Ma 148 c: Geometry and Physics of Information
Spring 2017, 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.
Notes of classes
Notes will be posted here.
- Tuesday April 4: Preliminaries on Convexity
- Thursday April 6: Shannon Entropy and Khinchin Axioms, Kullback-Leibler
divergence and Fisher-Rao metric
- Tuesday April 11: Thermodynamic semirings, Khinchin axioms
and algebraic properties
- Thursday April 13: Thermodynamic semirings for Renyi and Tsallis
entropy and Kullback-Leibler divergence, entropy operads
- Tuesday April 18: Mutual information and inclusion-exclusion,
Hochschild complex, mutual information as Hochschild cocycles,
finite field polylogarithms and Tsallis entropies
- Thursday April 20: Quantum versus classical probability: events, observables, probability distributions, expectation values, composite systems, dynamics
- Tuesday April 18: finite dimensional quantum mechanics as complex
projective geometry, pure states and projective spaces, tensor products and Segre embeddings
- Thursday April 20: pure states and Bloch vectors, complex projective
spaces as symmetric products, Fubini-Study metric and quantum information,
Wigner's theorem
- Tuesday April 25: geometric structure of the space of density matrices
(U(N) orbits, flag manifolds and simplexes), entropy in quantum information,
von Neumann entropy
- Thursday April 27: relative entropy in quantum probability, completely positive maps and quantum channels, measurement postulate, entropy and channel capacity
- Tuesday May 2: thermodynamic semirings, tropical and entropical
trace and von Neumann entropy, thermodynamic deformations of states on C* algebras, entropies of dynamical systems, quantum relative entropy and Fisher metric
- Thursday May 4: Classical codes, code parameters, rate and minimum distance, spoiling operations, existence of the asymptotic bound, singleton bound
- Tuesday May 9:
- Thursday May 11:
- Tuesday May 16:
- Thursday May 18:
- Tuesday May 23:
- Thursday May 25:
- Tuesday May 30:
- Thursday June 1: student presentations
Slides
Slides of classes will be added here:
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
- Philippe Elbaz-Vincent, Herbert Gangl, "Finite
polylogarithms, their multiple analogues, and the
Shannon entropy", pdf
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
- P.Jizba, "Information theory and generalized statistics"
pdf
- Pierre Baudot, Daniel Bennequin, "The Homological Nature of Entropy"
pdf
Syllabus
Requirements for the class: one oral
presentation on reading materials chosen from the list
provided above (first come first serve) and attendance
of (most) lectures.
Assigned student presentations:
The schedule of assigned persentations will be added here