Quantum versus Classical Measures of Complexity on Classical Information Lock Yue Chew, PhD Division of Physics and Applied Physics School of Physical & Mathematical Sciences & Complexity Institute Nanyang Technological University
Astronomical Scale Macroscopic Scale Mesoscopic 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, 5 (989).
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. Optimal Predictor of the System s Process. 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, 7 ().
Concept of Ɛ-Machine Symbolic Sequences:.. S - S - S S.. Futures : Past : St StSt S S t S t S S t3 t t Partitioning the set S of all histories into causal states S i. Within each partition, we have P S s PS s' () where sand s are two different individual histories in the same partition. Equation () is related to subtree similarity discussed later. J. P. Crutchfield and C. R. Shalizi, Physical Review E 59, 75 (999).
Ɛ Machine - Preliminary In physical processes, measurements are taken as time evolution of the system. The time series is converted into a sequence of symbols s = s, s,..., s i,.... 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 = {,,, k -}. Then, each s i ϵ A.
Ɛ-Machine Reconstruction S =,,,,,,,,,,,,.. From symbol sequence -> build a tree structure with nodes and links. Period-3 sequence The links are given symbols according to the symbol sequence with A = {, }.
Ɛ-Machine Reconstruction 3 4 3 5 3 5 4 Level Level Level 3 5 5 5 5 5 4 4 4 4 4 3 3 3 3 3 3 3 5 4 5 3 4 3 4
Ɛ-Machine Reconstruction S =,,,,,,,,,,,,.. ( p = /3) ( p = /3) Probabilistic Structure Based on Orbit Ensemble ( p = /) 3 ( p = /) ( p = ) ( p = ) 4 ( p = ) 5 Combination of both deterministic and random computational resources But the Probabilistic Structure is transient.
Ɛ-Machine Simplest Cases All Head or All Process s =,,,,,,.. Purely Random Process Fair Coin Process ( p = ) ( p = /) ( p = /) These are the extreme cases of Complexity. They are structurally simple. We expect their Complexity Measure to be zero.
Ɛ-Machine: Complexity from Deterministic Dynamics 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. T n 3 3-level sub-tree of n contains all nodes that can be reached within 3 links
The Logistic Map Logistic equation : x n+ = rx n x n Starts period-doubling; at r = 3 : period- Chaotic at r = 3.569946 Partitioning : s i =, x i <.5, x i.5
Ɛ-Machine: Complexity from Deterministic Dynamics Construction of ε-machine: Level Sub-tree Similarity Transitions Level 3 Level Sub-tree Similarity 3 3
Ɛ-Machine: Complexity from Deterministic Dynamics Transitions Transient State
Ɛ-Machine: Complexity from Deterministic Dynamics s =,,,,,,, The ε-machine captures the patterns of the process.
Ɛ-Machine: Causal State Splitting Reconstruction Total = 9996 P( *) = 339/9996.33 A P( *) = 6687/9996.67 P( *).5 C P( *) = 654/339.5 P( *) =655/339.5 P( *).49 P( *) = C D P( *) = B C P( *) = 53/6687.75 P( *) = 655/6687.5 A P( *).49 P( *).5 P( *).33 P( *).67 P( *).49 C D C C D C B P( *) P( *) P( *).49 P( *).49 P( *) P( *).5 P( *).75 P( *).5 P( *) P( *) P( *).5 P( *).5 P( *) P( *).49 P( *).5 C. R. Shalizi, K. L. Shalizi and J. P. Crutchfield, arxiv preprint cs/5.
Ɛ-Machine: Causal State Splitting Reconstruction -The Even Process ( p =.67) A ( p =.33) ( p =.5) ( p =.75) C B ( p =.5) ( p =.5) ( p = ) D
Statistical Complexity and Metric Entropy Statistical Complexity is defined as C p log p Bits It serves to measure the computational resources required to reproduce a data stream. Metric Entropy is defined as h p m m s s m log Bits/Symbols It serves to measure the diversity of observed patterns in a data stream. p s m J. P. Crutchfield, Nature Physics 8, 7 ().
The Logistic Map Linear region in the periodic window Decreasing trend at chaotic region Phase transition, H.8 : edge of chaos
The Arc-Fractal Systems
The Arc-Fractal Systems
The Arc-Fractal Systems Out-Out (Levy): Out-In (Heighway): Out-In-Out (Crab): In-Out-In (Arrowhead):
The Arc-Fractal Systems Sequence -> each arc is given an index according to its angle of orientation -> Sequence :,7,3 Out-In-Out rule, level 3 of construction: 7,3,,7,,3,,7,3,,3,7,,7,3,7,,3,,7,3,,3,7,3,,7 In base-3:,,,,,,,,,,,,,,,,,,,,,,,,,,
The Arc-Fractal Systems The fractals generated by the arc-fractal systems can be associated with symbolic sequences,. H. N. Huynh and L. Y. Chew, Fractals 9, 4 (). H. N. Huynh, A. Pradana and L. Y. Chew, PLoS ONE (), e7365 (5)
The Arc-Fractal Systems H. N. Huynh, A. Pradana and L. Y. Chew, PLoS ONE (), e7365 (5)
Complex Symbolic Sequence and Neurobehavior
Quantum Ɛ-Machine () T, () T,3 () T,3 () T,3 General Classical ε-machine () T, () T, () () T, T,3 () () () S S T, T, T 3,3 S3 () T, () T, () T 3, () T 3, () T 3, () T 3, () T 3,3 S j n k r T ( r) j, k S n Causal State of Quantum ε- Machine for r j,,,n k M. Gu, K. Wiesner, E. Rieper and V. Vedral, Nature Communications 3, 76 ()
Quantum Causal State : Quantum Complexity S j n k r T r k ( r) j, k for j,,, n Quantum Complexity : Cq Tr log Bits where the density matrix p S S and pi is the probability of the quantum causal state. i i i i S i Note that C q where p log p Bits C C R. Tan, D. R. Terno, J. Thompson, V. Vedral and M. Gu, European Physical Journal Plus 9(9), - (4)
Complexity Quantum versus Classical Complexity The Logistic Map 6 5 C C q 4 3...3.4.5.6.7.8.9. H(6)/6
Ɛ-Machine at Point Edge of Chaos R 3.569946... C 5.4836 C q 5.376835
Ɛ-Machine at Point Chaotic Region R 3.75 3. 3468 C 3. 93695 C q
Ɛ-Machine at Point 3 Chaotic Region R 3.96 C.364846. 96989 C q
Quantum Ɛ-Machine -Change of Measurement Basis General Qubit Measurement Basis: a b * b a where and are new orthogonal measurement basis such that: * () T, S S () T, * a b S b * a S
Quantum Ɛ-Machine -Change of Measurement Basis S b * a S a * b T T ( ), b ( ), a S S T ( ), b Deterministic ε - Machine Stochastic ε - Machine T ( ), a
Quantum Ɛ-Machine -Change of Measurement Basis T (), q T (), q S S T (), q T (), q Stochastic ε - Machine S S q q q q
Quantum Ɛ-Machine -Change of Measurement Basis * * * * a q b q b q a q S a q b q b q a q S If the quantum ε machine is prepared in the quantum causal state S and measurement yields, the quantum ε machine now possesses the causal state: * * S b q S a q 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?
Neil Huynh Andri Pradana Matthew Ho Thank You