From our physics color commentator Rob Sheldon:
a) Classical information is local. It is like beans in a bag, one, say, for each bushel of wheat. They are not connected to each other, each is independent of the other.
b) Quantum information is non-local. It depends on the orientation of the other beans. It is like beads on an abacus, or digits “in the 100’s column” that count differently than digits “in the one’s column”.
The information in (a) is calculated by combinations. The information in (b) is calculated with permutations.
If I have 3 identical beans, then the number of combinations is 0, 1, 2, 3, so it represents 2 “bits” of base-2 information. But if the positions matter, say, the beans are different colors, then I have 3! possibilities=> 3x2x1 = 6 pieces of information or 3 “bits” of base-2 information. The difference doesn’t seem like much but it grows really fast as the number of beans (and bins) grows.
One more example. The matter of the visible universe is equivalent to 10^80 protons. If we assume it comes with 10^80 electrons and 10^80 neutrons, we have 3×10^8 beans to play with. The classical information in that is roughly 10^80. Seth Lloyd of MIT does a calculation that gets something like 10^100 for classical information.
If every amino acid in a 100-chain peptide has 20 possibilities, that is 20^100 possibilities or much, much greater than the classical information in the entire universe.
If, however, the position of those protons matters, so A->B->C is not the same as A->C->B, then the information is closer to quantum, or (10^80)! = n! where n! is the number of permutations possible for n-items. Using Stirling’s formula, log(n!) ~ n log n – n. So exponentiating, we have n! ~ n^(n-1) => (10^80)^(10^80) which is quite a bit larger than all the peptides the universe might be able to manufacture in its lifetime. Therefore the answer to the question:
“Can the Universe have enough front-loaded information to explain the Cambrian Explosion?”
“Yes, if it is quantum information”.
What about QM information in DNA? Well, the sequence is pretty much fixed, so it would have to be located elsewhere–perhaps the methylation patterns that interact non-locally. Whatever it is, it would have to depend on permutations rather than combinations.
See also: Rob Sheldon: “Naturalness” in physics is dead, says Sabine Hossenfelder, and that’s a good thing