true randomness? (finite vs infinite sequences)

[José Manuel Rodríguez Caballero]

 Sam wrote

> I second Harvey here: as far as I know, quantum mechanics can provide

finite datasets that are better than classical ?state of the art?
> pseudorandom
> sources.  *However*, when talking about randomness in computability theory,
> the definition is concerned with infinite sequences.


Which notion are we dealing with (in theory and practise)?


Assuming the Physical Church Turing Thesis (Wolfram's version), each result
of a quantum measurement is recorded by an observer, which is assumed to be
a machine with memory (Hugh Everett's definition). Assuming an observer
with arbitrarily large memory (a finite memory that increases with time),
we have a potential Turing machine to record an infinite sequence of
quantum measurements. Hence, we can consider whether or not the sequence
recorded by the potentially infinite memory of the observer can be
generated by a deterministic algorithm.

To be completely rigorous, it should be proved that an observer with
potentially infinite memory is possible. Indeed, we could imagine a
universe that collapses after some time and no observer is possible
anymore. This possibility, considered for the first time by Lord Kelvin
(William Thomson)

Thomson, William (1862). "On the Age of the Sun's Heat". Macmillan's
Magazine. Vol. 5. pp. 388–393.

is known as the heat death of the universe. Freeman Dyson

Freeman J. Dyson, "Time without end: Physics and biology in an open
universe," Reviews of Modern Physics, Vol. 51, Issue 3 (July 1979), pp.
447-460; doi:10.1103/RevModPhys.51.447.
https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.51.447

 and Frank J. Tipler

Tipler, Frank J (June 1986), "Cosmological Limits on Computation",
International Journal of Theoretical Physics, 25 (6): 617–61,
Bibcode:1986IJTP...25..617T, doi:10.1007/BF00670475, S2CID 59578961
https://link.springer.com/content/pdf/10.1007/BF00670475.pdf

proposed methods to construct an eternal observer in an open (but not
accelerating) and a closed (contracting universe), respectively. But, the
current scientific consensus is that our universe is accelerating. Hence,
if it is possible to construct an eternal observer who increases its memory
in time, it cannot be done following Freeman Dyson and Frank J. Tipler
proposals. Therefore, the answer to your question remains open for our
universe, and it is positive for the universes where either Dyson's or
Tipler's constructions can be done. This problem belongs to a new approach
to physics known as constructor theory, developed by David Deutsch, Chiara
Marletto, and collaborators in Oxford Physics:

Constructor Theory is a new approach to formulating fundamental laws in
> physics. Instead of describing the world in terms of trajectories, initial
> conditions and dynamical laws, in constructor theory laws are about which
> physical transformations are possible and which are impossible, and why.
> This powerful switch has the potential to bring all sorts of interesting
> fields, currently regarded as inherently approximative, into fundamental
> physics. These include the theories of information, knowledge,
> thermodynamics, and life.


https://www.constructortheory.org/

Kind regards,
Jose M.
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