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

2026-04-28

AI Summary

Traditional quantum error-correction metrics struggle to characterize the error structure and entanglement properties in heterogeneous quantum networks, such as qubit-qudit hybrid systems. This work proposes a mathematical framework based on dimension multisets, generalizing scalar weights to multisets to precisely capture the physical composition of error supports in mixed-dimensional Hilbert spaces. It establishes, for the first time, a mixed-dimensional quantum MacWilliams identity, from which a shadow identity and a new tight Singleton bound are derived. By further integrating combinatorial tiling techniques, the paper explicitly constructs mixed-dimensional tripartite absolutely maximally entangled states. The study thus provides rigorous constraints on the parameters of mixed-dimensional quantum error-correcting codes and enables effective criteria for the existence and explicit construction of such entangled states.

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.

Problem

• quantum error correction
• heterogeneous systems
• mixed-dimensional Hilbert spaces
• absolutely maximally entangled states
• quantum MacWilliams identity

Innovation

• mixed-dimensional quantum codes
• quantum MacWilliams identity
• absolutely maximally entangled states
• dimension multisets
• shadow inequalities