sharpening occam’s razor with quantum mechanics sissa journal club matteo marcuzzi 8th april, 2011
DESCRIPTION
Johannes KeplerTRANSCRIPT
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Sharpening Occam’s razor with Quantum
MechanicsSISSA Journal Club
Matteo Marcuzzi 8th April, 2011
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Niclas Koppernigck(Copernicus)
Clausius Ptolemaeus(Ptolemy)
Tyge Brahe(Tychonis)
Describing Systems
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Describing Systems
Johannes Kepler
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Describing Systems
.3
2
cRT
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Describing Systems
Algorithmic Abstraction
212
21
rmmGF
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Describing Systems
Algorithmic AbstractionSame output
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Describing Systems
Same outputDifferent intrinsic information!
Solar system
celestial objects
Sun FlaresPlanet Orography
MeteorologyPeople behaviour
Compton Scattering
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Describing Systems
Same outputDifferent intrinsic information!
Much more memory
required!OCCAM’S RAZOR
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Describing SystemsN Spin Chain
Up parity
1 spin-flip per second10 if even
if odd
0
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Describing SystemsN Spin Chain
Up parity10 if even
if odd
0
1 spin-flip per second
1
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Describing SystemsN Spin Chain
Up parity10 if even
if odd
0
1 spin-flip per second
1 0
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Describing SystemsN Spin Chain
Up parity10 if even
if odd
0
1 spin-flip per second
1 0 1
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Describing SystemsN Spin Chain
Up parity10 if even
if odd
0
1 spin-flip per second
1 0 1 0
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Describing SystemsN Spin Chain
Up parity10 if even
if odd
0
1 spin-flip per second
1 0 1 0 1
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Describing SystemsN Spin Chain
Up parity10 if even
if odd
0
1 spin-flip per second
1 0 1 0 1 0
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Describing SystemsN Spin Chain
Up parity10 if even
if odd
0
1 spin-flip per second
1 0 1 0 1 0 1
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Describing SystemsN Spin Chain
Up parity10 if even
if odd
0
1 spin-flip per second
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
N bits needed
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Describing SystemsHidden System
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
read x
return (x+1) mod 21-bit only!
Statistically equivalent output
N bits
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Computational Mechanics
• Statistical equivalence
•Measure of complexity
• Pattern identification
0m
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Computational Mechanics
0m 0stm
• Statistical equivalence
•Measure of complexity
• Pattern identification
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Computational Mechanics
0m 0stm 04 m
• Statistical equivalence
•Measure of complexity
• Pattern identification
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Computational Mechanics
0m 0stm 04 m ?
• Statistical equivalence
•Measure of complexity
• Pattern identification
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Computational Mechanics
�
21012 SSSSSS Stochastic Process
DiscreteStationary
iSRandom Variables A: Alphabet
edcbaA ,,,,
1,0'A
GCTAA ,,,''
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Computational Mechanics
�
21012 SSSSSS Stochastic Process
DiscreteStationary
iSRandom Variables A: Alphabet
123 SSSS �
Pasts
210 SSSS Futures
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Computational Mechanics
�
21012 SSSSSS Stochastic Process
DiscreteStationary
iSRandom Variables A: Alphabet
123 ssssA ��
Set of histories
210 ssssA Set of future strings
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Computational Mechanics
�
21012 SSSSSS Stochastic Process
DiscreteStationary
Machine
000101000101110101101…
Statistical Equivalence
)()( sSsS MS
� PP
1,0A
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Computational Mechanics
�
21012 SSSSSS Stochastic Process
DiscreteStationary
Machine
A�
1,0A…010100010
…1100111
…01010101
States
Partition R
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Computational Mechanics
�
21012 SSSSSS Stochastic Process
DiscreteStationary
Machine
A�
States
Partition R
1R
2R
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ijS RsRas ���P
a
Computational Mechanics
�
21012 SSSSSS Stochastic Process
DiscreteStationary
Machine
A�
1R
2R ijM
aij RRaT ,)( P Transition Rates
)(11
aT
)(12
aT
Rj
Ri
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Computational Mechanics
�
21012 SSSSSS Stochastic Process
DiscreteStationary
OCCAM POOL
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Computational MechanicsA little information theory
spspSHAs
log Shannon entropy
XSH Conditional entropy YXSH ,
XSHSHXSI : Mutual information
SSIE�
: Excess entropy
SH
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Computational Mechanics
Machine Cannot distinguish between them
SSHSH�
R
Partition RA�
We want to preserve information
SSHSH�
R
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Computational Mechanics
Machine
SSHSH�
R
Partition RA�
We want to preserve information
SSHSH�
R
with the least possible memory
0C Log(# states)minimize
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Computational Mechanics
Machine
SSHSH�
R
Partition RA�
We want to preserve information
SSHSH�
R
with the least possible memory
minimize RHC Statisticalcomplexity
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Computational Mechanics
SSHSH�
R
We want to preserve information
SSHSH�
R
with the least possible memory
minimize RHC Statisticalcomplexity
OCCAM POOL Optimal partition
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Computational Mechanics
Optimal partition
We want to preserve informationwith the least possible memory
minimize RHC Statisticalcomplexity
SSHSH�
R
)'()( ssss SS
�� �� PP
ε-machine
ε
'~ ss ��if
Causal States
(unique)
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Computational Mechanics: Examples
2-periodic sequence
2-periodic, ends with
2-periodic, ends with
1p1p
A
B
I
initial state
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Computational Mechanics: Examples
2-periodic sequence
1p1p
A
B
I
initial state21p
21precurrent transient
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Computational Mechanics: Examples1D Ising model
p p
p
p
ijV transfer matrix
jij
i Vuujip
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Computational Mechanics: Examples1D Next-nearest-neighbours Ising
p
p
p
p
p
p p
p
31 J
12 J
2.0T
2
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Computational Mechanics: Examples1D Next-nearest-neighbours Ising
p
p
p
p
p
p p
p
31 J
12 J
2.0T
23
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Computational Mechanics: Examples1D Next-nearest-neighbours Ising
p
p
p
p
p
p p
p
31 J
12 J
2.0T
23
1
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Computational Mechanics: Examples1D Next-nearest-neighbours Ising
p
p
p
p
p
p p
p
31 J
12 J
2.0T
23
1
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Computational Mechanics: Examples1D Next-nearest-neighbours Ising
p
p
p
p
p
p p
p
31 J
12 J
2.0T
negligible
1
1
8B
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Computational Mechanics: Examples1D Next-nearest-neighbours Ising
p
p
31 J
12 J
2.0T
1
1
period 3
period 18B
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Sharpening the razor with QM
EC Statistical complexit
y
Excess entropy SSI
�: RH
EC Ideal system
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Sharpening the razor with QM
,,AA�
ε
ε-machines are deterministic
ε
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,,AA�
ε
Sharpening the razor with QM
1R
3R2R
4R
0,, )(44
)(34
)(24 blueblueblue TTT EC
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,,AA�
ε
Sharpening the razor with QM
EC 0)( cijT
fixed i,c unique j
fixed j,c unique i
ideal
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Sharpening the razor with QM
ε qεcausal state Ri system state i
symbol “s” symbol state s
siTSis
skik
,
)()(sijT
q-machine states
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qεsystem state i
symbol state s
siTSis
skik
,
)(
q-machine states
Sharpening the razor with QM
CLASSICAL
QUANTUM
Prepare kS
Measure C.S.j t
2kSjt )(t
kjTProbability
tjS
)( iiq RS PP ip
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qεsystem state i
symbol state s
siTSis
skik
,
)(
q-machine states
Sharpening the razor with QM
CLASSICAL
QUANTUM )( iiq RS PP ip
ii
i ppC log
logtrCq
iii
i SSp
qCC
E
E
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qεsystem state i
symbol state s
siTSis
skik
,
)(
q-machine states
Sharpening the razor with QM
CLASSICAL
QUANTUM
ii
i ppC log
logtrCq
iii
i SSp
qCC
ijji SS
E
E
ks
sjk
sik TT
,
)()(Ideal system
E
E
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qεsystem state i
symbol state s
siTSis
skik
,
)(
q-machine states
Sharpening the razor with QM
CLASSICAL
QUANTUM
qCC
ijji SS
Non-ideal systems
Quantum mechanics improves efficiency
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Sharpening the razor with QM
single spinp
21
p
p
p
p1 p1 21
p
?21
21
C
E
qC
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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)