On Translation-Invariant Matrix Product States and advances in MPS representations of the $W$-state

TL;DR

This paper investigates translation-invariant matrix product state representations of quantum states with periodic boundary conditions, introducing new construction methods, analyzing their optimality, and applying these to the $W$-state with numerical verification.

Contribution

It presents new methods for constructing TI MPS representations, analyzes their optimality, and provides a deterministic algorithm to find the minimal bond dimension for arbitrary states.

Findings

Constructed a TI MPS for the $W$-state with bond dimension $loor{n/2}+1$.

Showed that a bond dimension of $n$ is always achievable for certain classes of states.

Verified the optimality of the $W$-state MPS representation for small $n$ through numerical methods.

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 $\left \lfloor \dfrac{n}{2} \right \rfloor +1$ 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 $d(\psi)$ for a given state $\psi$. In particular we introduce a deterministic algorithm for the search of $d(\psi)$ for an arbitary state. Using numerical methods, we verify the optimality of our previous construction for the $n$-party $W$-state for small $n$.