Phase Squeezing of Quantum Hypergraph States

Abstract

Corresponding to a hypergraph $G$ with $d$ vertices, a quantum hypergraph state is defined by $|G⟩=\frac{1}{\sqrt{2^d}}{\sum }_{n=0}^{2^d-1}(-1{)}^{f(n)}|n⟩$, 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 $n$-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 $\{|n⟩:n=0,1,\dots (2^d-1)\}$. 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.

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