Genuinely multipartite entangled states and orthogonal arrays
Dardo Goyeneche, Karol Życzkowski · Physical Review A · 2014
A pure quantum state of $N$ subsystems with $d$ levels each is called $k$-multipartite maximally entangled state, which we call a $k$-uniform state, if all its reductions to $k$ qudits are maximally mixed. These states form a natural generalization of $N$-qudit Greenberger-Horne-Zeilinger states which belong to the class 1-uniform states. We establish a link between the combinatorial notion of orthogonal arrays and $k$-uniform states and prove the existence of several classes of such states for $N$-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of $N>5$ qubits. We show that finding a different class of 2-uniform states would imply the Hadamard conjecture, so the full classification of 2-uniform states seems to be currently out of reach. Furthermore, we establish links between the existence of $k$-uniform states and classical and quantum error correction codes and provide a graph representation for such states.