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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/231693
The mixed-dimensional quantum MacWilliams identity
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As emerging quantum architectures evolve into heterogeneous networks combining different physical substrates, such as qubits for logic and higherdimensional 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’s 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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Màster Oficial de Ciència i Tecnologia Quàntiques / Quantum Science and Technology, Facultat de Física, Universitat de Barcelona. Curs: 2025-2026. Tutor: Simeon Ball
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Matèries (anglès)
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GONZÁLEZ LOCIGA, David. The mixed-dimensional quantum MacWilliams identity. [consulted: 2 of October of 2026]. Available at: https://hdl.handle.net/2445/231693