Quantum entanglement and quantum nonlocality for $N$-photon entangled states

Abstract

Quantum entanglement and quantum nonlocality of $N$-photon entangled states $|\psi_{N m}> =\mathcal{N}_{m}[\cos\gamma|N-m>_{1}|m>_{2} +e^{i\theta_{m}}\sin\gamma|m>_{1}|N-m>_{2}]$ and their superpositions are studied. We indicate that the relative phase $\theta_{m}$ affects quantum nonlocality but not quantum entanglement for the state $|\psi_{N m}>$. We show that quantum nonlocality can be controlled and manipulated by adjusting the state parameters of $|\psi_{N m}>$, superposition coefficients, and the azimuthal angles of the Bell operator. We also show that the violation of the Bell inequality can reach its maximal value under certain conditions. It is found that quantum superpositions based on $|\psi_{N m}>$ can increase the amount of entanglement, and give more ways to reach the maximal violation of the Bell inequality.