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

• URL: http://arxiv.org/abs/2606.01943v1
• Date: Mon, 01 Jun 2026 09:06:51 GMT
• Title: An Explicit Scott-Type Bound for Absolutely Maximally Entangled States with Arbitrary Defect
• Abstract summary: Absolutely maximally entangled (AME) states and, more generally, $k$-uniform states in $(Cq)otimes n$ are central objects in multipartite entanglement theory.<n>In the extremal case $k=lfloor n/2rfloor, Scott (2004) proved a sharp nonexistence bound showing that AME states cannot exist once the number of parties $n$ exceeds a threshold of order $2q2$.<n>In this paper, we establish a fully explicit Scott-type upper bound for AME states with arbitrary defect $lge 0$
• Score: 7.9630512253221175
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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