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