Friday, July 14, 2006

Physical limits on information processing

Physical limits on information processing

Authors: Stephen D.H. Hsu

We derive a fundamental upper bound on the rate at which a device can process information (i.e., the number of logical operations per unit time), arising from quantum mechanics and general relativity. In Planck units a device of volume V can execute no more than the cube root of V operations per unit time. We compare this to the rate of information processing performed by nature in the evolution of physical systems, and find a connection to black hole entropy and the holographic principle.


Anonymous said...

You missed a factor of root(pi) in equation 2.

Anonymous said...

actually it's sqrt(2*pi) :)

Steve Hsu said...

As I wrote in the paper, I'm not keeping track of O(1) factors, mainly because the hoop conjecture isn't formulated more precisely than that.

Anonymous said...

Interesting... I wonder what is the relation with

Anonymous said...

I once heard a Feynman story, some guy went up to him after a QFT lecture and said "I understand everything except the factors of 2 pi" and Feynman puportedly replied "Buddy, then you don't understand nothin!"

Steve Hsu said...

Oppenheimer purportedly could never get the 2's and pi's right. Here we just admit that the bound is only accurate at the order of magnitude level (or should be thought of as a bound on the scaling behavior). Nobody knows the numerical factor in the hoop conjecture, other than it is probably of order one, so there is no point in carrying the other O(1) coefficients through to the end.

Regarding that preprint, I think the decoherence effect they introduce is speculative.

Joe said...

Hi Steve,

I've just read the preprint and I have a few question. I hope you don't mind me asking it here rather than emailing you directly.

The first is whether you are justified in deglecting all black holes, even those which will evaporate over the course of the calculation? Is there a simple justification for this that I am missing?

The second is whether there is justification for applying results which stem from the schwarzchild radius to quantum systems. Lacking a decisive theory of quantum gravity does relying on a result from general relativity really make sense in the limits of a very small systme/volume? Again, perhaps there may be a simple reason for this that I am missing, and I hope that my comments don't show a complete misunderstanding of the paper.



Steve Hsu said...

Hi Joe,

Thanks for your interest in the paper. You can email me directly, which will probably get faster response than a comment here.

I don't need to neglect black holes as they simply saturate the bound.

Regarding small volumes (and quantum gravity), the hoop conjecture doesn't necessarily apply to very small (Planckian) volumes, as it is derived in the semiclassical approximation. (For the bound I have in mind a macroscopic "device" which is itself semiclassical.) Note that if the energies involved are much larger than the Planck energy, the Schwarzschild radius is much larger than the Planck length and quantum gravity effects are small.

The only place my analysis does not apply is to a single component of Planck mass and Planck length. In that case it's not clear what the bound is, but most likely not very different than V^{1/3} since the limit is probably smooth. If you aggregate a bunch of such components, the semiclassical result will nevertheless apply, as the total energy and Schwarzschild radius are getting large in Planck units.

Hope that helps!

PS I notice you are a judoka -- I'm a Brazilian jiujitsu guy myself :-)

Joe said...

Thanks for the reply. I did BJJ for a little while, but since moving to Oxford haven't been able to find anywhere to do it.

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