On the Completely Separable State for the Lohe Tensor Model

We study completely separable states of the Lohe tensor model and their asymptotic collective dynamics. Here, the completely separable state means that it is a tensor product of rank-1 tensors. For the generalized Lohe matrix model corresponding to the Lohe tensor model for rank-2 tensors with the same size, we observe that the component rank-1 tensors of the completely separable states satisfy the swarm double sphere model introduced in [Lohe in Physica D 412, 2020]. We also show that the swarm double sphere model can be represented as a gradient system with an analytic potential. Using this gradient flow formulation, we provide the swarm multisphere model on the product of multiple unit spheres with possibly different dimensions, and then we construct a completely separable state of the swarm multisphere model as a tensor product of rank-1 tensors which is a solution of the proposed swarm multisphere model. This concept of separability has been introduced in the theory of quantum information. Finally, we also provide a sufficient framework leading to the complete aggregation of completely separable states.