Tensor Network Methods for Invariant Theory

TL;DR

This paper introduces a tensor network graphical approach to represent and analyze polynomial invariants of quantum states, offering a simpler and insightful alternative to traditional polynomial equations, especially for matrix product states.

Contribution

It develops a tensor network method for representing polynomial invariants, providing structural insights and simplifying the study of invariants in quantum states.

Findings

Generated a complete set of local unitary invariants for matrix product states.

Demonstrated the use of tensor network techniques to analyze quantum invariants.

Provided a graphical framework that simplifies the understanding of polynomial invariants.

Abstract

Invariant theory is concerned with functions that do not change under the action of a given group. Here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states. This graphical approach provides an alternative to the polynomial equations that describe invariants, which often contain a large number of terms with coefficients raised to high powers. This approach also enables one to use known methods from tensor network theory (such as the matrix product state factorization) when studying polynomial invariants. As our main example, we consider invariants of matrix product states. We generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the R\'enyi entropies. We find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted, as well as an alternative, and sometimes much simpler, means to study polynomial invariants of quantum states. In addition, many tensor network methods, such as matrix product states, contain excellent tools that can be applied in the study of invariants.