This question was asked to Jeannette Wing after her PCAST presentation on computer science.
“From a physics perspective, is there a maximum volume of information we can have?” (a nice challenge question for the theoretical computer science community since I think it begs the question “What is information?”)
Beyond "What is information?" one should also figure out what "volume" means in this context? Perhaps maximum density of information is a better measure.
Lance, there is in fact a theorem which gives bounds on this. The Margolus-Levitin theorem bounds the rate of computation in terms of energy density. There is a nice trick which can then be played: If the local energy density exceeds a certain limit, a black hole will form causing an event horizon which will essentially prevent you from getting an answer by causally disconnecting that region of space-time from the rest of the universe. Seth Lloyd has a nice paper using this trick to estimate the computational power of the universe (Phys. Rev. Lett. 88, 237901 (2002), arXiv).
You can of course use similar reasoning on any finite region of space-time.
That comment in her article doesn't give a lot of context about what kind of answer she might be expecting. But certainly this is by now a well-known and venerable question about which much is already known. The Wikipedia page on the holographic principle has a good overview. The most counterintuitive thing about the holographic principle is that it says the information capacity of a region should be proportional to its surface area; if you think of information capacity in terms of how many tiny two-state devices you can pack in there, you'd expect the interior volume to be the limiting factor. That intuition holds true up to a certain point, but eventually the concentration of mass-energy, putting aside quantum miniaturization issues, becomes so great that a black hole forms. Roughly speaking, by a bit of dimensional analysis and the fact that gravity is an inverse-square law, it's radius squared (proportional to surface area) that's the relevant quantity here.
This is an interesting and somewhat amusing question but it's poorly formulated in its current form.
I'll take another stab/risk at an answer hoping that scoring will take the original difficulty and basic/inherent "soft" ambiguity of the question in mind and that based on current literature knowledge there are several possible avenues but arguably no "correct answer".
The main query seems to be "physics analogies in computer science" of which volume is one of them. Therefore it's highly related to this other question Physics results in TCS?
To answer this question I'll take a few different approaches which I think all have merit.
In this case strictly interpreted, volume is in the unit "space" or "length cubed". (Although note in physics sometimes the term "space" is measured in either terms length or length cubed.)
Therefore one could argue that in algorithmics the study of the space hierarchy is analogous to the study of "volume". A $O(n^3)$ space algorithm seems also to have the units "length cubed" where length is expressed in terms of input size.
The problem with this is that in algorithmics a $O(n^3)$ algorithm is merely a contrived limit, in a sense TCS has considered the "multidimensional space continuum" since its inception seen in the space hierarchy theorem.
Carrying this over to physics, it might seem absurd to consider anything other than 3-d space, but of course advanced physics takes into consideration all kinds of higher dimensional space realms all the time, via c-dimensional Hilbert [vector] spaces. So it would seem that in some sense an $O(n^c)$ space algorithm can be construed to operate in a c-dimensional hilbert space. Seems like mulmuleys GCT program would be a premiere example of this.
Another approach to a volume analogy (and other physics quantities) in TCS is as follows, as discussed in the other question. It is known that SAT has a transition point extremely analogous to the transition point in physics/thermodynamics, Which happens eg with ideal gases under compression from one phase to another eg gas to liquid. This happens under a decrease in volume (say of the container of the gas). Now in SAT with random inputs the main two parameters on input size are clauses and variables. (Another parameter is number of variables in clauses, although that is often fixed at 3 for 3-SAT.)
Adjusting either the clauses or variables while keeping the other fixed pushes the problem difficulty through the easy-hard-easy transition point. Therefore it seems that these parameters are somehow analogous to Volume although I haven't seen the specifics mapped out. Digging into some of the deep papers on the statistical physics of SAT may turn up the analog of Volume. See [5] for a basic mapping of SAT onto statistical physics terminology.
[5] Analytic and Algorithmic
Solution of Random
Satisfiability Problems by Mezard, Parisi, Zechina
http://dynamics.org/Altenberg/UH_ICS/EC_REFS/K-SAT/Mezard.Science.297_812.pdf
The other approach might be related to the prior responses talking about high density information processing in physical space leading to black holes. It would use the Planck limit which has units of time, space, mass etcetera. There is a Planck Length unit $l_p$ defined on that Wikipedia page. See units in this table.
I don't know the exact physical interpretation of the Planck length (it's arguably an active area of physics research), but I believe it can be interpreted in some sense as "smallest possible length of space". In other words $l_p^3$ might be the size of a smallest possible physical bit. It is interesting and perhaps a discontinuity of the analogy that in physics a smallest bit apparently must actually have volume (ie units length cubed) in contrast to computer science in which bits seem to have a dimension of unit "length" only.
