COT6600 Quantum Computation



COT 6602 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

o The Intuitive Concept of Information

o The Shannon and Von Neumann Entropy

o Properties of Entropy and Entropy Inequalities

o Distinguishing Quantum States and the Accessible Information

o Distance Measures for Quantum Information

o Entanglement as a Physical Resource

• Measurements of Quantum Systems

o Born’s Rule

o Measurement Operators

o von Neumann-type Projective Measurements

o Positive Operator Valued Measurements

o Newmark’s Theorem

o Pure and Mixed States

o Bipartite Systems; Schmidt Decomposition; Measurements of Bipartite Systems

o Purification of Mixed States

o Measurements of Quantum Circuits

o EPR

o Bell’s and CHSH Inequalities

• Applications

o Quantum Teleportation

o Superdense Coding

• Noiseless Quantum Shannon Theory

o Classical and Quantum Data Compression

o Quantum-Classical Trade-Off Coding

o Remote State Preparation

o Generalized Remote State Preparation

• Noisy Quantum Shannon Theory

o Shannon's Noisy Channel Coding Theorem

o Classical Information Transmission over Noisy Quantum Channels

o Entanglement Assisted Quantum Communication (The Mother Protocol)

▪ Quantum Information Transmission over Noisy Quantum Channels

▪ Entanglement Assisted Classical Information Transmission over Noisy Quantum Channels

o Entanglement Distillation Assisted by Quantum Communication (The Father Protocol)

▪ Entanglement Distillation Assisted by Classical Communication

▪ Noisy Teleportation

▪ Noisy Superdense Coding

o 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

o Block codes

o Hamming distance

o Linear Codes

o Bounds (Hamming, Singleton, Gilbert-Varsharmov, Plotkin, BCH)

• Quantum Error Correction

o A Necessary Condition for the Existence of a Quantum Code

o Quantum Hamming Bound

o Repetitive Codes for a Single Bit-Flip/Phase-Flip Errors

o Shor, Steane, and Calderbank-Shor-Steane (CSS), Codes

o Stabilizer Codes

o Perfect Quantum Codes

• Quantum Fault-Tolerance

o 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,” Cambridge University 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 Through a 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 Unknown State 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, quant-ph/9803072 v1, November 2000.

• D.Gottesman, “Stabilizer Codes and Quantum Error Correction”, Ph.D. Thesis, California Institute of Technology}, Preprint, quant-ph/9705052 v1, May 1997.

• D. Gottesman, “An Introduction to Quantum Error Correction,” Proceedings Symposium in Applied Mathematics, Preprint, 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, 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. Oxford University Press, 1997. Also: Preprint, quant-ph/9707034 v1, 1997.

• R. Jozsa, “Illustrating the Concept of Quantum Information,” Preprint quant-ph/0305114 v1, 2003.

• M. Keyl, “Fundamentals of Quantum Information,” Reprint 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, 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, quant-ph/0304102 v1, April 2003.

• A.M.Steane, “Multiple Particle Interference and Quantum Error Correction,” Preprint, 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, quant-ph/0307120 v1, July 2003.

• V.Vedral, “The Role of Entropy in Quantum Information Theory,” Preprint, 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%

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