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

## TL;DR

This paper investigates how the fixed total weight constraint in many-body spin states imposes strong restrictions on their local density matrices, impacting entanglement and applications in quantum information and quantum gravity.

## Contribution

It characterizes the combinatorial constraints on reduced density matrices of states in constant weight subspaces, revealing limitations on entanglement structures.

## Key findings

- Constraints on reduced density matrices due to constant weight condition
- Implications for quantum marginal problems and error-correcting codes
- Relevance to spin-network structures in quantum gravity

## Abstract

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.