Local density matrices of many-body states in the constant weight subspaces

Understand

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 $|\psi\rangle$.