Phase Squeezing of Quantum Hypergraph States

Corresponding to a hypergraph \(G\) with \(d\) vertices, a quantum hypergraph state is defined by \(|G\rangle = \frac{1}{\sqrt{2^d}}\sum_{n = 0}^{2^d - 1} (-1)^{f(n)} |n \rangle\), where \(f\) is a \(d\)-variable Boolean function depending on the hypergraph \(G\), and \(|n \rangle\) denotes a binary vector of length \(2^d\) with \(1\) at \(n\)-th position for \(n = 0, 1, \dots (2^d - 1)\). The non-classical properties of these states are studied. We consider annihilation and creation operator on the Hilbert space of dimension \(2^d\) acting on the number states \(\{|n \rangle: n = 0, 1, \dots (2^d - 1)\}\). 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 non-classicality, which only depends on the number of vertices of the hypergraphs. We also point out that coherence is observed in the phase quadrature.