Qudit hypergraph states

We generalize the class of hypergraph states to multipartite systems of qudits, by means of constructions based on the $d$-dimensional Pauli group and its normalizer. For simple hypergraphs, the different equivalence classes under local operations are shown to be governed by a greatest-common-divisor hierarchy. Moreover, the special cases of three qutrits and three ququarts is analyzed in detail.

Figure 1

Example of a graph state represented by the multigraph $G=(V,E)$, where $V=\{1,2,3,4\}$ and $E=\left\{\right\{1,2\},\{1,3\},\{1,3\},\{1,3\},\{1,4\},\{1,4\},\{2\},\{2\},\{3\left\}\right\}$. The graph state in this case is $|G\rangle =Z_{12}{Z}_{13}^3{Z}_{14}^2{Z}_2^2{Z}_3{|+\rangle }^V$.

Figure 2

Hypergraph state represented by the multihypergraph $H=(V,E)$, with $V=\{1,2,3,4,5,6\}$ and $E=\left\{\right\{1,2,3\},\{1,2,3\},\{1,6\},\{1,6\},\{5\},\{5\},\{3,4,5,6\left\}\right\}$. The corresponding hypergraph state is then $|H\rangle =Z_{123}^2{Z}_{16}^2{Z}_5^2{Z}_{3456}{|+\rangle }^V$.

Figure 3

Measurements in $Z$ basis on the hypergraph state $|H\rangle =Z_{123}^2{Z}_{13}^2{|+\rangle }^V$ ($d=4$). Measurement on qudit 2 results in (a) for outcomes $m_0$ or $m_2$ and (b) for outcomes $m_1$ or $m_3$. Now, measuring on qudit 1 results in (c) for outcomes $m_0$ or $m_2$ and (d) for outcomes $m_1$ or $m_3$. It is clear that any reduced state has rank 2.

Figure 4

SLOCC equivalence among representatives of the same class.