Tensor Terminology Conventions

Conventions for describing tensors and tensor networks:

• The range of values a single index can take is its dimension.

• The number of indices of a tensor is its order (a matrix is an order-2 tensor).

• The tensors making up a tensor network are factors or factor tensors.

• For a given tensor network, the contracted indices between factor tensors are internal indices. The uncontracted indices corresponding to the indices of the tensor the network represents are external indices.

• The term rank refers to the minimum dimension of a factorization of a tensor with respect to some bipartition of its indices.