$k$-uniform mixed states.

We investigate the maximum purity that can be achieved by k-uniform mixed states of N parties. Such N-party states have the property that all their k-party reduced states are maximally mixed. A scheme to construct explicitly k-uniform states using a set of specific N-qubit Pauli matrices is proposed. We provide several different examples of such states and demonstrate that in some cases the state corresponds to a particular orthogonal array. The obtained states, despite being mixed, reveal strong non-classical properties such as genuine multipartite entanglement or violation of Bell inequalities.