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.
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.
In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the MacWilliams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the MacWilliams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.
A unified Fourier framework for quantum weight enumerators of prime-power-dimensional qudits is developed in this paper. We introduce the notions of Weyl power spectrum and Weyl-conjugation autocorrelation for the Weyl-operator quantities nderlying the Shor--Laflamme weight enumerators. Representing these quantities as formal vectors, we show that they form a symplectic Fourier transform pair, from which the quantum MacWilliams identity follows by applying the Hamming-weight functional. This framework extends to a family of twisted Fourier transforms, including the Rains shadow transform. We show that identical-copy Bell sampling corresponds to a twisted autocorrelation related to the shadow enumerator through Weyl translation and Hamming projection. We further prove that all twisted Fourier transforms are Weyl translations of either the ordinary MacWilliams transform or the Rains shadow transform. This two-class reduction implies that the twists yield no additional independent coefficientwise positivity constraints under compatible projections. Nevertheless, the framework leads to refined MacWilliams and shadow identities that retain more Fourier information than ordinary Hamming weight enumerators. We employ these refined identities in linear programming, together with simplex mapping and shadow-coset counting, to exclude surviving candidate weight distributions. These techniques yield tighter bounds for binary and qutrit quantum codes, including nonexistence results for specific code parameters.