1. Sharpening Occam’s razor with Quantum Mechanics SISSA Journal Club Matteo Marcuzzi 8th April, 2011

2. Niclas Koppernigck (Copernicus) Clausius Ptolemaeus (Ptolemy) Tyge Brahe (Tychonis) Describing Systems

3. Johannes Kepler

4. Describing Systems

5. Algorithmic Abstraction

6. Describing Systems Algorithmic Abstraction Same output

7. Describing Systems Same output Different intrinsic information! Solar system celestial objects Sun Flares Planet Orography Meteorology People behaviour Compton Scattering

8. Describing Systems Same output Different intrinsic information! Much more memory required! OCCAM’S RAZOR

9. Describing Systems N Spin Chain Up parity 1 spin-flip per second if even if odd 0

10. Describing Systems N Spin Chain Up parity if even if odd 0 1 spin-flip per second 1

11. Describing Systems N Spin Chain Up parity if even if odd 0 1 spin-flip per second 10

12. Describing Systems N Spin Chain Up parity if even if odd 0 1 spin-flip per second 101

13. Describing Systems N Spin Chain Up parity if even if odd 0 1 spin-flip per second 1010

14. Describing Systems N Spin Chain Up parity if even if odd 0 1 spin-flip per second 10101

15. Describing Systems N Spin Chain Up parity if even if odd 0 1 spin-flip per second 101010

16. Describing Systems N Spin Chain Up parity if even if odd 0 1 spin-flip per second 1010101

17. Describing Systems N Spin Chain Up parity if even if odd 0 1 spin-flip per second 10101010101010101 N bits needed

18. Describing Systems Hidden System 010101010101010101 read x return (x+1) mod 2 1-bit only! Statistically equivalent output N bits

19. Computational Mechanics Statistical equivalence Measure of complexity Pattern identification

20. Computational Mechanics Statistical equivalence Measure of complexity Pattern identification

21. Computational Mechanics Statistical equivalence Measure of complexity Pattern identification

22. Computational Mechanics ? Statistical equivalence Measure of complexity Pattern identification

23. Computational Mechanics Stochastic Process Discrete Stationary Random Variables Alphabet

24. Computational Mechanics Stochastic Process Discrete Stationary Random Variables Alphabet Pasts Futures

25. Computational Mechanics Stochastic Process Discrete Stationary Random Variables Alphabet Set of histories Set of future strings

26. Computational Mechanics Stochastic Process Discrete Stationary Machine 000101000101110101101… Statistical Equivalence

27. Computational Mechanics Stochastic Process Discrete Stationary Machine …010100010 …1100111 …01010101 States Partition R

28. Computational Mechanics Stochastic Process Discrete Stationary Machine States Partition R

29. a Computational Mechanics Stochastic Process Discrete Stationary Machine Transition Rates

30. Computational Mechanics Stochastic Process Discrete Stationary OCCAM POOL

31. Computational Mechanics A little information theory Shannon entropy Conditional entropy Mutual information Excess entropy

32. Computational Mechanics MachineCannot distinguish between them Partition R We want to preserve information

33. Computational Mechanics Machine Partition R We want to preserve information with the least possible memory Log(# states)minimize

34. Computational Mechanics Machine Partition R We want to preserve information with the least possible memory minimize Statistical complexity

35. Computational Mechanics We want to preserve information with the least possible memory minimize Statistical complexity OCCAM POOLOptimal partition

36. Computational Mechanics Optimal partition We want to preserve information with the least possible memory minimize Statistical complexity ε-machine ε if Causal States (unique)

37. Computational Mechanics: Examples 2-periodic sequence 2-periodic, ends with A B I initial state

38. Computational Mechanics: Examples 2-periodic sequence A B I initial state recurrent transient

39. Computational Mechanics: Examples 1 D Ising model transfer matrix

40. Computational Mechanics: Examples 1 D Next-nearest-neighbours Ising 2

41. Computational Mechanics: Examples 1 D Next-nearest-neighbours Ising 2 3

42. Computational Mechanics: Examples 1 D Next-nearest-neighbours Ising 2 3 1

43. Computational Mechanics: Examples 1 D Next-nearest-neighbours Ising 2 3 1

44. Computational Mechanics: Examples 1 D Next-nearest-neighbours Ising negligible

45. Computational Mechanics: Examples 1 D Next-nearest-neighbours Ising period 3 period 1

46. Sharpening the razor with QM Statistical complexity Excess entropy Ideal system

47. Sharpening the razor with QM ε ε-machines are deterministic ε

48. ε Sharpening the razor with QM

49. ε fixed i,cunique j fixed j,cunique i ideal

50. Sharpening the razor with QM ε qεqε causal state Risystem state symbol “s”symbol state q-machine states

51. qεqε system state symbol state q-machine states Sharpening the razor with QM CLASSICAL QUANTUM Prepare Measure C.S. Probability t

52. qεqε system state symbol state q-machine states Sharpening the razor with QM CLASSICAL QUANTUM

53. qεqε system state symbol state q-machine states Sharpening the razor with QM CLASSICAL QUANTUM Ideal system

54. qεqε system state symbol state q-machine states Sharpening the razor with QM CLASSICAL QUANTUM Non-ideal systems Quantum mechanics improves efficiency

55. Sharpening the razor with QM single spin ?

56. References M. Gu, K. Wiesner, E. Rieper & V. Vedral - "Sharpening Occam's razor with Quantum Mechanics" - arXiv: quant-ph/1102.1994v2 (2011) C. R. Shalizi & J. P. Crutchfield - "Computational Mechanics: Pattern and Prediction, Structure and Simplicity" - arXiv: cond-mat/990717v2 (2008) D. P. Feldman & J. P. Crutchfield - "Discovering Noncritical Organization: Statistical Mechanical, Information Theoretic, and Computational Views of Patterns in One-Dimensional Spin Systems" - Santa Fe Institute Working Paper 98-04-026 (1998)