In quantum mechanics, tensor network representations like the matrix product state (MPS) simplify the handling of complex wave functions by breaking them down into a series of linked tensors, each representing a system site. These tensors use auxiliary indices to represent entanglement, with more entangled systems requiring larger indices. For mixed states, the entanglement of purification (EP) indicates the necessary size of auxiliary indices for accurate representation, but is challenging to calculate precisely, so estimates of its bounds are used instead.
