Entanglement Classification via Operator Size

TL;DR

This paper introduces a polynomial-based method for classifying multipartite entanglement by relating operator size to entanglement structure, enabling polynomial factorization to identify entanglement features.

Contribution

It defines the entanglement polynomial using operator size and subspace dimensions, providing a new algebraic tool for entanglement classification and analysis.

Findings

Entanglement polynomial characterizes entanglement equivalence classes.

Polynomial factorization reveals entanglement building blocks.

Renormalized states facilitate practical computation of the polynomial.

Abstract

In this work, multipartite entanglement is classified by polynomials. I show that the operator size is closely related to the entanglement structure. Given a generic quantum state, I define a series of subspaces generated by operators of different sizes acting on it. The information about the entanglement is encoded into these subspaces. With the dimension of these subspaces as coefficients, I define a polynomial which I call the entanglement polynomial. The entanglement polynomial induces a homomorphism from quantum states to polynomials. It implies that we can characterize and find the building blocks of entanglement by polynomial factorization. Two states share the same entanglement polynomial if they are equivalent under the stochastic local operations and classical communication. To calculate the entanglement polynomial practically, I construct a series of states, called renormalized states, whose ranks are related to the coefficients of the entanglement polynomial.