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.

翻译：随着新型量子架构演进为结合不同物理基底的异质网络（例如用于逻辑运算的量子比特和用于鲁棒通信的高维量子态），传统量子纠错的标量度量方法已显不足。为此，我们提出基于维度多重集的数学框架，用于刻画混合维度希尔伯特空间中的量子纠错码（QECC）与绝对最大纠缠（AME）态。通过用多重集替代标量权重，我们精确捕获这些异质系统中错误支撑的物理组成。核心成果是混合维度量子MacWilliams恒等式，该恒等式建立了Shor-Laflamme枚举子与酉权重枚举子之间的形式代数关系。基于此，我们推导出混合维度影子恒等式，并导出参数约束的严格广义界——明确建立混合维度量子Hamming界、Singleton界和Scott界，同时开发线性规划系统以系统评估码可行性。针对纯混合维度码，我们推导出比Singleton界更紧且无同质对应版本的界。最后，运用该枚举子体系全面分析AME态：利用影子不等式约束其存在性，并引入组合网格方法用于显式构造混合维度三部分AME态。