Qudit hypergraph states and their properties

Hypergraph states, a generalization of graph states, constitute a large class of quantum states with intriguing nonlocal properties, and they have promising applications in quantum information science and technology. In this paper, we study some features of an independently proposed generalization of hypergraph states to qudit hypergraph states, i.e., each vertex in the generalized hypergraph (multi-hypergraph) represents a $d$-level system instead of a two-level one. It is shown that multi-hypergraphs and $d$-level hypergraph states have a one-to-one correspondence, and the structure of a multi-hypergraph exhibits the entanglement property of the corresponding quantum state. We discuss their relationship with some well-known state classes, e.g., real equally weighted states and stabilizer states. The Bell nonlocality, an important resource in fulfilling many quantum information tasks, is also investigated.

Figure 1

Examples of hypergraphs. (a) A common hypergraph with $E=\{\varnothing ,\{1\},\{2,3\},\{1,2,3\left\}\right\}$ (the red circle in the left represents an empty hyperedge). The corresponding qubit hypergraph state is $|\psi \rangle =(-|000\rangle -|001\rangle -|010\rangle +|011\rangle +|100\rangle +|101\rangle +|110\rangle +|111\rangle )/\sqrt{8}$, whose stabilizer group is generated by $X_1{C}_{\varnothing }{C}_{\{2,3\}}, X_2{C}_{\left\{3\right\}}{C}_{\{1,3\}}$, and $X_3{C}_{\left\{2\right\}}{C}_{\{1,2\}}$. (b) Hypergraph with $E=\left\{\right\{1,2,3\},\{2,3,4\left\}\right\}$. As the cardinalities of the hyperedges are both 3, this hypergraph is a three-uniform hypergraph, a generalization of a conventional graph. (c) A hypergraph with $E=\left\{\right\{1,2\},\{2,3\left\}\right\}$. Because the hyperedges are both of cardinality 2, this hypergraph reduces to a conventional graph and the corresponding qubit hypergraph state becomes a conventional graph state.

Figure 2

Examples of multi-hypergraphs. Hyperedges with different multiplicities are drawn in different colors. (a) A common multi-hypergraph with $E=\{\varnothing ,\{1\},\{2,3\},\{1,2,3\},\{1,2,3\left\}\right\}$, i.e., $m_{\varnothing }=1, m_{\left\{1\right\}}=1, m_{\{2,3\}}=1, m_{\{1,2,3\}}=2$, otherwise $m_e=0$. (The red circle in the left represents an empty hyperedge.) Suppose each vertex represents a qutrit, then the corresponding quantum state is $|H_3\rangle ={\sum }_{i_1,i_2,i_3=0}^2{\omega }_3^{1+i_1+i_1{i}_2+2i_1{i}_2{i}_3}|i_1{i}_2{i}_3\rangle /\sqrt{27}$, where ${\omega }_3=e^{i2\pi /3}$. The stabilizer group is generated by $X_1{C}_{\varnothing }^{†}{\left(C_{\{2,3\}}^{†}\right)}^2, X_2{C}_{\left\{3\right\}}^{†}{\left(C_{\{1,3\}}^{†}\right)}^2$, and $X_3{C}_{\left\{2\right\}}^{†}{\left(C_{\{1,2\}}^{†}\right)}^2$. (b) An $N$-vertex multi-hypergraph with $m_{\{1,2,\cdots ,N\}}=3$ (otherwise, $m_e=0$). This multi-hypergraph is symmetric in the permutation of vertices. When encoding this multi-hypergraph into a quantum state, each vertex represents a quantum system whose dimension is larger than 3. (c) A multi-hypergraph with $E=\left\{\right\{1,2\},\{1,2\},\{2,3\left\}\right\}$. Because all the hyperedges in $E$ are of cardinality 2, this multi-hypergraph is in fact a conventional multigraph that can be encoded into qudit graph states.

Figure 3

Relationship between “qudit hypergraph states” and “generalized real equally weighted states.” (a) When $d=2$, “qudit hypergraph states” reduce to qubit hypergraph states, “generalized real equally weighted states” reduce to real equally weighted states, and the two sets are equivalent. (b) When $d>2$, qudit hypergraph states form a proper subset of generalized real equally weighted states.

Figure 4

Relationship among qudit hypergraph states ($A$), qudit graph states ($B$), and stabilizer states ($C$). $B$ is a proper subset of $A\cap C$, because there are qudit hypergraph states that are qudit graph states acted upon by single-vertex hyperedge operations and zero-vertex hyperedge operations, i.e., they are stabilizer states but not qudit graph states.

Figure 5

Violation of the CHSH inequality for various combinations of $d$ and $N$ in our measurement scheme, where $d$ is the dimension of each vertex (notice that $d$ is assumed to be prime) and $N$ is the number of vertices in the multi-hypergraph. The data points are connected for revealing the monotonicity of $C$ with respect to $N$ ($d$).