Let $V=\bigotimes_{k=1}^{N} V_{k}$ be the $N$ spin-$j$ Hilbert space with $d=2j+1$-dimensional single particle space. We fix an orthonormal basis $\{|m_i\rangle\}$ for each $V_{k}$, with weight $m_i\in \{-j,\ldots j\}$. Let $V_{(w)}$ be the subspace of $V$ with a constant weight $w$, with an orthonormal basis $\{|m_1,\ldots,m_N\rangle\}$ subject to $\sum_k m_k=w$. We show that the combinatorial properties of the constant weight condition imposes strong constraints on the reduced density matrices for any vector $|\psi\rangle$ in the constant weight subspace, which limits the possible entanglement structures of $|\psi\rangle$. Our results find applications in the overlapping quantum marginal problems, quantum error-correcting codes, and the spin-network structures in quantum gravity.

中文翻译：

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恒重子空间中多体状态的局部密度矩阵

令 $V=\bigotimes_{k=1}^{N} V_{k}$ 为 $N$ 自旋-$j$ Hilbert 空间，$d=2j+1$ 维单粒子空间。我们为每个 $V_{k}$ 固定一个正交基 $\{|m_i\rangle\}$，权重 $m_i\in \{-j,\ldots j\}$。设 $V_{(w)}$ 是 $V$ 的子空间，权重为 $w$，标准正交基 $\{|m_1,\ldots,m_N\rangle\}$ 服从 $\sum_k m_k= w$。我们表明，恒重条件的组合特性对恒重子空间中任何向量 $|\psi\rangle$ 的约简密度矩阵施加了强约束，这限制了 $|\psi\rangle$ 可能的纠缠结构。我们的结果在重叠量子边际问题、量子纠错码和量子引力中的自旋网络结构中找到了应用。