Transition probabilities subject to Lloyd's finite information limit?

In summary: 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 summary, the information capacity of the universe is infinite, but it is limited by our inability to measure certain outcomes.
  • #1
Bill H
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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
 
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  • #2
Bill H said:
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
 
  • #3
Bill H said:
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 [tex] f(x), [/tex]for some outcome that is some real number [tex]x=\omega,[/tex]then sure, knowing that real number outcome gives you infinite information, since the self-information associated with that outcome is
[tex] I(\omega) = -\log(Pr(\omega)) = \lim_{\delta\omega \to 0} -\log(\int_\omega^{\omega+\delta\omega}f(x)dx) = -\log(0) [/tex] 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?
 
  • #4
kurros said:
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
 
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FAQ: Transition probabilities subject to Lloyd's finite information limit?

What is Lloyd's finite information limit?

Lloyd's finite information limit is a concept in information theory that states that there is a maximum amount of information that can be transmitted in a given communication channel. This limit is dependent on the channel's bandwidth and the signal-to-noise ratio.

How does this limit affect transition probabilities?

This limit impacts transition probabilities because it restricts the amount of information that can be transmitted about the current state of a system. As a result, the accuracy of predicting future states decreases as the system approaches Lloyd's limit.

Can Lloyd's limit be overcome?

No, Lloyd's limit is a fundamental limit in information theory and cannot be surpassed. However, it can be approached by using advanced coding and decoding techniques.

How do transition probabilities factor into this limit?

Transition probabilities play a crucial role in Lloyd's limit as they represent the likelihood of a system transitioning from one state to another. As the amount of information that can be transmitted decreases, the accuracy of these probabilities also decreases, which can impact the overall performance of the system.

How do scientists work with Lloyd's limit in their research?

Scientists take Lloyd's limit into account when designing communication systems or analyzing data from complex systems. They may also use techniques such as error-correcting codes to improve the accuracy of transition probabilities subject to this limit.

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