Are information conservation and energy conservation related?

Question

as evident from the title, are both, conservation of energy and conservation of information two sides of the same coin??

Is there something more to the hypothesis of hawking's radiation other than the fact that information cannot be lost? or can I say energy cannot be lost??

Answer 1

First of all I do not think that conservation of information is an established statement. It seems to be an open problem still as far as black holes go.

Even if true, it is a different type of conservation, analogous to the unitarity requirements of a system of functions or phase space considerations.

From the conclusions of a paper by Hawking :

In this paper, I have argued that quantum gravity is unitary and information is preserved in black hole formation and evaporation. I assume the evolution is given by a Euclidean path integral over metrics of all topologies. The integral over topologically trivial metrics can be done by dividing the time interval into thin slices and using a linear interpolation to the metric in each slice. The integral over each slice will be unitary and so the whole path integral will be unitary. On the other hand, the path integral over topologically non trivial metrics will lose information and will be asymptotically independent of its initial conditions. Thus the total path integral will be unitary and quantum mechanics is safe.

How does information get out of a black hole? My work with Hartle[8] showed the radiation could be thought of as tunnelling out from inside the black hole. It was therefore not unreasonable to suppose that it could carry information out of the black hole. This explains how a black hole can form and then give out the information about what is inside it while remaining topologically trivial. There is no baby universe branching off, as I once thought. The information remains firmly in our universe. I’m sorry to disappoint science fiction fans, but if information is preserved, there is no possibility of using black holes to travel to other universes. If you jump into a black hole, your mass energy will be returned to our universe but in a mangled form which contains the information about what you were like but in a state where it can not be easily recognized. It is like burning an encyclopedia. Information is not lost, if one keeps the smoke and the ashes. But it is difficult to read. In practice, it would be too difficult to re-build a macroscopic object like an encyclopedia that fell inside a black hole from information in the radiation, but the information preserving result is important for microscopic processes involving virtual black holes. If these had not been unitary, there would have been observable effects, like the decay of baryons.

Energy is a conserved quantity because of Noether's theorem: wherever it holds, energy is conserved. In extreme General Relativity scenaria energy itself loses its meaning, whereas phase space and unitarity may hold and if Hawking is correct, information is conserved.

So energy conservation and possible conservation of information are two unconnected effects.

Answer 2

Conservation of mass-energy is an extremely well-defined and exhaustively proven concept.

However, as was aptly noted in the earlier answer to your question, "conservation of information" has a far less solid status. For example, some interpretations of quantum mechanics would assert that you can retract or erase information under very carefully defined circumstances. In that view, the idea of conservation of information would at the very least have to be defined with great care.

So to avoid getting into issues of terminology, let me suggest instead a somewhat different phrasing of your question:

For a given level of mass-energy in some region of space, what is the maximum amount of information that can be stored without loss (conserved) within that space?

One end of that question is particularly easy to address: If a region of space has zero mass-energy for the time over which it is to store information, it also has zero ability to store information. It would make a really bad memory cell! I should mention for clarity that doing something like adding a particle to that region to create a "1" would be cheating, since it would mean your real memory cell consisted of the empty space plus wherever you were stashing the spare particle.

Alas, anything above that simple minimum of zero information storage for zero mass gets a bit more complicated, pun intended.

One complication is that low-mass particles just don't want to stay in place within a limited region. So if you for example you tried to code a "1" by putting an electron on one side of small cube of space, you would find that over time the location of your electron (and thus your memory cell) would become increasingly uncertain.

That is not something you can engineer away, at least not for a single particle in a vacuum. That's because the location of small masses must be described by wave functions, and wave functions always spread out over time. The problem is not that different from pouring a cup of water onto the surface of a pond and hoping for some reason it will just stay on the surface. Waves are waves, and wave functions like to spread.

Atoms get around this problem by steering the electron waves around in very tight loops (via electrostatic attraction), so perhaps atoms could solve the problem? Alas, what you would find by replacing electrons with atoms is that the atoms also spread like waves, albeit far more slowly than the electrons did. In fact, you can't completely win on this problem. More mass slows the spread, but even a baseball sitting in an impossibly perfect, absolutely pristine, and radiation-free vacuum would slowly start to spread like a wave, and thus very slowly lose track of its position over enough time. You can also use other masses or particles to keep bouncing the memory part around to keep it in place, but of course the very act of banging things into your memory will introduce some dangers of losing information. The "banging" nonetheless works pretty well for classical systems, and that is what we do with most real memories: We embed them in solid objects that keep them mostly in place. But there is a limit even there, since solid memories still rely ultimately on basic particles. The wave function drift issue has a real impact, for example, on the design of flash memories. Those work by trying to keep small numbers of electrons corralled into very small spaces. One of the reasons why flash drives specify limits (e.g. 10 years) on how long they can retain information is because those electrons tend to forget where they are, even with all of the barriers that the solid flash chips put into effect to try to keep them corralled.

So let's accept that as given, something perhaps for a bumper sticker: Drift Happens.

So, ignoring drift, can you code anything with certainty onto a very light particle?

Oddly enough, the answer is yes. The smallest possible bit of matter that is willing to stay in one place for a decent span of time is the electron (or its antimatter equivalent, the positron). This simple particle has a quantum form of angular momentum called spin, and it has it in the absolute smallest unit that quantum mechanics allows. That's because angular momentum starts getting very "chunky" (quantum) at its lower limits, refusing to have any other than a small number of precisely defined values. The value in the case of the electron is 1/2 spin unit. (Why "1/2" is an interesting question all by itself.)

It turns out that spin enables you to use such a particle as a perfect memory -- one that will not forget even over very long periods of time -- for exactly one quantity: An axis of rotation, called a spin vector, that points in whatever direction you last used to "read" the electron. Mind you, it's up to you and your memory reading machinery to make sure you remember the coordinate system that you used to set the electron. But since you cannot really talk about or even define information clearly without some kind of a reader apparatus, assuming such a reader is pretty much a necessity anyway.

This is somewhat surprising, given quantum mechanics' reputation for making everything fuzzy. If your reader can only handle one axis in space for setting and reading electrons, then every electron can store exactly one bit of information, since the electron can rotate in either of two directions around that axis.

Using individual electron spin axes to store bits pushes hard on quantum physics, yet also gives an experimentally meaningful ratio of bits-to-mass: one bit per electron mass. Since the electron is the smallest stable particle that can stay in place for a while to act as a memory, it's a decent ballpark for maximum realizable storage.

However, an electron can be pointed along any axis and remember it perfectly. So the idea can be asked: Why not store more information by pointing the electrons in many different directions?

Well... you can certainly set the electrons up along different axes, but then quantum mechanics come back in to bite you, hard, when you try to read them. You see, the 100% certainty guarantee for reading a bit applies only if you are extremely careful to measure the electron along exactly the same axis that you used to set it. Deviate even a tiny bit from that axis, and quantum uncertainty starts to creep in. In fact, if you try measuring the electron at an angle 90 degree off from the axis you used to set it, the stored data becomes 100% irretrievable -- you just get random noise! Your memory has "forgotten" everything.

So, short of building a separate reader for every axis and thereby defeating the whole goal of achieving high bit-to-mass ratios, trying to use multiple axes just causes your memory to become less reliable. Only at zero deviation from the axis used to set the electron does it act like a reliable memory. You can use statistical methods to try to fix things up, but what you'll find if with that route is that you can never exceed than the one-bit-per-electron storage density of perfect alignment.

So at least for this approach -- there may be others -- that's the best I can do to answer your question. The relationship between conservation of mass-energy and conservation (as in storage) of information is that you can reliably store one bit of information, more-or-less indefinitely, for every electron mass in your storage medium.

And how much data does that work out to be? About $1.1 × 10^{30}$ bits, or $1.2 × 10^{14}$ petabytes, per kilogram.

That's a lot of bits!