The Entanglement perspectives on the quantum many body problem

量子多体系统的背景和状态量子多体系统是由几个组分组成的系统，例如由原子制成的材料，或由光子或核子制成的光。量子多体系统的复杂性量子多体系统由复波函数来描述，其参数数量随着成分的数量呈指数增加，从而有效地限制了我们研究它们的能力。在特定情况下，例如当研究用局部足够的哈密顿量描述的多体系统的平衡状态时，张量网络提供了对多体量子系统的指数大状态的多项式描述。张量网络是由以特定模式收缩的小的基本张量构成的，该模式由网络的几何形状决定。张量网络的几何形状完全决定了它所描述的状态下的关联模式。由于这个原因，在平衡态的纠缠结构和张量网络的几何结构之间存在着联系，张量网络最适合描述局域关联。在非平衡演化过程中，短程关联辐射到长程关联，因此传统的张量网络可以很好地描述多体波函数。研究假设与目的我们的目标是用张量网络来描述多体量子系统的非平衡动力学。1)我们将应用这样的直觉：尽管在非平衡演化过程中产生了长程纠缠，但关联仍然是短程的。通过这种方式，我们将设计一种策略来执行多体量子态的近似非平衡，这将允许得到局域可观测的正确演化。2)表征和理解高级张量网络结构，如谱张量网络和分支Mera，并使它们适合于执行量子系统的非平衡演化。

Background and state of the ArtQuantum many body systems are systems made by several constituent, such as for example materials, made by atoms, or light made by photons or nuclei made by nucleons.Complexity of quantum many body systems Quantum many body systems are described by complex wave functions whose number of parameters increases exponentially with the number of constituents thus effectively limiting our capability to study them.Tensor networks In specific cases, like for example when studying the equilibrium state of many body systems described with local enough Hamiltonians, tensor networks provide a polynomial description of the exponentially large state of many body quantum systems. Tensor networks are constructed from small elementary tensors contracted in a specific pattern dictated by the geometry of the network.Geometry of tensor networks the geometry of the network completely determines the patterns of correlations in the state it describes. For this reason there is a connection between the structure of entanglement at equilibrium and the geometry of the tensor network and tensor networks are best suited to describe local correlations. Out of equilibrium, During the out-of-equilibrium evolution short range correlations radiate into long range correlations and thus traditional tensor networks chase to provide good description of the many body wave function. Research hypothesis and objectives Our objective is to find a possible descriptions of the out-of-equilibrium dynamics of a many body quantum systems in term of tensor networks.We will1) apply the intuition that despite the fact that during the out-of-equilibrium evolution long range entanglement is created, the correlations are still short ranged. In this way we will devise a strategy to perform approximate out of equilibrium of many body quantum states that will allow to get the correct evolutions of local observables.2) characterize and understand advanced tensor network structures such as the spectral tensor networks and the branching MERAs and adapt them to perform the out of equilibrium evolution of quantum systems.