COT 6602 Quantum Information Theory and Quantum
Error Correcting Codes
Credit: 3 units
Offered: Spring semester
Instructors: Dan Marinescu and Pawel Wocjan
Class outline
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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
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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:
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D. C. Marinescu and G. M. Marinescu, “Approaching Quantum Information Theory
and Error Correction,” http://www.eecs.ucf.edu/~dcm/QCV2.pdf
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
http://arxiv.org/abs/quant-ph/ maintained by Los Alamos National Laboratory.
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H. Barnum, M. A. Nielsen, and B. Schumacher, “Information Transmission Through
a Noisy Quantum Channel,” Physical Review A, 57(6):4153--4175, 1998.
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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, 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/quantph/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, arxiv.org/quantph/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/quantph/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/quantph/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.
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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, http://www.cs.uwaterloo.ca/~watrous/lecture-notes.html, 2007.
Grading policy:
Homework 35%
Midterm
25%
Final exam 40%