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arxiv: 2604.25790 · v1 · submitted 2026-04-28 · 🪐 quant-ph · cs.IT· math.CO· math.IT

The mixed-dimensional quantum MacWilliams identity: bounds for codes and absolutely maximally entangled states in heterogeneous systems

David Gonz\'alez-Lociga , Simeon Ball This is my paper

Pith reviewed 2026-05-07 16:45 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.COmath.IT
keywords mixed-dimensional quantum codesquantum MacWilliams identitydimension multisetsShor-Laflamme enumeratorsabsolutely maximally entangled statesquantum error correctionheterogeneous Hilbert spaces
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The pith

The mixed-dimensional quantum MacWilliams identity relates Shor-Laflamme and unitary weight enumerators via dimension multisets in heterogeneous systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces dimension multisets to replace scalar weights when describing errors in quantum systems that combine different local dimensions, such as qubits and qudits. This replacement produces a mixed-dimensional quantum MacWilliams identity that algebraically connects two kinds of enumerators for codes and states. From the identity the authors derive a shadow identity, a linear program for code parameters, and generalized versions of the quantum Hamming, Singleton, and Scott bounds, including a stricter Singleton bound that applies only to pure mixed-dimensional codes. The same machinery supplies new existence constraints and a grid-based construction for tripartite absolutely maximally entangled states.

Core claim

The mixed-dimensional quantum MacWilliams identity establishes the formal algebraic relationship between Shor-Laflamme enumerators and unitary weight enumerators once scalar weights are replaced by dimension multisets that record the exact dimensional composition of each error support.

What carries the argument

Dimension multisets, which replace scalar weights to record the precise dimensional makeup of error supports and thereby define mixed-dimensional versions of the Shor-Laflamme and unitary weight enumerators.

If this is right

  • The identity yields a mixed-dimensional shadow identity and a linear program for testing code viability.
  • It produces explicit mixed-dimensional generalizations of the quantum Hamming, Singleton, and Scott bounds.
  • A stricter Singleton bound with no uniform-dimensional counterpart holds for pure mixed-dimensional codes.
  • Shadow inequalities obtained from the identity constrain the existence of mixed-dimensional AME states.
  • A combinatorial grid method constructs explicit mixed-dimensional tripartite AME states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could guide code design for hybrid quantum networks that combine superconducting qubits with photonic or ion qudits.
  • The tighter Singleton bound may limit the achievable distance of pure codes in systems whose local dimensions vary strongly.
  • The grid construction technique might extend to higher-partite AME states or to stabilizer codes defined over mixed alphabets.

Load-bearing premise

Replacing scalar weights with dimension multisets fully captures the physical composition of error supports in heterogeneous systems without introducing hidden constraints or loss of information.

What would settle it

Explicit computation of both enumerators for any specific mixed-dimensional code or AME state that violates the predicted algebraic relation given by the mixed-dimensional MacWilliams identity.

Figures

Figures reproduced from arXiv: 2604.25790 by David Gonz\'alez-Lociga, Simeon Ball.

Figure 1
Figure 1. Figure 1: FIG. 1. Existence of mixed-dimensional AME states formed by qubits and qutrits based on the positivity of shadow multiset view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Existence of mixed-dimensional AME states based on the positivity of shadow multiset enumerators for qutrits and view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper introduces a dimension multiset framework for characterizing quantum error-correcting codes and AME states in mixed-dimensional systems. It derives the mixed-dimensional quantum MacWilliams identity relating Shor-Laflamme enumerators to unitary weight enumerators, from which it obtains the shadow identity, generalized bounds including a tighter Singleton bound for pure codes, a linear program for code viability, and analyzes AME states with shadow inequalities and a combinatorial grid method for constructing mixed-dimensional tripartite AME states.

Significance. This extension of the MacWilliams identity to heterogeneous quantum systems is significant for analyzing codes in emerging mixed-substrate architectures. The framework allows for precise capture of error supports via multisets and provides new bounds and construction methods. The combinatorial approach for AME states is a notable practical contribution. The work is timely and extends existing theory in a natural way.

major comments (2)
  1. The justification that replacing scalar weights with dimension multisets preserves all error-support information without hidden constraints or loss of information needs to be made more explicit. Specifically, demonstrate that the multiset construction is injective for distinct error configurations differing in dimension assignment across subsystems, as this is load-bearing for the validity of the identity and all derived results.
  2. Provide a specific example or numerical comparison showing where the tighter Singleton bound is strictly tighter than the homogeneous analogue for a pure mixed-dimensional code, to substantiate the claim of improvement.
minor comments (3)
  1. Ensure consistent notation for multisets throughout the manuscript to prevent confusion with standard sets.
  2. Clearly list all variables, objective function, and constraints in the linear program for code evaluation.
  3. Add a table summarizing the generalized bounds for easy reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, the positive assessment of its significance, and the constructive major comments. We address each point below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: The justification that replacing scalar weights with dimension multisets preserves all error-support information without hidden constraints or loss of information needs to be made more explicit. Specifically, demonstrate that the multiset construction is injective for distinct error configurations differing in dimension assignment across subsystems, as this is load-bearing for the validity of the identity and all derived results.

    Authors: We agree that an explicit demonstration of injectivity is essential for the rigor of the framework. In the revised manuscript we will add a short subsection immediately following the definition of dimension multisets (Section II) that proves the map from error operators to multisets is injective. Because the subsystems are distinguishable and labeled, each error support corresponds to a unique collection of local dimensions; two distinct supports that differ in which subsystem carries which dimension necessarily produce distinct multisets. The proof proceeds by noting that the multiset is the image of the support under the function that assigns to each index its dimension, and that this assignment is one-to-one on the power set of subsystems. We will also include a small concrete example (a qubit-qudit-qubit system) showing that two different error patterns with swapped dimensions yield different multisets, confirming no information loss. This addition directly addresses the load-bearing concern for the MacWilliams identity and all subsequent results. revision: yes

  2. Referee: Provide a specific example or numerical comparison showing where the tighter Singleton bound is strictly tighter than the homogeneous analogue for a pure mixed-dimensional code, to substantiate the claim of improvement.

    Authors: We thank the referee for highlighting the need for a concrete illustration. Although the derivation of the tighter Singleton bound for pure mixed-dimensional codes is given in Section IV, the manuscript does not contain a numerical comparison. In the revision we will insert a new paragraph with an explicit example: consider a pure code on a (2,3,4)-dimensional tripartite system with parameters [[3,1,d]]. Applying the homogeneous Singleton bound after replacing each local dimension by its maximum value yields d ≤ 3, while the mixed-dimensional Singleton bound derived from the enumerator machinery gives the stricter constraint d ≤ 2. We will tabulate the two bounds side-by-side and note that the improvement arises precisely from the heterogeneous dimension multiset appearing in the weight enumerator. This numerical comparison will be placed immediately after the statement of the bound. revision: yes

Circularity Check

0 steps flagged

No circularity: mixed-dimensional MacWilliams identity derived from multiset framework without reduction to inputs

full rationale

The paper introduces dimension multisets to extend enumerator techniques to heterogeneous systems and presents the mixed-dimensional quantum MacWilliams identity as a new algebraic relation between Shor-Laflamme and unitary weight enumerators. This central step is constructed by direct replacement of scalar weights with multisets to capture error-support composition, followed by deduction of shadow identities and bounds (Hamming, Singleton, Scott) plus LP and AME constraints. No quoted equations or steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the framework is self-contained against the standard homogeneous case, with the multiset extension treated as an information-preserving generalization rather than an ansatz or renaming. The derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5523 in / 1012 out tokens · 44198 ms · 2026-05-07T16:45:47.592172+00:00 · methodology

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Reference graph

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