Quantizing Delsarte Theory
Abstract
We develop Delsarte theory for quantum association schemes, allowing both composition and the quantum Schur product to be noncommutative. A MacWilliams identity and positivity of the inner and dual distributions yield conic upper bounds for codes and lower bounds for designs in terms of effective size, which extends subset cardinality. These bounds recover classical linear programs for symmetric association schemes and semidefinite programs for homogeneous coherent configurations. We also quantize Schurian schemes using finite-group representations and characterize quantum block designs in quantum Johnson schemes arising from irreducible stabilizer representations by averaging over subgroups fixing prescribed points individually. In a quantum Johnson scheme on M2(ℂ)⊕10, we prove that the minimum effective size of a nonzero positive element satisfying the strength-two design condition is $20$, attained by complementary projections outside the center of the underlying algebra. Every such element in the center has effective size $40$, while the conic lower bound is 40/3, strictly below the actual minimum.
Related
- Schrijver-Delsarte rigidity in association schemes and undecidability of quantum graph homomorphism
- The Rate of the Delsarte Bound for Binary Codes and Spheres
- Thoughts on Quantum Events, Quantizing Gravity and von Neumann Algebras
- A dual theory of Zhu's theory
- On the internal homotopy theory of motivic categories
BibTeX
@misc{misawa2026quantizing,
title = {{Quantizing Delsarte Theory}},
author = {Ryutaro Misawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09900},
eprint = {2610.09900},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09900},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantizing Delsarte Theory
AU - Ryutaro Misawa
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09900
UR - https://arxiv.org/abs/2610.09900
AB - We develop Delsarte theory for quantum association schemes, allowing both composition and the quantum Schur product to be noncommutative. A MacWilliams identity and positivity of the inner and dual distributions yield conic upper bounds for codes and lower bounds for designs in terms of effective size, which extends subset cardinality. These bounds recover classical linear programs for symmetric association schemes and semidefinite programs for homogeneous coherent configurations. We also quantize Schurian schemes using finite-group representations and characterize quantum block designs in quantum Johnson schemes arising from irreducible stabilizer representations by averaging over subgroups fixing prescribed points individually. In a quantum Johnson scheme on M_2(ℂ)^⊕10, we prove that the minimum effective size of a nonzero positive element satisfying the strength-two design condition is $20$, attained by complementary projections outside the center of the underlying algebra. Every such element in the center has effective size $40$, while the conic lower bound is 40/3, strictly below the actual minimum.
ER -