来自最大距离可分码的真正不可扩展乘积基
Genuinely Unextendible Product Bases from Maximum Distance Separable Codes
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中文总结 AI 辅助
本文利用经典最大距离可分码，为N≥3的参与者构造出真正不可扩展乘积基，得到真正纠缠子空间与多体束缚纠缠态，建立纠错码与多体纠缠的联系，提供认证真正多体束缚纠缠的代数途径。
中文摘要 AI 辅助
真正不可扩展乘积基（GUPB）是一类不完全正交的完全乘积态集合，其正交补在任意二划分下均不包含任何乘积向量，该集合的存在性长期以来是一个开放性问题。本文中，我们利用经典最大距离可分（MDS）码，为任意数量N≥3的参与者构造了GUPB。MDS特性对任意二划分下诱导的乘积铺砌施加了刚性约束；结合傅里叶模式删除与一个 stopper 态，该刚性约束确保了真正不可扩展性。因此，每个GUPB的正交补是一个真正纠缠的子空间，其归一化投影算子在任意二划分下的部分转置下保持不变，从而产生了一个显式的多体束缚纠缠态族。我们进一步构造了GME见证，该见证能够检测这些态，而任何完全可分解的见证都无法做到这一点。此外，所得的不可区分性在任意有限张量幂和任意二划分下可分的测量下均保持。这些结果建立了纠错码与多体纠缠之间的直接联系，并为认证真正多体束缚纠缠提供了一条代数途径。
英文摘要
The existence of genuinely unextendible product bases (GUPBs), incomplete orthogonal sets of fully product states whose orthogonal complements contain no product vector across any bipartition, has remained an open problem. Here we construct GUPBs for any number $N\geq3$ of parties using classical maximum distance separable (MDS) codes. The MDS property imposes a rigidity on the induced product tiling across every bipartition; combined with Fourier mode deletion and a stopper state, this rigidity enforces genuine unextendibility. Consequently, the orthogonal complement of each GUPB is a genuinely entangled subspace whose normalized projector is invariant under partial transposition across every bipartition, yielding an explicit family of multipartite bound entangled states. We further construct GME witnesses that detect these states even though no fully decomposable witness can do so. Moreover, the resulting indistinguishability persists under arbitrary finite tensor powers and measurements separable across any bipartition. These results establish a direct connection between error-correcting codes and multipartite entanglement and provide an algebraic route to certifying genuinely multipartite bound entanglement.
