Transition probabilities subject to Lloyd's finite information limit?

[Bill H]

This is a question about The Computational Capacity of the Universe by Seth Lloyd.

It seems to me that arbitrary real numbers cannot be part of the state of the universe, since they carry an infinite amount of information. There are transition probabilities from the current state of the universe to future states. If these probabilities are arbitrary real numbers between 0 and 1, then they carry an infinite amount of information.

Can these probabilities be arbitrary reals, or are they subject to the finite information capacity and hence constrained to a finite subset of the reals?

Seth Lloyd computes the information capacity of the universe as proportional to the age of the universe squared. But I have read that quantum information is conserved - it cannot be created or destroyed. How can the information capacity increase if information is conserved?

Thank you.

Bill H

 

[bhobba]

It seems to me that arbitrary real numbers cannot be part of the state of the universe, since they carry an infinite amount of information.

Its a model with limitations it is thought at about the plank scale. What it says below that is currently unknown - it may be that the universe has finite information carrying capacity - or not - we just don't know. But pushing our current models to say the real numbers they use can be infinitely subdivided into domains beyond which they are applicable is not valid.

Thanks
Bill

 

[kurros]

It seems to me that arbitrary real numbers cannot be part of the state of the universe, since they carry an infinite amount of information. There are transition probabilities from the current state of the universe to future states. If these probabilities are arbitrary real numbers between 0 and 1, then they carry an infinite amount of information.

In what sense does a single real number contain an infinite amount of information? So far as I know, information is defined relative to some probability distribution, and ok say we have a continuous probability density function $$f(x),$$for some outcome that is some real number $$x=\omega,$$then sure, knowing that real number outcome gives you infinite information, since the self-information associated with that outcome is
$$I(\omega) = -\log(Pr(\omega)) = \lim_{\delta\omega \to 0} -\log(\int_\omega^{\omega+\delta\omega}f(x)dx) = -\log(0)$$ which is infinite.

But even if the outcome is theoretically a real number, we never measure such a thing. There is always uncertainty, so the probability of whatever outcome you measure is never actually zero. Doesn't this prevent any problem?

 

[bhobba]

In what sense does a single real number contain an infinite amount of information?

Any real interval can be put into 1-1 correspondence with any other interval, regardless of the size of the interval - which is the definition of infinite. Its not real numbers per se that contain infinite information, its functions (ie mappings - numbers by and of themself tells us nothing - except of course the number - it needs to be mapped to something) that can be defined on them ie a function defined on any interval contains exactly the same amount of information as one defined on any other interval - which, again from the definition of infinite, is infinite.

However, physically, it doesn't make sense to consider intervals below a certain size for all sorts of reasons eg as you point out, because we can't measure below a certain threshold, you can't exploit that infinity.

Thanks
Bill