Comment on "Matrix Pencils and Entanglement Classification"

TL;DR

This paper extends a polynomial-time algorithm to enumerate all SLOCC equivalence classes in 2⊗m⊗n quantum systems, using matrix pencil canonical forms, and provides explicit representatives for classes in 2⊗3⊗n systems.

Contribution

It introduces a straightforward modification of an existing algorithm to classify all SLOCC equivalence classes via Kronecker canonical forms in the specified quantum systems.

Findings

Enumeration of all SLOCC classes in 2⊗m⊗n systems.

Explicit representatives for each class in 2⊗3⊗n systems.

Algorithmic approach based on matrix pencil canonical forms.

Abstract

In our earlier posting "Matrix Pencils and Entanglement Classification", arXiv:0911.1803, we gave a polynomial-time algorithm for deciding if two states in a space of dimension $2\otimes m\otimes n$ are SLOCC equivalent. In this note, we point out that a straightforward modification of the algorithm gives a simple enumeration of all SLOCC equivalence classes in the same space, with the class representatives expressed in the Kronecker canonical normal form of matrix pencils. Thus, two states are equivalent if and only if they have the same canonical form. As an example, we present representatives in canonical form for each of the 26 equivalence classes in $2\otimes 3\otimes n$ systems.