Matrix Pencils and Entanglement Classification

Details Matrix Pencils and Entanglement Classification Matrix Pencils and Entanglement Classification

2009-11-09

arxiv.org/abs/0911.1803v1

J. Math. Phys. 51, 072205 (2010)

Physics

Quantum Physics

Scientific paper

10.1063/1.3459069

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.

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