Quantum Entanglement and Spectral Form Factor

We replace a Hamiltonian by a modular Hamiltonian in the spectral form factor and the level spacing distribution function. This establishes a connection between quantities within quantum entanglement and quantum chaos. To have a universal study for quantum entanglement, we consider the Gaussian random 2-qubit model. For a generic 2-qubit model, a larger value of entanglement entropy gives a larger maximum violation of Bells inequality. We first provide an analytical estimation of the relation between quantum entanglement quantities and the dip when a subregion only has one qubit. Our numerical result confirms the analytical estimation that the occurring time of the first dip in the spectral form factor is further delayed should imply a larger value of entanglement entropy. We observe a classical chaotic behavior that dynamics in a subregion is independent of the choice of the initial state at a late time. The simulation shows that the level spacing distribution is not random matrix theory at a late time. In the end, we develop a technique within QFT to the spectral form factor for its relation to an $n$-sheet manifold. We apply the technology to a single interval in 2d CFT and also the spherical entangling surface in ${cal N}=4$ super Yang-Mills theory. The result is one for both theories, but the Renyi entropy can depend on the Renyi index. It indicates the difference between the continuum and discrete spectrum, and the dependence is then not a suitable criterion for showing whether a state is maximally entangled in QFT. The spectral form factor with a modular Hamiltonian also gives a strong constraint to the entanglement spectrum of QFT, which is useful in the context of AdS/CFT correspondence.