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Lock Yue Chew, PhDDivision of Physics and Applied Physics
School of Physical & Mathematical Sciences
&
Complexity Institute
Nanyang Technological University
Quantum versus Classical Measures of
Complexity on Classical Information
Astronomical
Scale
Mesoscopic
Scale
Macroscopic
Scale
What is Complexity?
• Complexity is related to “Pattern” and “Organization”.
Nature inherently organizes;
Pattern is the fabric of Life.
James P. Crutchfield
• There are two extreme forms of Pattern: generated by a
Clock and a Coin Flip.
• The former encapsulates the notion of Determinism, while the
latter Randomness.
• Complexity is said to lie between these extremes.
J. P. Crutchfield and K. Young, Physical Review Letters 63, 105 (1989).
Can Complexity be Measured?
• To measure Complexity means to measure a system’s
structural organization. How can that be done?
• Conventional Measures:
• Difficulty in Description (in bits) – Entropy; Kolmogorov-Chaitin
Complexity; Fractal Dimension.
• Difficulty in Creation (in time, energy etc.) – Computational Complexity;
Logical Depth; Thermodynamic Depth.
• Degree of organization – Mutual Information; Topological ϵ-
machine; Sophistication.
S. Lloyd, “Measures of Complexity: a Non-Exhaustive List”.
Topological Ɛ-Machine
1. Optimal Predictor of the System’s Process
2. Minimal Representation – Ockham’s Razor
3. It is Unique.
4. It gives rise to a new measure of Complexity known as
Statistical Complexity to account for the degree of
organization.
5. The Statistical Complexity has an essential kind of
Representational Independence.
J. P. Crutchfield, Nature 8, 17 (2012).
Concept of Ɛ-Machine
• Within each partition, we have
• Equation (1) is related to sub-
tree similarity discussed later.
J. P. Crutchfield and C. R. Shalizi, Physical Review E 59, 275 (1999).
Symbolic Sequences: ….. S-2S-1S0S1 …..
)1('|| sSPsSP
Futures :
21 tttt SSSS
Past :123 tttt SSSS
Partitioning the set 𝑺 of all histories into
causal states 𝑆𝑖.
where 𝑠and 𝑠′ are two different
individual histories in the same
partition.
• In physical processes, measurements are taken as time
evolution of the system.
• The time series is converted into a sequence of symbols s
= s1, s2, . . . , si, . . . . at time interval τ.
ε
• Measurement phase space M is segmented or partitioned
into cells of size ε.
• Each cell can be assigned a label leading to a list of
alphabets A = {0, 1, 2, … k -1}.
• Then, each si ϵ A.
Ɛ Machine - Preliminary
Ɛ-Machine – ReconstructionS = 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, …..
0
1
0
0
0
1
0
0
1
0
0
0
0
1
1
1
0
0
1
1
0
0
0
1
0
0
• From symbol sequence -> build a tree
structure with nodes and links .
• The links are given symbols according to
the symbol sequence with A = {0, 1}.
Period-3 sequence
0
1
0
0
0
1
0
0
1
0
0
0
0
1
1
1
0
0
1
1
0
0
0
1
0
0
Ɛ-Machine – Reconstruction
1
2 3
4 3 5
3 5 4
Level 1
Level 2
Level 3
5
5
5
5
5
4
4
4
4
43
3
3
3
3
1
2
10
3
3 50
450
2
34
0 1
341
S = 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, …..
Ɛ-Machine – Reconstruction
1
3
54
0 ( p = 2/3) 1 ( p = 1/3)
1 ( p = 1/2)
0 ( p = 1/2)1 ( p = 1)
0 ( p = 1)
0 ( p = 1)
Probabilistic
Structure
Based on Orbit
Ensemble
Combination of both deterministic and random
computational resources
But the Probabilistic Structure
is transient.
2
Ɛ-Machine – Simplest Cases
All Head or All “1” Process
s = 1, 1, 1, 1, 1, 1, …..
Purely Random Process
Fair Coin Process
• These are the extreme cases of Complexity.
• They are structurally simple.
• We expect their Complexity Measure to be zero.
1 ( p = 1)
1 ( p = 1/2)
0 ( p = 1/2)
Ɛ-Machine:
Complexity from Deterministic Dynamics
𝑇𝑛3 3-level sub-tree of n contains all nodes
that can be reached within 3 links
• The Symbol Sequence is derived from the Logistic Map.
• The parameter r is set to the band merging regime r =
3.67859 …
• For a finite sequence, a tree length Tlen and a machine length
Mlen need to be defined.
The Logistic Map
Logistic equation : 𝑥𝑛+1 = 𝑟𝑥𝑛 1 − 𝑥𝑛
Starts period-doubling; at 𝑟 = 3 : period-2
Chaotic at 𝑟 = 3.569946…
Partitioning : 𝑠𝑖 = 0, 𝑥𝑖 < 0.51, 𝑥𝑖 ≥ 0.5
Construction of ε-machine:
Transitions
Sub-tree
Similarity
Sub-tree
Similarity
1
12
12
2 2
3
123
Level 1
Level 2
Level 3
Ɛ-Machine:
Complexity from Deterministic Dynamics
Transitions
Ɛ-Machine:
Complexity from Deterministic Dynamics
Transient State
s = 1,0,1,1,1,0,1,…
• The ε-machine captures the patterns of the process.
Ɛ-Machine:
Complexity from Deterministic Dynamics
0 1
0
1 0
1P(0|*0)
= 1654/3309
≈ 0.5 P(1|*0)
=1655/3309
≈ 0.5
P(0|*1)
= 1655/6687
≈ 0.25
P(1|*1)
= 5032/6687
≈ 0.75
P(0|*00)
≈ 0.51
0
P(1|*00)
≈ 0.49
1
P(0|*01)
= 0
0
P(1|*01)
= 11
P(0|*10)
≈ 0.49
P(1|*10)
≈ 0.51
0 1
P(0|*11)
≈ 0.33
P(1|*11)
≈ 0.670
1
C
P(0|*000)
≈ 0.49
0
P(1|*000)
≈ 0.51
1
P(0|*001)
≈ 0
0
P(1|*001)
≈ 1
P(0|*010)
≈ 0
01 1
P(1|*010)
≈ 0
P(0|*011)
≈ 0.49
P(1|*011)
≈ 0.51
0 1
P(0|*100)
≈ 0.51
P(1|*100)
≈ 0.49
0 1P(0|*101)
≈ 0
P(1|*101)
≈ 1
P(0|*110)
≈ 0.49
P(1|*110)
≈ 0.51
P(1|*111)
≈ 0.75
P(0|*111)
≈ 0.25
0 0 01 1 1
A
B
Total = 9996 P(0|*)
= 3309/9996
≈ 0.33
P(1|*)
= 6687/9996
≈ 0.67
AC CD
C D C C D C B
Ɛ-Machine: Causal State Splitting Reconstruction
C. R. Shalizi, K. L. Shalizi and J. P. Crutchfield, arXiv preprint cs/0210025.
Ɛ-Machine:
Causal State Splitting Reconstruction
-The Even Process
A
C
D
0 ( p = 0.33)
1 ( p = 0.67)
1 ( p = 0.5)
0 ( p = 0.5)
1 ( p = 0.75)
0 ( p = 0.25)
1 ( p = 1)
B
Statistical Complexity and Metric Entropy
• Statistical Complexity is defined as
It serves to measure the computational resources required to reproduce a
data stream.
• Metric Entropy is defined as
Bits/Symbols
It serves to measure the diversity of observed patterns in a data stream.
J. P. Crutchfield, Nature Physics 8, 17 (2012).
Bits
ppC 2log
m
s
m spspm
hm
2log1
The Logistic Map
Linear region in the periodic window
Decreasing trend at chaotic region
Phase transition, 𝐻∗ ≈ 0.28 : “edge of chaos”
The Arc-Fractal Systems
The Arc-Fractal Systems
Out-Out (Levy):
Out-In-Out (Crab):
Out-In (Heighway):
In-Out-In (Arrowhead):
The Arc-Fractal Systems
Sequence -> each arc is given an index according to its angle of orientation
-> Sequence : 11,7,3
Out-In-Out rule, level 3 of construction:7,3,11,7,11,3,11,7,3,11,3,7,11,7,3,7,11,3,11,7,3,11,3,7,3,11,7
In base-3:1,0,2,1,2,0,2,1,0,2,0,1,2,1,0,1,2,0,2,1,0,2,0,1,0,2,1
The Arc-Fractal Systems
The Arc-Fractal Systems
• The fractals generated by the arc-fractal systems can be
associated with symbolic sequences1,2.
1H. N. Huynh and L. Y. Chew, Fractals 19, 141 (2011).2H. N. Huynh, A. Pradana and L. Y. Chew, PLoS ONE 10(2), e0117365 (2015)
The Arc-Fractal Systems
H. N. Huynh, A. Pradana and L. Y. Chew, PLoS ONE 10(2), e0117365 (2015)
Complex Symbolic Sequence and Neuro-
behavior
)0(
1,2T
1S 2S 3S
)0(
2,1T
)1(
2,1T
)1(
1,2T
)0(
1,1T
)0(
1,1T
)1(
3,2T
)0(
2,3T
)0(
3,2T
)0(
2,2T
)1(
2,3T
)1(
2,2T
)0(
3,1T
)0(
3,3T
)1(
3,1T
)1(
3,3T
)0(
1,3T
)1(
1,3T
nS
Quantum Ɛ-Machine
krTSn
k r
r
kjj
1
1
0
)(
,
nj ,,2,1 for
General Classical ε-Machine
Causal State of Quantum ε-
Machine
M. Gu, K. Wiesner, E. Rieper and V. Vedral, Nature Communications 3, 762 (2012)
Quantum Complexity
krTSn
k r
r
kjj
1
1
0
)(
, nj ,,2,1 for
Quantum Causal State :
Quantum Complexity :
where the density matrix ii
i
i SSp and ip is the
probability of the quantum causal state .iS
Note that CCq where Bits
ppC 2log
logTrqC Bits
R. Tan, D. R. Terno, J. Thompson, V. Vedral and M. Gu, European Physical Journal Plus 129(9), 1-10 (2014)
1
3
2
4
5
6
0.50.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 1.0
C
qC
Com
plex
ity
H(16)/16
Quantum versus Classical Complexity
The Logistic Map
Ɛ-Machine at Point 1
– Edge of Chaos
....569946.3R
480236.5C
376835.5qC
Ɛ-Machine at Point 2
– Chaotic Region
715.3R 341681.3C 093695.3qC
Ɛ-Machine at Point 3
– Chaotic Region
960.3R
364846.1C 906989.0qC
Quantum Ɛ-Machine
-Change of Measurement Basis
1S2S
1)1(
2,1 T
1)0(
1,2 T
General Qubit Measurement Basis:
10
10
** ab
ba
where and are new orthogonal
measurement basis such that:
ab
ba
*
*
1
0 10
21
2
1
S
S
Quantum Ɛ-Machine
-Change of Measurement Basis
11
22
*
2
*
1
baS
abS
1S2S
2)(
2,1 aT
2)(
2,1 bT
2)(
1,2 bT
2)(
1,2 aT
Deterministic ε - Machine
Stochastic ε - Machine
Quantum Ɛ-Machine
-Change of Measurement Basis
1S2S
0
)1(
2,1 qT
0
)0(
1,1 1 qT
1
)0(
2,1 qT
1
)1(
2,2 1 qT
Stochastic ε - Machine
21110
21101
112
001
qqS
qqS
Quantum Ɛ-Machine
-Change of Measurement Basis
211211
211211
11
*
1
*
12
00
*
0
*
01
aqbqbqaqS
aqbqbqaqS
If the quantum ε – machine is prepared in the quantum causal state 1S
and measurement yields , the quantum ε – machine now possesses the
causal state:
2
*
01
*
01 SbqSaq
which is a superposition of the quantum resource. Further measurements
in the and basis leads to further iterations within the above results
due to the inherent entangling features in the quantum causal state structure.
Ideas to Proceed
Mandelbrot Set
- Complexity within Simplicity
Question:
Could there be infinite regress in
terms of information processing
within quantum ε – Machine?
Question:
Could classical ε – Machine acts as a
mind-reading machine in the sense of
Claude Shannon’s? Would a quantum
version do better?
Thank You
Neil Huynh Andri PradanaMatthew Ho
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