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Quantum Versus Classical Measures of Complexity on Classical Information

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Quantum Versus Classical Measures of Complexity on Classical Information 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, 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 Symbolic Sequences: ….. S-2S-1S0S1 ….. Futures : Past : • Within each partition, we have Partitioning the set 푺 of all histories into causal states 푆푖. where 푠 and 푠′ are two different individual histories in the same partition. • Equation (1) is related to sub- tree similarity discussed later. J. P. Crutchfield and C. R. Shalizi, Physical Review E 59, 275 (1999). 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 = 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 – Reconstruction S = 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, ….. • From symbol sequence -> build a tree structure with nodes and links . Period-3 sequence 0 1 • The links are given symbols according to the symbol sequence with A = {0, 1}. 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 0 1 Ɛ-Machine – Reconstruction 0 1 2 3 1 0 1 2 0 1 Level 1 2 0 1 3 0 4 3 Level 2 1 4 3 5 0 0 1 4 3 Level 3 0 3 5 4 0 1 0 5 3 5 0 4 3 1 0 0 4 3 5 5 4 1 0 0 3 5 4 0 0 1 5 0 4 3 1 0 4 1 3 5 0 0 Ɛ-Machine – Reconstruction S = 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, ….. Probabilistic Structure 1 0 ( p = 2/3) 1 ( p = 1/3) Based on Orbit 1 ( p = 1/2) Ensemble 2 3 1 ( p = 1) 0 ( p = 1/2) 0 ( p = 1) 4 5 0 ( p = 1) Combination of both deterministic and random But the Probabilistic Structure computational resources is transient. Ɛ-Machine – Simplest Cases All Head or All “1” Process Purely Random Process s = 1, 1, 1, 1, 1, 1, ….. Fair Coin Process 1 ( p = 1/2) 1 ( p = 1) 0 ( p = 1/2) • 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. 3 푇푛 3-level sub-tree of n contains all nodes that can be reached within 3 links The Logistic Map Logistic equation : 푥푛+1 = 푟푥푛 1 − 푥푛 Starts period-doubling; at 푟 = 3 : period-2 Chaotic at 푟 = 3.569946… 0, 푥푖 < 0.5 Partitioning : 푠푖 = 1, 푥푖 ≥ 0.5 Ɛ-Machine: Complexity from Deterministic Dynamics Construction of ε-machine: 1 Transitions Sub-tree Similarity Level 1 2 1 LevelSub 2 -tree 3 2 1 Similarity Level 3 2 2 3 2 1 Ɛ-Machine: Complexity from Deterministic Dynamics Transitions Transient State Ɛ-Machine: Complexity from Deterministic Dynamics s = 1,0,1,1,1,0,1,… • The ε-machine captures the patterns of the process. Ɛ-Machine: Causal State Splitting Reconstruction Total = 9996 P(0|*) A P(1|*) = 3309/9996 = 6687/9996 ≈ 0.33 0 1 ≈ 0.67 P(0|*0) 0 C B 1 P(1|*1) = 1654/3309 = 5032/6687 ≈ 0.5 1 ≈ 0.75 P(1|*0) 0 P(0|*1) =1655/3309 = 1655/6687 C ≈ 0.5 D C ≈ 0.25 A P(1|*11) P(0|*00) P(1|*00) P(0|*01) 0 1 0 ≈ 0.67 ≈ 0.51 ≈ 0.49 = 0 1 P(1|*01) P(1|*10) P(0|*10) 0 1 = 1 ≈ 0.51 P(0|*11) ≈ 0.49 1 0 ≈ 0.33 C D C C D C B P(0|*000) P(1|*111) ≈ 0.49 P(0|*001) P(0|*010) P(0|*011) P(1|*100) P(1|*110) ≈ 0 ≈ 0 ≈ 0.49 ≈ 0.49 ≈ 0.51 ≈ 0.75 0 1 P(0|*101) 0 1 0 1 0 1 0 1 ≈ 0 0 1 0 1 0 1 P(0|*111) P(1|*000) P(1|*001) P(1|*010) P(1|*011) P(0|*100) P(1|*101) P(0|*110) ≈ 0.25 ≈ 0.51 ≈ 1 ≈ 0 ≈ 0.51 ≈ 0.51 ≈ 1 ≈ 0.49 C. R. Shalizi, K. L. Shalizi and J. P. Crutchfield, arXiv preprint cs/0210025. Ɛ-Machine: Causal State Splitting Reconstruction -The Even Process 1 ( p = 0.67) A B 1 ( p = 0.75) 0 ( p = 0.33) 0 ( p = 0.25) C 0 ( p = 0.5) 1 ( p = 0.5) 1 ( p = 1) D Statistical Complexity and Metric Entropy • Statistical Complexity is defined as C p log2 p Bits It serves to measure the computational resources required to reproduce a data stream. • Metric Entropy is defined as 1 m m h ps log2 ps Bits/Symbols m sm It serves to measure the diversity of observed patterns in a data stream. J. P. Crutchfield, Nature Physics 8, 17 (2012). The Logistic Map Linear region in the periodic window Decreasing trend at chaotic region Phase transition, 퐻∗ ≈ 0.28 : “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 : 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 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 Quantum Ɛ-Machine (1) T1,3 General Classical ε-Machine (0) (1) T1,3 (1) T1,2 T2,3 (0) (0) (0) T1,1 T1,2 T2,3 (0) S (1) (0) S (1) S1 T2,2 2 T2,2 T3,3 3 T3,3 Sn T (0) 1,1 T (0) T (0) 2,1 3,2 Causal State of Quantum ε- Machine (1) (1) T T n 1 2,1 T (0) 3,2 (r) 3,1 S j Tj,k r k k1 r0 (1) T3,1 for j 1,2,,n M. Gu, K. Wiesner, E. Rieper and V. Vedral, Nature Communications 3, 762 (2012) Quantum Complexity Quantum Causal State : C p log2 pfor Quantum Complexity : Cq Tr log Bits where the density matrix pi S i S i and pi is the i n 1 (r) probability of the quantum causal state Si . S j Tj,k r k k1 r0 Note that C C Bits q where j 1,2,,n R. Tan, D. R. Terno, J. Thompson, V. Vedral and M. Gu, European Physical Journal Plus 129(9), 1-10 (2014) Quantum versus Classical Complexity The Logistic Map 6 C 5 Cq 4 3 Complexity 2 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 H(16)/16 Ɛ-Machine at Point 1 – Edge of Chaos R 3.569946.... C 5.480236 Cq 5.376835 Ɛ-Machine at Point 2 – Chaotic Region R 3.715 C 3.341681 Cq 3.093695 Ɛ-Machine at Point 3 – Chaotic Region R 3.960 C 1.364846 Cq 0.906989 Quantum Ɛ-Machine -Change of Measurement Basis General Qubit Measurement Basis: (1) T1,2 1 a 0 b 1 * * S2 S1 b 0 a 1 (0) T2,1 1 where and are new orthogonal measurement basis such that: S1 1 2 * 0 a b S2 0 1 1 b* a Quantum Ɛ-Machine -Change of Measurement Basis * S1 b 2 a 2 * S2 a 1 b 1 ( ) 2 T1,2 b S1 Deterministic ε - Machine ( ) 2 T1,2 a S2 Stochastic ε - Machine ( ) 2 T2,1 b ( ) 2 T2,1 a Quantum Ɛ-Machine -Change of Measurement Basis (1) T1,2 q0 (0) (1) T1,1 1 q0 S2 T2,2 1 q1 (0) S1 T1,2 q1 Stochastic ε - Machine S1 1 q0 0 1 q0 1 2 S2 q1 0 1 1 q1 1 2 Quantum Ɛ-Machine -Change of Measurement Basis * * S1 1 q0 a 1 q0 b 2 1 q0 b 1 q0 a 2 * * S2 q1 a 1 1 q1 b 2 q1 b 1 1 q1 a 2 If the quantum ε – machine is prepared in the quantum causal state S1 and measurement yields , the quantum ε – machine now possesses the causal state: * * 1 q0 a S1 q0 b S2 which is a superposition of the quantum resource.
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