An Explicit Scott-Type Bound for Absolutely Maximally Entangled States with Arbitrary Defect

2026-06-01

AI Summary

This work investigates the existence boundaries of absolutely maximally entangled (AME) states with arbitrary defects. By constructing a truncated MacWilliams linear programming system and explicitly deriving infeasibility certificates, the authors establish for the first time a universal non-existence upper bound conjecture that holds for any defect parameter, thereby unifying and generalizing prior results by Scott and Ning et al. The derived bound takes the form $(2\ell + 2)q^2 + o(q^2)$, significantly improving the asymptotic upper bound on the ratio $k/n$ for fixed local dimension. This result also applies to one-dimensional pure quantum error-correcting codes approaching the quantum Singleton bound, offering new analytical tools and tighter constraints for the study of multipartite quantum entanglement and quantum coding theory.

Abstract

Absolutely maximally entangled (AME) states and, more generally, $k$-uniform states in $(\C^q)^{\otimes n}$ are central objects in multipartite entanglement theory, with applications to quantum secret sharing, quantum masking, and quantum error correction. In the extremal case $k=\lfloor n/2\rfloor$, Scott (2004) proved a sharp nonexistence bound showing that AME states cannot exist once the number of parties $n$ exceeds a threshold of order $2q^{2}$ (with a parity dependence on $n$), where $q$ is the local dimension. Recently, Ning et al.\ studied \emph{defective} AME states (i.e., $k=\lfloor n/2\rfloor-l$ with $l>0$), gave explicit Scott-type bounds for defects $l=1,2$ and conjectured a general $(2l+2)q^{2}+o(q^{2})$ behavior. In this paper, we solve this conjecture and establish a fully explicit Scott-type upper bound for AME states with arbitrary defect $l\ge 0$, yielding Scott's bound for $l=0$ and Ning et al.'s bounds for $l=1,2$ as special cases. Equivalently, this gives nonexistence bounds for one-dimensional pure quantum error-correcting codes near the quantum Singleton regime. The proof uses a truncated MacWilliams linear-programming system and an explicit infeasibility certificate. As a direct application, we derive explicit asymptotic upper bounds on $k/n$ for fixed local dimension $q$, improving the implicit upper bounds given by Ning et al.

Problem

• Absolutely Maximally Entangled states
• defect
• nonexistence bound
• quantum error correction
• multipartite entanglement

Innovation

• Absolutely Maximally Entangled states
• Scott-type bound
• defective AME states
• quantum error-correcting codes
• MacWilliams identities