Quantum Mechanics. The single-bit limit.
Another addition to my 'crackpot' series.
Consider the assertion that "information is fundamental and finite". The idea is that it is fundamental in the same way as mass and charge, rather than just some useful mathematical concept. In Quantizing gravity, and why it is difficult, by Leonard Susskind, at 16:40 he says that the smallest possible black hole has a mass of 0.02 milligrams, known as the Planck mass, because this is the mass of black hole with an information content of one bit, and you can't have less than one bit. It is perhaps surprising that the minimum mass is about the same as a speck of dust, so not really very small.
But is total information finite or could it be infinite? The universe is very big, but there is an infinite difference between very big and infinite. We are accustomed to the idea that the universe is of finite size and age, having started at the 'big bang' and expanded ever since. But what about the content of space-time? If there are an infinite number of possible variations in the content then the information content of the universe may be infinite.
It is a curious fact that all electrons are identical. At least, no-one has yet found one with a different mass or charge. (The mu and tau mesons behave like more massive electrons, but two extra particles is not much closer to infinity) We could imagine a universe where every particle could have any rest mass or charge, and then the total information content could be infinite. It happens that our universe is more restricted, and it is interesting to enquire why this is so. Is finite information content a necessary property of our universe?
Suppose we start with the proposition that the information content of the universe is finite and see where this leads. There are several ways to define information, e.g. the number of yes/no questions which must be asked to fully specify the state of a system. I will use a classical definition of the information capacity of a signal. This is: Information = duration Dt x bandwidth Df x signal to noise ratio. If we send the signal twice as long, then we can send twice as much information. Similarly doubling the bandwidth means we can send twice the information in the same time. The signal to noise ratio tells us how many different signal amplitudes can be distinguished. Now, if the information in the universe is finite there must be a non-zero noise level. Alternatively we can say that amplitudes are quantized so that only a finite number of levels are possible. Either way, the minimum quantity of information is one 'bit' and occurs when the signal to noise ratio is about 1. Then we can just distinguish signal on or off, giving one binary digit or bit. Our equation then becomes 1 = Dt x Df. Frequency is related to energy by the equation E = h x frequency where h is Planck's constant. Substituting for frequency we arrive at DE x Dt = h which is well known as the Heisenberg Uncertainty Principle (almost, there is a factor of 2pi missing, but the signal to noise 'about 1' is not a very precise limiting factor).
Note: the Heisenberg principle for time and energy (or frequency) can be disputed, (example here), because of the Stone-von Neumann uniqueness theorem.The known physical phenomena of quantisation, identical elementary particles and uncertainty principles seem to match the proposition that information is finite.
There is an interesting experiment described by Feynman in one of his 'Lectures on Physics' (Vol.3 Section 3-3.)
The experiment involved scattering of particles from a crystal. Neutrons in this example are fired at a crystal and produce a sharp diffraction pattern at a detector. The pattern however became blurred if the neutrons had their spins aligned opposite to the crystal atoms, in which case it was possible for a neutron to exchange spin with an atom and so leave a record of which atom it interacted with. As in the double-slit diffraction experiment detecting the path taken destroys the interference between the wavefunctions corresponding to the different available paths. The point is that in this example it appears that no human observer is needed, it is just the fact that the information has been extracted that matters (maybe also the fact that it is stored, it would be interesting to know how long the information needs to be stored, if it was erased before the neutron reached the final detector what would happen?) This seems to avoid the problem about Schrodinger's cat, that something special happens when a conscious observer gets involved. What this is forgetting of course is that there could be a superposition of different interactions with the crystal and different patterns at the detector, which only become a single result when an observer looks.
This is related to a problem often referred to as Wigner's friend. The process responsible for making the large scale world behave in a more or less classical way is known as decoherence. As an example of how this works, suppose I am locked in a room observing the result of some quantum experiment which could have result A or B. Professor Wigner will unlock the room and ask to see the result in a few minutes, but until then, having seen the result to be A I start thinking about the vector potential, usually represented by the letter A. Suddenly this chain of thought leads to a brilliant idea how to solve a problem professor Wigner has been stuck on for weeks. I look forward to telling him my idea, but then I realise that I am just part of a superposition up until that door opens, so how can I make sure it is the version of me with the good idea who he finds? The obvious answer is to shout and bang on the door so that he opens it. If that succeeds then it can only be one version of me he finds, the other version would not be making such a noise. This is what decoherence is all about, once there is interaction, or exchange of information between one of the superposed states and the external environment then the other states become highly improbable. The point about large scale observers is that they are extremely difficult to isolate from the environment. Schroedinger's cat, it has been suggested, would have a gravitational field which differs for a live cat and a dead cat, and so even an enclosure with thick solid walls would not be sufficient to prevent observable interaction with the outside world.The expectation that an object or a particle in a superposition of states has a detectable gravitational field suggests the field must itself exist as a superposition, and this is sometimes used as an argument for gravity being governed by the same QM laws as particles. This however leaves a problem unsolved, there is no end to the chain of superposition, at what point can we say a final measurement has taken place? Attempts to construct a theory of quantum gravity have serious difficulties, so is there a way to keep gravity classical? One feature of gravity is that it is far weaker than the other known forces, so we maybe need to ask whether there is some limit below which QM effects are not observable? The point is that for decoherance to occur it is necessary to extract information, so a sufficient level of field should be needed to allow the extraction of a single bit. The information content of a region of space containing a given energy density is what we need to know, and one proposed value is known as the Bekenstein Bound. The need for the minimum 1 bit available information suggests decoherence via gravitational interaction occurs at and above a certain mass level, and so QM effects only occur at a sufficiently small mass scale where there is insufficient gravitational interaction to cause decoherence. In this way gravity could in effect remain classical, it is only detectable at mass levels above which QM effects are not detectable.
So, what is the largest object known to exhibit superposition. An example described in Quantum Ground State and Single Phonon Control of a Mechanical Resonator used an object containing about a trillion atoms and with size about 40um to achieve a superposition of states, so this at least is still below any size limit. Given the extreme weakness of gravity however it is not too surprising if it fails to extract sufficient information to prevent the effect.
The states are just two levels of vibration energy so we need to know the gravitational information difference between these states, and a (very) approximate calculation suggests the energy difference between the states is small enough to ensure the 'gravitational information' should be well under one bit, and no gravitational decoherence would be expected. More to the point no gravitational superposition of states is needed because there is no detectable difference in gravitational effects caused by the two states of the resonator.No doubt there are other reasons to believe gravity is quantum mechanical, but if we accept that information is finite then classical gravity would not be expected to cause decoherence in superposition experiments of the type described here, and therefore is not in conflict with these observations.
This is not a new idea, something similar was suggested by Feynman In 1995 in his 'Lectures on Gravitation':
"I would like to suggest that it is possible that quantum mechanics fails at large distances and for large objects. Now, mind you, I do not say that I think that quantum mechanics does fail at large distances, I only say that it is not inconsistent with what we do know. If this failure of quantum mechanics is connected with gravity, we might speculatively expect this to happen for masses such that GM2/hc = 1, of M near 10-5 grams, which corresponds to some 1018 particles."
Here the mass he refers to is known as the 'Planck mass' which is also the mass of the smallest black hole mentioned earlier, corresponding to a single bit of information. Something similar was also suggested by Penrose, but his version refers to space-time curvature exceeding some specified level. I have yet to find anyone mentioning the relevance of the single bit limit.Where this all goes wrong is that a field with finite information content is not really a classical field. The finite information is a quantum mechanical effect, so it seems we already have assumed quantum gravity as soon as we assumed finite information applies to gravity. That assumes there are only two possibilities, classical or quantum mechanical. Maybe a 'classical' field with finite information is somewhere between these extremes. It is not immediately obvious that finite information leads unavoidably to the entire theory of quantum mechanics. If it did this then we would have a good candidate for an 'explanation' of quantum mechanics.
Alternatively even if gravity was classical and did not have finite information content then it could be that matter can only interact with the field in single bits, so the field could vary continuously from zero upwards, but only interact with matter when it reached a level capable of transferring one bit or more. There would then possibly be some level of 'background' energy unavailable for interaction locally, but which would add up over a large volume so that for example on a galactic scale there would be significant extra energy...
Another question about information. There are a number of quantities such as electric charge which are conserved, and which exist in about equal quantities of positive and negative, so that the total in the universe could be zero. The quantity of information is also believed to be conserved, but what is the total, and why would it be one value rather than another? It would be easier to explain if there existed negative information so that the total could be zero. We appear to have the same problem with mass, or energy, but there is a possibility of gravitational potential energy being negative, and the total energy of the universe could be zero. To take a much simplified universe, suppose there are just two objects with a gravitational attraction. Using the energy of the objects to increase the separation, suppose their total energy is needed to separate them to infinity, then we are left with nothing, the positive and negative energies have exactly cancelled. But what happens to the information content of the initial state? The final 'nothing' can have no information, so it has either not been conserved, or we have another alternative that gravitational information is negative.
Ok, so what would that look like in practice, would there be some obvious observable effect to distinguish between positive and negative gravitational information? My guess is that there would be nothing immediately obvious to be seen, but I remain unconvinced.
