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

• URL: http://arxiv.org/abs/2604.25790v1
• Date: Tue, 28 Apr 2026 15:59:03 GMT
• Title: The mixed-dimensional quantum MacWilliams identity: bounds for codes and absolutely maximally entangled states in heterogeneous systems
• Abstract summary: We introduce a mathematical framework based on dimension multisets to characterize quantum error-correcting codes.<n>By replacing scalar weights with multisets, we accurately capture the exact physical composition of error supports across diverse systems.
• Score: 2.7214777196418645
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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