Critical sets of the total variance of state detect all SLOCC entanglement classes

TL;DR

This paper introduces a universal algorithm to classify pure multiparticle quantum states into SLOCC equivalence classes using critical sets of the total variance, applicable to both distinguishable and indistinguishable particles.

Contribution

It provides a novel parametrization of SLOCC classes through critical points of the total variance, including analysis of Morse indices, applicable to general quantum systems.

Findings

Algorithm successfully classifies all SLOCC classes in examples

Critical sets correspond to distinct entanglement classes

Morse indices reveal stability properties of entanglement classes

Abstract

We present a general algorithm for finding all classes of pure multiparticle states equivalent under Stochastic Local Operations and Classsical Communication (SLOCC). We parametrize all SLOCC classes by the critical sets of the total variance function. Our method works for arbitrary systems of distinguishable and indistinguishable particles. We also discuss the Morse indices of critical points which have the interpretation of the number of independent non-local perturbations increasing the variance and hence entanglement of a state. We illustrate our method by two examples.