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(Simple) Thoughts on Complexity
Vlatko Vedral
Oxford & Singapore
vlatko.vedral@qubit.org
Motivation
What is complexity?
What properties would we like?
Evolution of complexity vs disorder
Is it a function of the laws of physics?
Micro to macro?
SUMMARY OF TALK
Measures of Complexity
Ising model, biological complexity
Quantizing classical measures
How different is quantum complexity?
Beyond quantum?
Complexity of description
Information
Entropy
Algorithmic Complexity or Algorithmic Information Content
See eg. Lloyd, Measures of complexity
E.g. Entropy of the Universe
Bekenstein bound:
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(Current estimate: Lineweaver and Egan, 19orders of magnitude smaller)
Discretization of Bloch
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Think about a harmonic oscillator:
For an electron in the atom5010N
Buniy, Hsu, and Zee, 1999
Complexity of simulation
Harder to simulate defines complexity
Micro to macro?
More is different: there are macroscopic features whose evaluation would amount to solving the Halting problem!
(Mile Gu et al, 2008)
Complexity of structure
All about correlations
Interdependency of bits on each other
Usually quantified by some mutual information
Consequence of subadditivity of entropy
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Biological Complexity: Ising Model
Ising Model used to simulate many different systems innatural and social sciences;
Genetic Evolution as modelled by Ising
(Peliti, Statistical theory of Darwinian evolution, 1997)
Genetic evolution
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No of individualsWith genome s
Fitness
Individual label (1,2,…M)
“Second law” of evolution
Average fitness cannot decrease with time (proof for zero mutation rate)
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Too good to be true? Earth-Sun entropy production:
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More realistic…
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StochasticNature of mutations
Hammingdistance
Connections with Ising
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Use the transfer matrix expansion:
Sharp-peak landscape
“Complexity” need not increase with time (same as the existence of disordered phase):
If the rate of mutation is bigger than strength ofselection than no increase in fitness.
Landscape changing quickly leads to the sameresult.
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Properties
Want low complexity for both deterministic and random
Want continuity (when the process changes by a bit, the complexity also changes by a bit)
Non-additivity: whole “bigger” than sum of parts (as in entropy subadditivity)
Functionality?
A simple example of a measure
maxmax
1H
H
H
HC
Satisfies first two, does not satisfy superadditivity
Shiner, Davisonand, Landsberg, 1999 Phys. Rev. E
How to quantize?
Measures that depend on probabilities are typicallyquantized by choosing the optimal quantum measurement(e.g von Neumann entropy is the minimum of Shannonentropy over all projective measurements performed on agiven quantum state);
If the measure is based on Universal Turing machines (e.g.Kolmogorov), then we need to rephrase in terms of quantumTuring machines;
Others measures are more ambiguous, like thermodynamiccomplexity (e.g. is thermalization faster quantumly?).
Quantum simplicity: Ising Model
Von Neumann
Quantum simplicity is a consequence of that factthat
The quantum uncertainty in a state is smaller thanclassical uncertainty of any observable measuredon that state
)()( AHS
But…
With many (all?) definitions of complexitythere are extra (hidden) costs
Like for instance, the assumptions aboutthe existence of the universal Turingmachine, or some other devices thatperform more specific computations.
Beyond quantum
Generalized probabilistic theories and complexity (c.f. Cabello et al)
E.g. (Re)bits, i.e. Does the phase matter?
We know entropy depends on phase…
Can we match the excess entropy?
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