Ma 148 c: Geometry and Physics of Information
Spring 2016, Caltech Math Department, Tuesday-Thursday 10:30-11:55am SLN159,
Instructor:
Matilde Marcolli
Brief Course Description
This class will present various ways in which geometry and physics
play a role in information theory. Topics will include information geometry;
algebraic structures of entropy; geometric aspects of quantum information;
methods of statistical physics in coding theory; quantum field theory methods
in the theory of computation.
Suggested readings
- M.Mezard, A.Montanari, "Information, physics and computation", Oxford
University Press, 2009.
- I.Bengtsson, K.Zyczkowski, "Geometry of quantum states", Cambridge
University Press, 2006.
- K.R. Parthasarathy, "Quantum Computation, quantum error
correcting codes and information theory", Narosa, 2006.
- Gilles Pisier, K. R. Parthasarathy, Vern Paulsen and Andreas Winter,
"The Functional Analysis of Quantum Information Theory", Springer 2015.
- K. Fujii, "Quantum computation with topological codes" Springer, 2016.
- S.Amari, H.Nagaoka, "Methods of Information Geometry" AMS and Oxford, 2000
Additional suggested readings
Additional reading material will be added as the class
progresses. Papers listed here refer to some of the
material that will be covered in class:
- Matilde Marcolli, Ryan Thorngren, "Thermodynamic semirings",
arXiv:1108.2874
- Matilde Marcolli, Nicolas Tedeschi, "Entropy algebras and
Birkhoff factorization", arXiv:1412.0247
- Yuri Manin, Matilde Marcolli, "Error-correcting codes
and phase transitions", arXiv:0910.5135
- Yuri Manin, Matilde Marcolli, "Kolmogorov complexity
and the asymptotic bound for error-correcting codes", arXiv:1203.0653
- Matilde Marcolli, Christopher Perez, "Codes as fractals
and noncommutative spaces", arXiv:1107.5782
- Matilde Marcolli, John Napp, "Quantum computation and
real multiplication", arXiv:1312.3590
- Yuri Manin, "Renormalization and computation I:
motivation and background", arXiv:0904.4921,
- Yuri Manin, "Renormalization and Computation II:
Time Cut-off and the Halting Problem", arXiv:0908.3430.
- Colleen Delaney, Matilde Marcolli, "Dyson-Schwinger
equations in the theory of computation", arXiv:1302.5040
Suggested papers for student presentations:
- John C. Baez, Mike Stay, "Physics, Topology, Logic and
Computation: A Rosetta Stone", arXiv:0903.0340
- John C. Baez, Tobias Fritz, Tom Leinster, "A characterization of
entropy in terms of information loss", arXiv:1106.1791
- Yuri Manin, "A computability challenge: asymptotic
bounds and isolated error-correcting codes",
arXiv:1107.4246
- Yuri Manin, "Complexity vs Energy: Theory of Computation and
Theoretical Physics", arXiv:1302.6695
- Yuri Manin, "Zipf's law and L. Levin's probability distributions",
arXiv:1301.0427
- Yuri Manin, "Classical computing, quantum computing,
and Shor's factoring algorithm", arXiv:quant-ph/9903008
- S.Trebst, M.Troyer, Z.Wang, A.Ludwig, "A short introduction
to Fibonacci anyon models, arXiv:0902.3275
- Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, Zhenghan Wang,
"Topological quantum computation", arXiv:quant-ph/0101025
- C.Nayak, S.Simon, A.Stern, M.Freedman, S.Sarma,
"Non-Abelian Anyons and Topological Quantum Computation", arXiv:0707.1889
- M.Redei, S.J.Summers, "Quantum Probability Theory", arXiv:quant-ph/0601158
Notes of classes
Notes will be posted here.
- Tuesday March 29: convex spaces and probability measures,
Perron-Frobenius theory, stochastic matrices
- Thursday March 31: Shannon entropy, Kullback-Leibler relative entropy,
Fisher-Rao metric, Renyi and Tsallis entropies, Khinchin axioms
- Tuesday April 5: Thermodynamic semirings and algebraic
characterization of Shannon entropy and other entropy functionals
- Thursday April 7: Quantum versus classical probability, Pure states
and complex projective geometry, Geometry of the convex set of density
matrices in terms of flag manifolds
- Tuesday April 12: entropy in quantum information, von Neumann
entropy and generalizations, thermodynamic deformations of the tropical trace
- Thursday April 14: Classical codes and code parameters: transmission
rate and relative minimum distance, spoiling operations
- Tuesday April 19: Existence of the asymptotic bound for error
correcting codes; code points with finite and with infinite multiplicity
- Thursday April 21: The Shannon Random Code Ensemble and quantum
statistical mechanics of decoding: the Gilbert-Varshamov curve as
phase transition
- Tuesday April 26: Kolomogorov complexity and the asymptotic bound
for error correcting codes
- Thursday April 28: Quantum statistical mechanical systems
associated to codes and code points
- Tuesday May 3: Quantum error correcting codes
- Thursday May 5: Quantum error correcting codes
- Tuesday May 10: CSS algorithm relating self-orthogonal classical
linear codes and q-ary quantum stabilizer codes
- Thursday May 12: CSS algorithm and the geometry of
rational noncommutative tori
- Tuesday May 17: Manin's renormalization and computation
- Thursday May 19: Renormalization and computation
- Tuesday May 24: CLASS CANCELLED (conference travel)
- Thursday May 26: Student presentations
- Tuesday May 31: Student presentations
- Thursday June 2: Student presentations
The class times of Thursday May 26, Tuesday May 31 and
Thursday June 2 will be used for student prasentations.
Some files of notes
Syllabus
The course is offered Pass/Fail. Requirements: one oral
presentation on reading materials chosen from the list
provided above (first come first serve) and attendance
of (most) lectures.
Assigned student presentations:
Assigned persentations and presentation schedule:
- Thursday May 26: Eric Morgan, "Fibonacci anyon model"
- Thursday May 26: Milan Cvitkovic, "Quantum probability theory"
- Tuesday May 31: Aaron Chew, "Non-Abelian Anyons and
Topological Quantum Computation"
- Tuesday May 31: Emory Brown, "Entropy in terms of information loss"
- Thursday June 2: Charles Xu, "Physics, Topology, Logic and
Computation: A Rosetta Stone"