Genuinely multipartite entangled states and orthogonal arrays

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.

Figure 1

Orthogonal arrays of strength one (top left), two (bottom left), and three (right). Their symbolic expressions are OA(2,2,2,1), OA(4,3,2,2), and OA(8,4,2,3), respectively.

Figure 2

Connection between OAs and $k$-uniform states of $N$ qudits, where $k\le N-k$.

Figure 3

Graphic representation of $k$-uniform states. The regular polygons ${\mathcal{P}}_A$ and ${\mathcal{P}}_B$ represent the subsystems $A$ and $B$, respectively. The number of vertices of the polygons depend on the number $N$ of $d$-level subsystems and on the number $k\le k_{\max }=\lfloor N/2\rfloor$, whereas the edges characterize the degree of entanglement of the state.

Figure 4

Graphic representation of certain states of systems composed of (a) two and (b)–(d) three qubits, where (a) is a Bell state, (b) is the GHZ state, (c) is the W state, and (d) is the separable state given in Eq. (34). Auxiliary dashed lines do not belong to the graph as they represent polygons ${\mathcal{P}}_A$ and ${\mathcal{P}}_B$. States (a) and (b) represent 1-uniform states, while states (c) and (d) do not, as they do not satisfy the diagonality $\left(\mathrm{A}{}^{\prime }\right)$ and uniformity $\left(\mathrm{B}{}^{\prime }\right)$ conditions.

Figure 5

Lower bound for the number of terms $r_k$ of a $k$-uniform state of $N$ qubits imposed by the Rao bounds (10): $r_1=2,r_2=N+1,r_3=2N,r_4=N^2/2+N/2+1$, and $r_5=N^2-N+2$.

Figure 6

Overlap of the bounds $3\times 2^{n-2}+2\le N\le 3\times 2^{n-1}$ [see Eq. (51)].

Figure 7

Relationship between $k$-uniform states, QECCs, CECCs, and other relevant mathematical notions. Dashed (black) lines represent known connections, while solid (blue) lines denote the relations discussed in this paper.