Homology of Generic Stabilizer States

TL;DR

This paper studies the homological invariants of multi-party stabilizer states in quantum information, revealing that typical entanglement involves just over half of the parties as the number of qubits grows large.

Contribution

It introduces a probabilistic analysis of homological invariants for large stabilizer states using Bruhat decomposition, highlighting typical entanglement patterns.

Findings

Homological invariants concentrate around a generic expected value for large N.

Typical entanglement involves just over half of the parties.

The Bruhat decomposition is used as a key analytical tool.

Abstract

This work is concerned with multi-party stabilizer states in the sense of quantum information theory. We investigate the homological invariants for states of which each party holds a large equal number N of quantum bits. We show that in many cases there is a generic expected value of the invariants: for large N it is approximated with arbitrarily high probability if a stabilizer state is chosen at random. The result suggests that typical entanglement of stabilizer states involves but the sets comprising just over one half of the parties. Our main tool is the Bruhat decomposition from the theory of finite Chevalley groups.