Multi-invariants in stabilizer states

TL;DR

This paper develops tools and algorithms to compute multipartite entanglement measures, called multi-invariants, for stabilizer states, revealing connections to topology and simplifying calculations for certain models.

Contribution

It introduces an efficient numerical algorithm for multi-invariants, provides explicit formulas for tripartite states, and explores topological links in stabilizer states.

Findings

Efficient algorithm for computing multi-invariants.

Explicit formulas for tripartite stabilizer states.

Simplified calculations for ground states of models like toric code.

Abstract

Multipartite entanglement is a natural generalization of bipartite entanglement, but is relatively poorly understood. In this paper, we develop tools to calculate a class of multipartite entanglement measures - known as multi-invariants - for stabilizer states. We give an efficient numerical algorithm that computes multi-invariants for stabilizer states. For tripartite stabilizer states, we also obtain an explicit formula for any multi-invariant using the GHZ-extraction theorem. We then present a counting argument that calculates any Coxeter multi-invariant of a q-partite stabilizer state. We conjecture a closed form expression for the same. We uncover hints of an interesting connection between multi-invariants, stabilizer states and topology. We show how our formulas are further simplified for a restricted class of stabilizer states that appear as ground states of interesting models like the toric code and the X-cube model.