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

Abstract

Let $V={⨂}_{k=1}^N{V}_k$ be the $N$ spin-$j$ Hilbert space with $d=2j+1$-dimensional single particle space. We fix an orthonormal basis $\{|m_i⟩\}$ for each $V_k$, with weight $m_i\in \{-j,\dots j\}$. Let $V_{(w)}$ be the subspace of $V$ with a constant weight $w$, with an orthonormal basis $\{|m_1,\dots ,m_N⟩\}$ 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 $|\psi ⟩$ in the constant weight subspace, which limits the possible entanglement structures of $|\psi ⟩$. Our results find applications in the overlapping quantum marginal problems, quantum error-correcting codes, and the spin-network structures in quantum gravity.

comments welcome

References in corpus (7)

• Multi-party entanglement in graph states
• Criticality, the area law, and the computational power of PEPS
• Aspects of generic entanglement
• Experimental entanglement of a six-photon symmetric Dicke state
• Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy
• Constructing optimal quantum error correcting codes from absolute maximally entangled states
• Invariant Perfect Tensors