The resource theory of tensor networks

Tensor networks provide succinct representations of quantum many-body states and are an important computational tool for strongly correlated quantum systems. Their expressive and computational power is characterized by an underlying entanglement structure, on a lattice or more generally a (hyper)graph, with virtual entangled pairs or multipartite entangled states associated to (hyper)edges. Changing this underlying entanglement structure into another can lead to both theoretical and computational benefits. We study a natural resource theory which generalizes the notion of bond dimension to entanglement structures using multipartite entanglement. It is a direct extension of resource theories of tensors studied in the context of multipartite entanglement and algebraic complexity theory, allowing for the application of the sophisticated methods developed in these fields to tensor networks. The resource theory of tensor networks concerns both the local entanglement structure of a quantum many-body state and the (algebraic) complexity of tensor network contractions using this entanglement structure. We show that there are transformations between entanglement structures which go beyond edge-by-edge conversions, highlighting efficiency gains of our resource theory that mirror those obtained in the search for better matrix multiplication algorithms. We also provide obstructions to the existence of such transformations by extending a variety of methods originally developed in algebraic complexity theory for obtaining complexity lower bounds. The resource theory of tensor networks allows to compare different entanglement structures and should lead to more efficient tensor network representations and contraction algorithms.