Small $k$-pairable states

A $k$-pairable $n$-qubit state is a resource state that allows Local Operations and Classical Communication (LOCC) protocols to generate EPR-pairs among any $k$-disjoint pairs of the $n$ qubits. Bravyi et al. introduced a family of $k$-pairable $n$-qubit states, where $n$ grows exponentially with $k$. Our primary contribution is to establish the existence of 'small' pairable quantum states. Specifically, we present a family of $k$-pairable $n$-qubit graph states, where $n$ is polynomial in $k$, namely $n=O(k^3\ln^3k)$. Our construction relies on probabilistic methods. Furthermore, we provide an upper bound on the pairability of any arbitrary quantum state based on the support of any local unitary transformation that has the shared state as a fixed point. This lower bound implies that the pairability of a graph state is at most half of the minimum degree up to local complementation of the underlying graph, i.e., $k(|G \rangle)\le \lceil \delta_{loc}(G)/2\rceil$. We also investigate the related combinatorial problem of $k$-vertex-minor-universality: a graph $G$ is $k$-vertex-minor-universal if any graph on any $k$ of its vertices is a vertex-minor of $G$. When a graph is $2k$-vertex-minor-universal, the corresponding graph state is $k$-pairable. More precisely, one can create not only EPR-pairs but also any stabilizer state on any $2k$ qubits through local operations and classical communication. We establish the existence of $k$-vertex-minor-universal graphs of order $O(k^4 \ln k)$. Finally, we explore a natural extension of pairability in the presence of errors or malicious parties and show that vertex-minor-universality ensures a robust form of pairability.