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Journal of Mathematical Physics / Volume 51 / Issue 7 / ARTICLES / Quantum Information and Computation

J. Math. Phys. 51, 072205 (2010); http://dx.doi.org/10.1063/1.3459069 (16 pages)

Matrix pencils and entanglement classification

Eric Chitambar1, Carl A. Miller2, and Yaoyun Shi3

1Department of Physics, University of Michigan, 450 Church Street, Ann Arbor, Michigan 48109-1040, USA
2Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109-1043, USA
3Department of Electrical Engineering and Computer Science, University of Michigan, 2260 Hayward Street, Ann Arbor, Michigan 48109-2121, USA

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(Received 15 March 2010; accepted 22 May 2010; published online 22 July 2010)

Quantum entanglement plays a central role in quantum information processing. A main objective of the theory is to classify different types of entanglement according to their interconvertibility through manipulations that do not require additional entanglement to perform. While bipartite entanglement is well understood in this framework, the classification of entanglements among three or more subsystems is inherently much more difficult. In this paper, we study pure state entanglement in systems of dimension 2⊗m⊗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⊗m⊗n states are SLOCC equivalent. We then proceed to present a canonical form for all 2⊗m⊗n states based on the matrix pencil construction such that two states are equivalent if and only if they have the same canonical form. Besides recovering the previously known 26 distinct SLOCC equivalence classes in 2⊗3⊗n systems, we also determine the hierarchy between these classes.

© 2010 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. MATRIX PENCILS
  3. CONNECTION TO 2⊗m⊗n PURE STATES
  4. EQUIVALENCE ALGORITHM AND THE STATE KRONECKER CANONICAL FORM
    1. Equivalence algorithm
    2. The state Kronecker canonical form
  5. ALL TRIPARTITE SYSTEMS WITH A FINITE SLOCC EQUIVALENCE PARTITIONING
  6. NONINVERTIBLE TRANSFORMATIONS
  7. CONCLUSIONS AND FUTURE RESEARCH

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KEYWORDS and PACS

Keywords

quantum entanglement

PACS

  • 03.67.Mn

    Entanglement measures, witnesses, and other characterizations

  • 03.65.Ud

    Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3459069

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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