COT6602 Quantum Information Theory and Quantum Error Correcting Codes

Credit: 3 units

Offered:Spring semester

Instructors: Dan Marinescu and Pawel Wocjan

Class outline

  • Overview of Linear Algebra
  • Entropy and Information
  • The Intuitive Concept of Information
  • The Shannon and Von Neumann Entropy
  • Properties of Entropy and Entropy Inequalities
  • Distinguishing Quantum States and the Accessible Information
  • Distance Measures for Quantum Information
  • Entanglement as a Physical Resource
  • Measurements of Quantum Systems
  • Born’s Rule
  • Measurement Operators
  • von Neumann-type Projective Measurements
  • Positive Operator Valued Measurements
  • Newmark’s Theorem
  • Pure and Mixed States
  • Bipartite Systems; Schmidt Decomposition; Measurements of Bipartite Systems
  • Purification of Mixed States
  • Measurements of Quantum Circuits
  • EPR
  • Bell’s and CHSH Inequalities
  • Applications
  • Quantum Teleportation
  • Superdense Coding
  • Noiseless Quantum Shannon Theory
  • Classical and Quantum Data Compression
  • Quantum-Classical Trade-Off Coding
  • RemoteState Preparation
  • Generalized RemoteState Preparation
  • Noisy Quantum Shannon Theory
  • Shannon's Noisy Channel Coding Theorem
  • Classical Information Transmission over Noisy Quantum Channels
  • Entanglement Assisted Quantum Communication (The Mother Protocol)
  • Quantum Information Transmission over Noisy Quantum Channels
  • Entanglement Assisted Classical Information Transmission over Noisy Quantum Channels
  • Entanglement Distillation Assisted by Quantum Communication (The Father Protocol)
  • Entanglement Distillation Assisted by Classical Communication
  • Noisy Teleportation
  • Noisy Superdense Coding
  • The Fully Quantum Slepian-Wolf Theorem (FQSW)
  • State Merging and the Operation Meaning of Conditional Entropy
  • Distributed Quantum Source Compression
  • Introduction to Classical Error Correction
  • Block codes
  • Hamming distance
  • Linear Codes
  • Bounds (Hamming, Singleton, Gilbert-Varsharmov, Plotkin, BCH)
  • Quantum Error Correction
  • A Necessary Condition for the Existence of a Quantum Code
  • Quantum Hamming Bound
  • Repetitive Codes for a Single Bit-Flip/Phase-Flip Errors
  • Shor, Steane, and Calderbank-Shor-Steane (CSS), Codes
  • Stabilizer Codes
  • Perfect Quantum Codes
  • Quantum Fault-Tolerance
  • Threshold Theorem

References:

  • D. C. Marinescu and G. M. Marinescu, “Approaching Quantum Information Theory and Error Correction,”
  • M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information,”Cambridge, 2000
  • J.S.Bell,“Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy,”CambridgeUniversity Press, Cambridge, 1987.
  • T.M. Cover and J.A. Thomas,“Elements of Information Theory,” Wiley Series in Telecommunications, Wiley, New York,1991.
  • S.A.Vanstone and P.C. van Oorschot, “An Introduction to Error Correcting Codes with Applications,” Kluwer Academic Publishers, Boston, MA, 1989.

Literature:

Many research articles can be accessed through the quant-ph archive maintained by Los Alamos National Laboratory.

  • H. Barnum, M. A. Nielsen, and B. Schumacher, “Information Transmission Througha Noisy Quantum Channel,” Physical Review A, 57(6):4153--4175, 1998.
  • C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters. “Teleporting an UnknownState via Dual Classical and Einstein-Podolsky-Rosen Channels,” Physical Review Letters, 70(13): 1895 - 1899, 1993.
  • C.H. Bennett and P.W. Shor, “Quantum Information Theory,” IEEE Trans. on Information Theory, 44(6):2724 - 2742, 1998.
  • A.R. Calderbank and P.W. Shor, “Good Quantum Error-Correcting Codes Exist,” Physical Review A, 54(42): 1098 - 1105, 1996.
  • A.K.Ekert and R.Jozsa, “Quantum Algorithms: Entanglement Enhanced Information Processing,”Proceedings of the Royal Society London A, 356(1743): 1769 - 1782, 1998. Also: Preprint, arxiv.org/quant-ph/9803072 v1, November 2000.
  • D.Gottesman, “Stabilizer Codes and Quantum Error Correction”, Ph.D. Thesis, California Institute of Technology}, Preprint, arxiv.org/quant-ph/9705052 v1, May 1997.
  • D. Gottesman, “An Introduction to Quantum Error Correction,” Proceedings Symposium in Applied Mathematics, Preprint, arxiv.org/quant-ph/00040072 v1, April 2000.
  • P. Hausladen, R. Jozsa, B. Schumacher, M. Westmorland, and W. K. Wooters,“Classical Information Capacity of a Quantum Channel,” Phys. Rev. A. 54(1):1869--1876, 1996.
  • S. Holevo, “The Capacity of Quantum Channel with General Signal States,” IEEE Trans. on Inform. Theory, 44:269--273, 1998, also Preprint, arXiv.org/quant-ph/9601020.
  • R. Jozsa and B. Schumacher, “A new Proof of the Quantum Noiseless Coding Theorem,” Journal of Modern Optics, 41(12):2343-2349, 1994.
  • R. Jozsa, “Entanglement and Quantum Computation,” Geometric Issues in the Foundations of Science. S. Hugget, L. Mason, K.P. Tod, S.T. Tsou, and N. M. J. Woodhouse, Editors. OxfordUniversity Press, 1997. Also: Preprint, arxiv.org/quant-ph/9707034 v1, 1997.
  • R. Jozsa, “Illustrating the Concept of Quantum Information,” Preprint arxiv.org/quant-ph/0305114 v1, 2003.
  • M. Keyl, “Fundamentals of Quantum Information,” Reprint arxiv.org/quant-ph/0202122, 2002.
  • E.Knill, R.Laflame, and W.H.Zurek, “Resilient Quantum Computation: Error Models and Thresholds,” Proceedings of the Royal Society London A , 454: 365 - 384, 1998.
  • R.Laflame, C. Miquel, J.-P. Paz, and W.H.Zurek, “Perfect Quantum-Error Correcting Code,” Physical Review Letters, 77: 198 - 201, 1996, Preprint, arxiv.org/quant-ph/9602019, 1996.
  • S. Lloyd, “Capacity of a Noisy Communication Channel,” Physical Review A, 56: 1613--1622, 1997.
  • W. Schumacher,“Quantum Coding,”Physical Review A, 51(4): 2738 - 2747, 1995.
  • W. Schumacker, M. D. Westmorland and W. K. Wooters, “Limitations on the Amount of Accessible Information in a Quantum Channel,” Phys. Rev. Lett, 76:3452--3455, 1996.
  • B. W. Schumacher and M. D. Westmorland, “Sending Quantum Information via Noisy Quantum Channels,” Phys. Rev. A, 56(1):131--138, 1997.
  • C.E. Shannon, “A Mathematical Theory of Communication,”Bell Sys. Tech. Journal, 27:379--423 and 23--656, 1948.
  • P.W.Shor, “Fault-Tolerant Quantum Computation,” 37th Annual Symposium on Foundations of Computer Science, 56 - 65, IEEE Press, Piscataway, NJ, 1996.
  • P.W.Shor, “Capacities of Quantum Channels and How to Find Them,” Preprint, arxiv.org/quant-ph/0304102 v1, April 2003.
  • A.M.Steane, “Multiple Particle Interference and Quantum Error Correction,” Preprint, arxiv.org/quant-ph/9601029 v3, May 1996.
  • A.M. Steane, “Error Correcting Codes in Quantum Theory,” Phys. Rev. Lett. 77:793--797, 1997.
  • B. M. Terhal, “Is Entanglement Monogamous?” IBM Journal of Research and Development, 48(1): 71--78,2004. Also Preprint, arxiv.org/quant-ph/0307120 v1, July 2003.
  • V.Vedral, “The Role of Entropy in Quantum Information Theory,” Preprint, arxiv.org/quant-ph/0102094 v1,
  • J. Watrous, “Lecture Notes: Theory of Quantum Information,” University of Waterloo, 2007.

Grading policy:

Homework 35%

Midterm 25%

Final exam 40%