Ma 148 a: Geometry and Physics of Information
Fall 2014, Caltech Math Department, Monday-Wednesday-Friday,
1:00 - 1:55 pm (room to be announced).
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.
- 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 material that
has/will be covered in class:
- Matilde Marcolli, Ryan Thorngren, "Thermodynamic semirings",
arXiv:1108.2874
- 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
- 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.
- Mikhail Kapranov, "Thermodynamics and the moment map", arXiv:1108.3472.
- O.Viro, "Dequantization of real algebraic geometry on logarithmic paper",
arXiv:math/0005163
- I. Itenberg and G. Mikhalkin, "Geometry in tropical limit",
ArXiv: 1108.3111v2
- R. Frank and E.H. Lieb, "Monotonicity of a relative Renyi entropy",
arXiv:1306.5358
- 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
- A. R. Calderbank, E.M.Rains, P.W.Shor, N.J.A.Sloane,
"Quantum Error Correction via Codes over GF(4)", arXiv:quant-ph/9608006
- S.Trebst, M.Troyer, Z.Wang, A.Ludwig, "A short introduction
to Fibonacci anyon models, arXiv:0902.3275
- Matilde Marcolli, John Napp, "Quantum computation and
real multiplication", arXiv:1312.3590
- 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
Assigned presentations:
- Juan Adame: Geometry in tropical limit
- Sadaf Amouzegar: A characterization of entropy in terms of information
loss
- Gabriel Foster: Quantum computation and real multiplication
- Dmitri Gekhtman: Physics, Topology, Logic and Computation
- Jim Tao: Zipf's law and L. Levin's probability distributions
- Alex Vaschillo: Topological quantum computation
- Alex Port: Monotonicity of a relative Renyi entropy
- Nick Zolman: Fibonacci Anyon Models
- Marcus Bintz: Non-Abelian Anyons and Topological Quantum Computation
- Madeleine Youngs: A computability challenge, asymptotic bound
and isolated error-correcting codes
Schedule of presentations:
- Friday Dec 5: Marcus Bintz, Gabriel Foster
- Monday Dec 8: Alex Port, Jim Tao, Madeleine Youngs
- Wednesday Dec 10: Alex Vaschillo, Juan Adame, Dmitri Gekhtman
- Friday Dec 12: Sadaf Amouzegar, Nick Zolman
Notes of classes
Notes will be posted here.
- Monday September 29: convex spaces and probability measures
- Wednesday October 1: Perron-Frobenius theory, stochastic matrices
- Friday October 3: Shannon entropy, Kullback-Leibler relative entropy,
Fisher-Rao metric, Renyi and Tsallis entropies
- Monday October 6: Khinchin axioms and Shannon entropy
- Wednesday October 8: thermodynamic semirings and algebraic
characterization of Shannon entropy
- Friday October 10: thermodynamic semirings for other entropy
functionals (Renyi, Tsallis)
- Monday October 13: thermodynamic semirings with relative
entropy; successor function; entropy operads
- Wednesday October 15: Quantum versus classical probability;
a dictionary of terms: state space, events, observables, probability
distributions, expectation values, variance, extreme points, independent
systems, dynamics (Heisenberg and Schroedinger versions)
- Friday October 17: Pure states and complex projective geometry,
Fubini-Study metric and information geometry
- Monday October 20: Geometry of the convex set of density matrices,
U(N)-orbits, flag manifolds
- Wednesday October 22: entropy in quantum information, convexity,
von Neumann entropy and generalizations
- Friday October 24: quantum relative entropy; measurement postulate
- Monday October 27: logical gates: CNOT, swap, phase; EPR quantum
teleportation, Weyl operators, quantum Fourier transform
- Wednesday October 29: universal quantum gates: density result
- Friday October 31: quantum error-correcting codes and Weyl operators
- Monday November 3: t-error correcting quantum codes and their code
parameters
- Wednesday November 5: Classical codes and code parameters: transmission
rate and relative minimum distance
- Friday November 7: Existence of the asymptotic bound for
error correcting codes; code points with finite and with infinite
multiplicity
- Monday November 10: Singleton bound, Hamming bound,
Gilbert-Varshamov bound
- Wednesday November 12: The Shannon Random Code Ensemble and quantum statistical mechanics of decoding: the Gilbert-Varshamov curve as phase transition
- Friday November 14: Kolomogorov complexity and the asymptotic bound
for error correcting codes
- Monday November 17: Quantum statistical mechanical systems
associated to codes and code points
- Wednesday November 19: classical codes and quantum error correcting codes
- Friday November 21: CSS algorithm relating self-orthogonal classical linear codes and q-ary quantum stabilizer codes
- Monday November 24: group rings, group algebras, twist by multipliers,
noncommutative tori, rational noncommutative tori and matrix algebras,
almost commutative geometries
- Wednesday November 26: CSS algorithm relating classical and quantum
codes reinterpreted in terms of the geometry of rational noncommutative tori
- Monday December 1: Renormalization in perturbative quantum field theory
and in the theory of computation, Manin's renormalization and computation
program
- Wednesday December 3: Algebraic renormalization via Hopf algebras and
Rota-Baxter algebras
- Friday December 5: Manin's Hopf algebra of flow charts and
an approach to renormalization of the halting problem
Grading policy
Grading will depend on participation in
class and on an oral presentation based on assigned reading material.