In a previous paper, Shor and Laflamme (see Phys. Rev. Lett., vol.78, p.1600-02, 1997) define two "weight enumerators" for quantum error-correcting codes, connected by a MacWilliams (1977) transform, and use them to give a linear-programming bound for quantum codes. We extend their work by introducing another enumerator, based on the classical theory of shadow codes, that tightens their bounds significantly. In particular, nearly all of the codes known to be optimal among additive quantum codes (codes derived from orthogonal geometry) can be shown to be optimal among all quantum codes. We also use the shadow machinery to extend a bound on additive codes to general codes, obtaining as a consequence that any code of length, can correct at most [(n+1)/6] errors.
In a recent paper, Shor and Laflamme (see Phys. Rev. Lett., vol.78, p.1600-2, 1997) defined two "weight enumerators" for quantum error-correcting codes, connected by a MacWilliams transform, and used them to give a linear programming bound for quantum codes. We introduce two new enumerators which, while much less powerful at producing bounds, are useful tools nonetheless. The new enumerators are connected by a much simpler duality transform, clarifying the duality between Shor and Laflamme's enumerators. We also use the new enumerators to give a simpler condition for a quantum code to have specified minimum distance, and to extend the enumerator theory to codes with block size greater than 2.
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.
While quantum weight enumerators establish some of the best upper bounds on the minimum distance of quantum error-correcting codes, these bounds are not optimized to quantify the performance of quantum codes under the effect of arbitrary quantum channels that describe bespoke noise models. Herein, for any Kraus decomposition of any given quantum channel, we introduce corresponding quantum weight enumerators that naturally generalize the Shor-Laflamme quantum weight enumerators. We establish an indirect linear relationship between these generalized quantum weight enumerators by introducing an auxiliary exact weight enumerator that completely quantifies the quantum code’s projector, and is independent of the underlying noise process. By additionally working within the framework of approximate quantum error correction, we establish a general framework for constructing a linear program that is infeasible whenever approximate quantum error correcting codes with corresponding parameters do not exist. Our linear programming framework allows us to establish the non-existence of certain quantum codes that approximately correct amplitude damping errors, and obtain non-trivial upper bounds on the maximum dimension of a broad family of permutation-invariant quantum codes.
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.