# What is the real-world significance of the Bekenstein bound?

The Bekenstein bound sets the maximum amount of information that can be contained in a region of space/energy, and is usually referred to in the same way as computer storage density:

For example, a single hydrogen atom, if it were to code as much information as permitted by the Bekenstein Bound, would code about 4×106 bits of information, since the hydrogen atom is about one Ångström in radius, and has a mass of about 1.67×10−27 kilograms. (source)

Nature permits a surprising amount of information to be encoded before the Bekenstein bound is reached. For example, a hydrogen atom can encode about 1 Mb of information — most of a floppy disk. (source)

The "Bekenstein bound" leaves room for a million bits in a hydrogen atom (source)

But what does this really mean? How could any information be stored in a hydrogen atom?

## 2 Answers

How could any information be stored in a hydrogen atom?

Your sources aren't talking about storing information in a hydrogen atom. They're talking about storing information in an amount of space whose volume is the same as the volume of a hydrogen atom.

What is the real-world significance of the Bekenstein bound?

If "real world" means practical, then the answer is that the Bekenstein bound has no real-world significance. The WP article is being silly by applying it to computer science. They aren't referring to computer science in the sense of actual computer hardware. They're just saying that computer science deals with the storage and manipulation of information, and this is an ultimate bound on that.

If you take some matter and compress it so much that it forms a black hole, you've hidden away the information contained in that matter. It's behind an event horizon and can't be retrieved. A black hole exactly saturates the Bekenstein bound. If you want to take the same amount of information and compress it without making it inaccessible behind an event horizon, you're going to have to compress less than the limit specified by the Bekenstein bound.

• "They're talking about storing information in an amount of space whose volume is the same as the volume of a hydrogen atom" The calculation is based on volume and mass, so what else could be in that volume with that mass? – endolith May 6 '13 at 1:18
• @endolith: A microscopic black hole. The first two quotes in the question are actually not literally correct, because they imply that it would be an actual hydrogen atom. To saturate the Bekenstein bound, you need a black hole, and a black hole isn't a hydrogen atom. – Ben Crowell May 6 '13 at 2:57
• so is it a black hole with the mass of a hydrogen atom inside, or is that not possible and therefore the example is also impossible? – endolith May 6 '13 at 13:30
• It's a black hole whose event horizon's radius is the same as the radius of a hydrogen atom. Its mass is greater than the mass of a hydrogen atom by many orders of magnitide. – Ben Crowell May 6 '13 at 22:01

If I give you a hydrogen atom in an excited state, and there are $2^{1000000}$ different possible excited states of a hydrogen atom, then I've encoded 1 Mb of information in your hydrogen atom.

In reality, there are far fewer than $2^{1000000}$ possible excited states of a hydrogen atom, see e.g. this table. The Bekenstein bound provides an upper limit on the number of possible excited states of a hydrogen atom, so that is some real world significance.

Edit: This argument is wrong, see the comments below.

• If there are a million states the electron can be in, then that would only be log2(1000000) = 20 bits of information. – endolith May 5 '13 at 22:59
• Also, there are actually an infinite number of excited states for a hydrogen atom - though most of them are so closely spaced as to be indistinguishable. – David Z May 5 '13 at 23:03
• Both those statements are basically true: a state is specified by four quantum numbers (the fourth is the electron spin quantum number), and the difference in energy between the lowest bound state and a free electron is finite, $13.6\text{ eV}$. But there are an infinite number of states within that energy range. The principal quantum number $n$ ranges from 1 to infinity. – David Z May 5 '13 at 23:10
• Interesting! Since I've learned something from this, instead of deleting my answer I'll just make it clear that it is wrong, and to see the comments. – Alex L May 5 '13 at 23:16
• Note the size of a hydrogen atom in the $2^{1000000}$th state is much, much larger than the size of a hydrogen atom in the ground state, so the Bekenstein bound holds. – Peter Shor May 6 '13 at 0:16