An Entangled Tale of Quantum Entanglement

This paper gives an overview from the perspective of Lie group theory of some of the recent advances in the rapidly expanding research area of quantum entanglement. This paper is a written version of the last of eight one hour lectures given in the American Mathematical Society (AMS) Short Course on Quantum Computation held in conjunction with the Annual Meeting of the AMS in Washington, DC, USA in January 2000. More information about the AMS Short Course can be found at this http URL

[1]  M. Nielsen Characterizing mixing and measurement in quantum mechanics , 2000, quant-ph/0008073.

[2]  David Deutsch,et al.  Information flow in entangled quantum systems , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[3]  Jr.,et al.  A Rosetta Stone for Quantum Mechanics with an Introduction to Quantum Computation , 2000, quant-ph/0007045.

[4]  D. Sattinger,et al.  Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics , 1986 .

[5]  Hoi-Kwong Lo,et al.  Introduction to Quantum Computation Information , 2002 .

[6]  D. Bruss,et al.  Separability and distillability in composite quantum systems-a primer , 2000 .

[7]  Eric M. Rains Polynomial invariants of quantum codes , 2000, IEEE Trans. Inf. Theory.

[8]  P. Horodecki,et al.  Schmidt number for density matrices , 1999, quant-ph/9911117.

[9]  B. Englert,et al.  Separability of entangled q-bit pairs , 1999, quant-ph/9912089.

[10]  M. Nielsen Conditions for a Class of Entanglement Transformations , 1998, quant-ph/9811053.

[11]  S Popescu,et al.  Multi-particle entanglement , 1998 .

[12]  KuÅ,et al.  Geometry of entangled states , 2001 .

[13]  M. Horodecki,et al.  Limits for entanglement measures. , 1999, Physical review letters.

[14]  Yuriy Makhlin Nonlocal Properties of Two-Qubit Gates and Mixed States, and the Optimization of Quantum Computations , 2002, Quantum Inf. Process..

[15]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[16]  I. Chuang,et al.  Quantum Computation and Quantum Information: Bibliography , 2010 .

[17]  J. Bell,et al.  Speakable and Unspeakable in Quatum Mechanics , 1988 .

[18]  S. Popescu,et al.  Good dynamics versus bad kinematics: is entanglement needed for quantum computation? , 1999, Physical review letters.

[19]  M. Plenio,et al.  Entanglement-Assisted Local Manipulation of Pure Quantum States , 1999, quant-ph/9905071.

[20]  R. Goodman,et al.  Representations and Invariants of the Classical Groups , 1998 .

[21]  A. Sudbery,et al.  Non-local properties of multi-particle density matrices , 1998, quant-ph/9801076.

[22]  Charles H. Bennett,et al.  Concentrating partial entanglement by local operations. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[23]  C. H. Bennett,et al.  Unextendible product bases and bound entanglement , 1998, quant-ph/9808030.

[24]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[25]  Nicolas J. Cerf,et al.  Quantum information theory of entanglement , 1996 .

[26]  M. Spivak A comprehensive introduction to differential geometry , 1979 .

[27]  Eisert,et al.  Catalysis of entanglement manipulation for mixed states , 2000, Physical review letters.

[28]  Michal Horodecki,et al.  Binding entanglement channels , 1999, quant-ph/9904092.

[29]  M. Horodecki,et al.  Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps , 2000, quant-ph/0006071.

[30]  C. H. Bennett,et al.  Quantum nonlocality without entanglement , 1998, quant-ph/9804053.

[31]  Jozef Gruska,et al.  Quantum Computing , 2008, Wiley Encyclopedia of Computer Science and Engineering.

[32]  S. Virmani,et al.  Ordering states with entanglement measures , 1999, quant-ph/9911119.

[33]  M. Nielsen Continuity bounds for entanglement , 1999, quant-ph/9908086.

[34]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .