Fig. 3
Tensor network state construction and entanglement renormalisation analysis. a The many-body wave function is decomposed into a system of individual tensor objects associated to each environment mode (A) or the system (S). Black lines represent a physical tensor index running over the possible states of the system, i.e., nb Fock states for vibrations. For a given physical state, an A tensor is a matrix of dimension D2, and the amplitude of a complete configuration of the wave function is given by the contraction (matrix multiplication) indicated in the figure (only two environments are shown for simplicity). The tensor for the system is 8th-order, due to its connections to seven environments and scales as D7 in bond dimension. b By analysing the entanglement entropy of different system–bath partitions, it is possible to decompose the system tensor into a network of third-order auxillary tensors (without physical indices), reducing the local connectivity in the network in an optimised way. Discrepancies in entropy S between multiple tensor decompositions (purple) reveal the tensor’s entanglement structure (blue) and guide the ER-network design (see main text)
