Matrix Pencils and Entanglement Classification

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.

References in corpus (5)

• Quantum entanglement
• Entanglement detection
• Tripartite entanglement transformations and tensor rank
• Classification of multipartite entanglement containing infinitely many kinds of states
• SLOCC Convertibility between Two-Qubit States

Cited by in corpus (15)

• Characterizing multipartite entanglement classes via higher-dimensional embeddings
• Multi-copy and stochastic transformation of multipartite pure states
• Qubit-qudit states with positive partial transpose
• Description of rank four PPT entangled states of two qutrits
• How entangled can a multi-party system possibly be?
• Matrix Product States: Entanglement, symmetries, and state transformations
• Classification of arbitrary-dimensional multipartite pure states under stochastic local operations and classical communication using the rank of coefficient matrix
• The Parametric Symmetry and Numbers of the Entangled Class of 2 \times M \times N System
• Quantifying entanglement of arbitrary-dimensional multipartite pure states in terms of the singular values of coefficient matrices
• Entanglement manipulation and distillability beyond LOCC
• The Entanglement Hierarchy 2 x m x n Systems
• A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems
• Many-body entanglement: Permutations and equivalence classes
• Symmetries and local transformations of translationally invariant Matrix Product States
• Special core tensors of multi-qubit states and the concurrency of three lines