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

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

Johannes Kepler

Describing Systems

Algorithmic Abstraction

Describing Systems Algorithmic Abstraction Same output

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

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

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

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

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

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

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

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

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

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

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

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

Computational Mechanics Statistical equivalence Measure of complexity Pattern identification

Computational Mechanics Statistical equivalence Measure of complexity Pattern identification

Computational Mechanics Statistical equivalence Measure of complexity Pattern identification

Computational Mechanics ? Statistical equivalence Measure of complexity Pattern identification

Computational Mechanics Stochastic Process Discrete Stationary Random Variables Alphabet

Computational Mechanics Stochastic Process Discrete Stationary Random Variables Alphabet Pasts Futures

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

Computational Mechanics Stochastic Process Discrete Stationary Machine … Statistical Equivalence

Computational Mechanics Stochastic Process Discrete Stationary Machine … … … States Partition R

Computational Mechanics Stochastic Process Discrete Stationary Machine States Partition R

a Computational Mechanics Stochastic Process Discrete Stationary Machine Transition Rates

Computational Mechanics Stochastic Process Discrete Stationary OCCAM POOL

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

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

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

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

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

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

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

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

Computational Mechanics: Examples 1 D Ising model transfer matrix

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

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

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

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

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

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

Sharpening the razor with QM Statistical complexity Excess entropy Ideal system

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

ε Sharpening the razor with QM

ε fixed i,cunique j fixed j,cunique i ideal

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

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

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

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

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

Sharpening the razor with QM single spin ?

References M. Gu, K. Wiesner, E. Rieper & V. Vedral - "Sharpening Occam's razor with Quantum Mechanics" - arXiv: quant-ph/ v2 (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 (1998)