Quantum $k$-uniform states from quantum orthogonal arrays

Abstract

The quantum orthogonal arrays define remarkable classes of multipartite entangled states called $k$-uniform states whose every reductions to $k$ parties are maximally mixed. We present constructions of quantum orthogonal arrays of strength 2 with levels of prime power, as well as some constructions of strength 3. As a consequence, we give infinite classes of 2-uniform states of $N$ systems with dimension of prime power $d\ge 2$ for arbitrary $N\ge 5$; 3-uniform states of $N$-qubit systems for arbitrary $N\ge 6$ and $N\neq 7,8,9,11$;3-uniformstatesof$N$systemswithdimensionofprimepower$d\geq 7$forarbitrary$N\geq 7$.