In 1997, Shor and Laflamme defined the weight enumerators for quantum error-correcting codes and derived a MacWilliams identity. We extend their work by introducing our double weight enumerators and complete weight enumerators for qubit codes and then investigate the MacWilliams identities for these enumerators. Based on the generalized MacWilliams identities, we solve an open problem, namely, the Singleton-type bound for asymmetric quantum codes (AQCs). Besides, the Hamming-type and the first linear-programming-type bounds for the AQCs are deduced similarly.
This paper develops a MacWilliams-type enumerator theory and semidefinite programming bounds for intrinsic quantum codes. An intrinsic code is a subspace
C⊂V of a finite-dimensional unitary representation
V of a group
G, with errors organized into the isotypic components of
L(V) under conjugation. The Knill-Laflamme conditions persist under every
G-equivariant realization of
V, aallowing intrinsic bounds to apply across diverse physical encodings. Without a group action, we first construct projector and twirl quadratic enumerators for an arbitrary orthogonal decomposition of
L(V), analogous respectively to the
A and
B enumerators of Shor-Laflamme theory. These enumerators satisfy positivity, normalization, and Knill-Laflamme inequalities, but the two superoperator families need not span the same space and therefore admit no canonical duality. A group action places their generalized versions in the same equivariant intertwiner algebra, and the change of basis between them defines the intrinsic MacWilliams transform. When the conjugation representation has multiplicities, this algebra is noncommutative and the enumerators are matrix-valued; positivity and detection are encoded by Loewner-order constraints, making the natural programming bound a semidefinite program. The multiplicity-free specialization recovers the known scalar MacWilliams transform and linear-programming framework. For symmetric powers, intrinsic depth equals ordinary qudit distance, yielding extremality certificates for permutation-invariant qudit codes. Finally, for the
27-dimensional
SU(3) representation
V=(2,2), an exact dual certificate proves the sharp bound
K≤5 for codes detecting the full adjoint isotypic component. An explicit
5-dimensional code attains the bound and, under the canonical embedding, gives a four-qutrit distance-
2 code.
We derive a relationship between two different notions of fidelity (entanglement fidelity and average fidelity) for a completely depolarizing quantum channel. This relationship gives rise to a quantum analog of the MacWilliams identities in classical coding theory. These identities relate the weight enumerator of a code to the one of its dual and, with linear programming techniques, provided a powerful tool to investigate the possible existence of codes. The same techniques can be adapted to the quantum case. We give examples of their power.
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.