Randomness, Quantum Mechanics - and Cryptography

[Victor Duchovni]

On Tue, Sep 07, 2010 at 10:22:57PM -0400, Jerry Leichter wrote:

> But there isn't actually such a thing as classical thermodynamical 
> randomness!  Classical physics is fully deterministic.  Thermodynamics uses 
> a probabilistic model as a way to deal with situations where the necessary 
> information is just too difficult to gather.  Classically, you could in 
> principle measure the positions and momenta of all the atoms in a cubic 
> liter of air, and then produce completely detailed analyses of the future 
> behavior of the system.  There would be no random component at all.  In 
> practice, even classically, you can't hope to get even a fraction of the 
> necessary information - so you instead look at aggregate properties and, 
> voila, thermodynamics.  There's no randomness assumption - much less an 
> unpredictability assumption - for the micro-level quantities.  What you 
> need is some uniformity assumptions.  If I had access to the full micro 
> details of that liter of air, your calculations of the macro quantities 
> would be completely undisturbed.

This glosses over the *fundamental* complexity of non-linear classical
dynamics. It is a leap to claim that the underlying determinism of a
classical dynamical system leads one to conclude that it is even in
principle "predictable", in the presence of chaos.

We should not short-change classical "chaos" which is an emergent property
of complex deterministic systems.

    http://www-chaos.umd.edu/research.html

    ...

    Riddled Basins  

    The notion of determinism in classical dynamics has eroded since
    Poincaré's work led to recognition that dynamical systems can exhibit
    chaos: small perturbations grow exponentially fast. Hence, physically
    ubiquitous measurement errors, noise, and computer roundoff strongly
    limit the time over which, given an initial condition, one can
    predict the detailed state of a chaotic system. Practically speaking,
    such systems are nondeterministic. Notwithstanding the quantitative
    uncertainty caused by perturbations, the system state is confined
    in phase space (on an "attractor") so at least its qualitative
    behavior is predictable. Another challenge to determinism arises
    when systems have competing attractors. With a boundary (possibly
    geometrically convoluted ) between sets of initial conditions tending
    to distinct attractors ("basins of attraction"), perturbations
    make it difficult to determine the fate of initial conditions near
    the boundary. Recently, mathematical mappings were found that are
    still worse: an attractor's entire basin is riddled with holes on
    arbitrarily fine scales. Here, perturbations globally render even
    qualitative outcomes uncertain; experiments lose reproducibility.

	J.C. Sommerer and E. Ott, "A Qualitatively Nondeterministic
	Physical System", Nature, 365, 135 (1993).

-- 
	Viktor.

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