Matrix Pencils and Entanglement Classification

TL;DR

This paper introduces a novel approach connecting matrix pencils to entanglement classification in quantum systems, providing a polynomial-time algorithm to determine SLOCC equivalence and revealing the hierarchy of entanglement classes.

Contribution

It establishes a new matrix pencil method for classifying entanglement in 2⊗m⊗n systems, enabling efficient SLOCC equivalence determination and hierarchy analysis.

Findings

First polynomial-time algorithm for SLOCC equivalence in 2⊗m⊗n systems

Revealed hierarchy among 26 entanglement classes in 2⊗3⊗n systems

Unified entanglement classification using matrix pencils

Abstract

In this paper, we study pure state entanglement in systems of dimension $2\otimes m\otimes n$. Two states are considered equivalent if they can be reversibly converted from one to the other with a nonzero probability using only local quantum resources and classical communication (SLOCC). We introduce a connection between entanglement manipulations in these systems and the well-studied theory of matrix pencils. All previous attempts to study general SLOCC equivalence in such systems have relied on somewhat contrived techniques which fail to reveal the elegant structure of the problem that can be seen from the matrix pencil approach. Based on this method, we report the first polynomial-time algorithm for deciding when two $2\otimes m\otimes n$ states are SLOCC equivalent. Besides recovering the previously known 26 distinct SLOCC equivalence classes in $2\otimes 3\otimes n$ systems, we also determine the hierarchy between these classes.