Phase squeezing of quantum hypergraph states

Corresponding to a hypergraph G with d vertices, a quantum hypergraph state is defined by $|G⟩=\frac{1}{\sqrt{2^d}}{\sum }_{n=0}^{2^d-1}{(-1)}^{f\left(n\right)}|n⟩$, where f is a d-variable Boolean function depending on the hypergraph G, and $|n⟩$ denotes a binary vector of length 2^d with 1 at the nth position for n = 0, 1, …(2^d − 1). The nonclassical properties of these states are studied. We consider the annihilation and creation operator on the Hilbert space of dimension 2^d acting on the number states $\{|n⟩:n=0,1,\dots (2^d-1\left)\right\}$. The Hermitian number and phase operators, in finite dimensions, are constructed. The number-phase uncertainty for these states leads to the idea of phase squeezing. We establish that these states are squeezed in the phase quadrature only and satisfy the Agarwal–Tara criterion for nonclassicality, which only depends on the number of vertices of the hypergraphs. We also point out that coherence is observed in the phase quadrature. (paper)