., 28 Apr 2026), the same paradigm yields heterogeneous analogues of the quantum Hamming bound, the quantum Singleton bound, a stricter Singleton-type bound for pure codes, and a Scott-type bound for AME states.

The mixed-dimensional Hamming bound counts correctable errors by dimension multiset rather than scalar weight. If a non-degenerate code has correctable threshold $\mathcal{S}$8 with $\mathcal{S}$9, then

$\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$00

(González-Lociga et al., 28 Apr 2026). The mixed-dimensional Singleton theorem states that if $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$01 with $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$02 and $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$03, then

$\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$04

(González-Lociga et al., 28 Apr 2026). For pure codes, the stronger inequality

$\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$05

holds for any $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$06 with $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$07, and the paper states that this tighter bound has no homogeneous analogue (González-Lociga et al., 28 Apr 2026). This suggests that heterogeneity introduces genuinely new extremal phenomena rather than merely restating the homogeneous theory with extra indices.

The application to AME states is especially notable. In the mixed-dimensional definition adopted in (González-Lociga et al., 28 Apr 2026), a pure state is AME if every subsystem $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$08 with $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$09 is maximally mixed upon tracing out the complement. For such states, the unitary coefficients are fixed explicitly by subsystem dimensions, and the Shor–Laflamme coefficients vanish for all nonempty multisets $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$10 with $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$11 (González-Lociga et al., 28 Apr 2026). Shadow inequalities then exclude many candidate heterogeneous AME configurations; for example, the paper gives nonexistence results for several qubit–qutrit and qutrit–ququart families, and constructs explicit tripartite heterogeneous AME states by a combinatorial grid method (González-Lociga et al., 28 Apr 2026). Earlier homogeneous AME work had already shown how quantum MacWilliams and shadow identities constrain AME existence in equal local dimensions (Huber et al., 2017); the mixed-dimensional theory generalizes this enumerator machinery to heterogeneous systems.

A common misconception is to identify “mixed-dimensional quantum MacWilliams identity” with any MacWilliams relation used in quantum coding. The literature distinguishes at least three layers. First, there are classical mixed-alphabet MacWilliams identities for additive codes over heterogeneous rings, which only indirectly yield quantum consequences through Gray maps and CSS (Debnath et al., 2022). Second, there are homogeneous quantum MacWilliams identities for Shor–Laflamme, double, and complete enumerators in equal local dimension, including asymmetric Pauli refinements (Hu et al., 2018). Third, there is the explicit heterogeneous quantum theory based on dimension multisets and local-dimension-indexed variables, where the code itself inhabits a mixed-dimensional Hilbert space and the transform explicitly depends on each $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$12 (González-Lociga et al., 28 Apr 2026). Conflating these obscures the genuine novelty of the mixed-dimensional formulation.

A plausible implication is that mixed-dimensional MacWilliams theory will continue to branch in two directions. One is algebraic generalization to broader product alphabets and longer nilpotent chains, already suggested for $\mathcal{R}=\mathbb{Z}_p[u]/\langle u^2\rangle$13 in the mixed-alphabet ring context (Debnath et al., 2022). The other is structural generalization beyond tensor-product weight, as in intrinsic quantum codes where enumerators become representation-theoretic and, in the presence of multiplicities, matrix-valued (Kubischta et al., 17 Apr 2026). Together these developments indicate that the mixed-dimensional quantum MacWilliams identity is best understood not as a single isolated formula, but as a unifying principle: duality of error statistics survives under nonuniform alphabets, nonuniform local dimensions, and nonuniform symmetry sectors, provided one chooses the correct enumerator variables and the correct transform basis.