Abstract
This work is devoted to the study Translation-Invariant (TI) Matrix Product State (MPS) representations of quantum states with periodic boundary conditions (PBC). We pursue two directions: we introduce new methods for constructing TI MPS representations of a certain class of TI states and study their optimality in terms of their bond dimension. We pay particular attention to the n-party W-state and construct a TI MPS representation of bond dimension n/2 +1 for it. We further study properties of this class and show that we can can always achieve a bond dimension of n for TI MPS representation of states in this class. In the framework of studying optimality of TI MPS representations with PBC, we study the optimal bond dimension d(ψ) for a given state ψ. In particular we introduce a deterministic algorithm for the search of d(ψ) for an arbitary state. Using numerical methods, we verify the optimality of our previous construction for the n-party W-state for small n.