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

TL;DR

This paper develops a new mathematical framework using dimension multisets to analyze quantum error correction and entangled states in heterogeneous systems, leading to generalized bounds and construction methods.

Contribution

It introduces the mixed-dimensional quantum MacWilliams identity and related bounds, extending quantum error correction theory to heterogeneous, mixed-dimensional systems.

Findings

Derived the mixed-dimensional quantum MacWilliams identity.

Established generalized quantum bounds like Hamming, Singleton, and Scott bounds.

Developed a linear program for evaluating code viability in mixed-dimensional systems.

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.