From Entanglement Witness to Generalized Catalan Numbers

TL;DR

This paper links entanglement detection in spin systems to generalized Catalan numbers, introducing a probabilistic measurement method and the concept of sterile witnesses that minimally disturb the system.

Contribution

It establishes a novel connection between entanglement witness degeneracies, $SO(3)$ representations, and lattice walks, introducing sterile witnesses for minimally invasive entanglement detection.

Findings

Degeneracies are given by generalized Catalan numbers.

A probabilistic measurement can distinguish entangled states.

Introduction of the sterile entanglement witness concept.

Abstract

The problem of entanglement detection for arbitrary spin systems is analyzed. We demonstrate how a single measurement of the squared total spin can probabilistically discern separable from entangled many-particle states. For achieving this goal, we construct a tripartite analogy between the degeneracy of entanglement witness eigenstates, tensor products of $SO(3)$ representations and classical lattice walks with special constraints. Within this framework, degeneracies are naturally given by generalized Catalan numbers and determine the fraction of states that are decidedly entangled and also known to be somewhat protected against decoherence. In addition, we introduce the concept of a "sterile entanglement witness", which for large enough systems detects entanglement without affecting much the system's state. We discuss when our proposed entanglement witness can be regarded as a sterile one.