# What is the difference between classical and quantum information?

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?”

is,

“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

## 17 Replies to “What is the difference between classical and quantum information?”

1. 1
J-Mac says:

Quantum information conservation law says that information is neither created nor destroyed… So where did the information come from or how was it created in the first place…
Or more precisely I should probably ask:

How come the information was never created???

2. 2
jdk says:

First, if I have three beans, whether they are identical or not, and draw them out of the bag, there is only one way to do that, which is to draw all three beans. There is only one combination: 3C3 = 1.

I don’t understand where the 0, 1, 2, 3 comes from. That seems to be the answer to a different question: how many beans can I choose to draw from the bag, not how many ways can I draw three beans.

So I don’t think the example of beans in respect to “local” information is using the term “combinations” correctly.

I also don’t think that the phrase “quantum information” is being used properly here. Quantum information in physics is related to information held in quantum states: the key characteristic, I think, is that the information is not discrete, but continuous over a probabilistic spectrum. Permutations of discrete objects are not examples of quantum information.

Let’s accept 3 x 10^80 as the number of elementary particles in the world. If I have a deck of 60 different cards, they can be arranged in more ways than that.

There is nothing “quantum” about this. If there is a lesson here, it is the organization of items that is important in exploring the number of possibilities in a situation, and not just the raw number of items involved.

So, I think the OP is not being accurate to call the examples given an illustration between classical and quantum information. Perhaps Rob or someone can show me a source that would support this as accepted terminology, but I don’t think it is.

3. 3
J-Mac says:

jdk,

I think you make some valid points…
The way I would try to explain quantum information is this:
What information would have to be teleported if I were to teleport some halibut?

Wouldn’t be the information about the quantum arrangement of the subparticles that make up the halibut and their quantum states?

4. 4
jdk says:

Thanks, J-Mac. I’m glad you think there is some merit to my post.

Also, as far as my small amount of educated layperson’s knowledge goes, I think transporting halibut is way beyond the current state of investigation into the nature and use of quantum information.

5. 5
J-Mac says:

jdk

Also, as far as my small amount of educated layperson’s knowledge goes, I think transporting halibut is way beyond the current state of investigation into the nature and use of quantum information.

Of course… I was just using halubut as an example…
At least in theory it is still possible… So what is really teleported is the the quantum info via entanglement which is the arrangement of subparticles and their quantum states…

6. 6
Seversky says:

I see a lot about how information can be represented and stored but I don’t see a definition of what information actually is.

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.

So it is astronomically improbable that a specific “100-chain peptide” could spring into existence fully-formed in one step? That seems reasonable but so what? Is there any part of evolutionary theory that claims such a thing did or even could happen?

You and I are unique arrangements of uncounted trillions of sub-atomic particles. That such arrangements came into existence at this place and this time in the history of the Universe is astronomically improbable, So, therefore design? Well, you may have been but, unless my parents lied to me, I came about through the natural process of human procreation. So, what price design?

7. 7
J-Mac says:

Seversky,

…That seems reasonable but so what?
If I were you, I would change my login thingy… If I were our opponent, I wouldn’t waste my time with you…I’m glad you have admitted that… GBF

8. 8
EricMH says:

It isn’t clear to me how QM information serves the same purpose in Seth Lloyd’s calculation. In his calculation, there are 10^90 different bits that can be either 0 or 1. There are 10^120 different opportunities to configure one of these 10^90 bits. As such, we have 10^90*10^120=10^210=2^697.6 bits to play with.

If we have a bitstring of 700 bits, then there are 2^700 different sequences of 0s and 1s we can create that are 700 bits long.

So, with Seth Lloyd’s count of bits in the universe, we can enumerate about a quarter of the 0 and 1 sequences that can be represented with 700 bits. If the target we are trying to find is 700 bits long, then we have a 25% chance of finding it by brute force enumeration of sequences.

From my rudimentary knowledge of QM entanglement, it would seem we would have *fewer* bits to work with than in Seth Lloyd’s estimate, since particles are no longer capable of being oriented independently from each other. Instead, one particle’s orientation necessitates another particle has the opposite orientation. So, in an extreme case, if every particle is entangled with every other particle in the universe, then the entire universe would only provide 1 bit of information.

Can someone provide a similar example where if we have a 700 bit target, how we can use the plentiful QM information in our universe to find the target? From the OP, it sounds like QM information makes it much easier to enumerate bitstrings, but I cannot make the connection currently.

9. 9
J-Mac says:

EricMH,

Have you been able to look up the recent developments in quantum mechanics? It’s not a sin if you haven’t…
Just keep searching!

10. 10
EricMH says:

@J-Mac, can you list what you mean? I know of Bell’s inequality, and experiments demonstrating it is violated. Implying there is not some hidden state that is determining particle spin. There is also Conway’s free will argument proving that particle states are completely undetermined by the past, even in a probabilistic sense, if the observer is undetermined. But I do not see the connection of how these facts increase our information resources for finding targets. Entanglement appears to decrease information resources.

11. 11
jdk says:

I also am interested in what recent developments j-mac might be referring to. Care to elaborate, j-mac?

12. 12
jdk says:

Hi Eric. You write,

So, in an extreme case, if every particle is entangled with every other particle in the universe, then the entire universe would only provide 1 bit of information.

My understanding (although I could be wrong, and would like to know more if I am) is that particles are entangled when they are created at the same time from some source. As far as current theory goes, two particles in completely different places, with no recent common causal history, such as an electron in New York and one in India, are not going to be entangled. Entanglement and non-locality are fascinating properties that go counter against traditional, non-quantum particle physics, but they don’t pertain to large groups of particles out and about in different parts of the world.

This is my understanding, at least

13. 13
EricMH says:

Yes, that makes sense. It was only an example to show that entanglement reduces available Shannon information, instead of increasing available information.

14. 14
critical rationalist says:

I didn’t realize being a “color commentator” meant presenting bad analogies.

How can we distinguish classical and quantum information? By distinguishing which tasks are necessarily possible for information in classical and quantum systems.

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.

Shannon information is classical information. It requires specify tasks that are possible in classical systems. For example..

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”.

Beads in an abacus are classical information. The entire abacus has a counterfactual nature, which includes the possible existence of beads in a particular column and row. An abacus uses people as a kind of computer, that runs an algorithm and comes up with the result. This is entirely classical in nature because the algorithm can be preformed by a classical computer.

The information in (a) is calculated by combinations. The information in (b) is calculated with permutations.

Permutations are perfectly possible and common in classical systems. So, it’s unclear how this is unique to quantum information.

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.

Again, this is a counterfactual nature about what states something could have been in. A cog could be rotated in one of two positions, three positions, four positions, etc. However, in practice, there is a limit as to how fine a rotation can be made with errors coming into play. That is why cogs snap to specific rotations. Whether there are two positions or 1000, the behavior of snapping to a specific place, even when not precisely rotated to that place, represents digital information. Again, this still in the realm of classical information.

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.

Except, the type of information in a system does not change based on available resources. Specifically, information theory assumes infinite resources and how many “beans we get to play with” would depend on how many resources we have. If the universe was larger, would less information be quantum?

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.

Every amino acid can have 20 possibilities because it can exist in 20 different classically physical configurations. That is a counterfactual statement about amino acids and is completely classical in nature.

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.

Quantum information is not a function of the size of a bean or how it is “related” to other beans. Rather, at the very small scale, “beans” behave differently. Specifically, there are tasks that are not possible at the scale of quantum mechanics. So, you can’t just assume a greater number of beans will result in a linear increase in information density.

“Can the Universe have enough front-loaded information to explain the Cambrian Explosion?” is,
“Yes, if it is quantum information”.

This simply doesn’t make sense. Hopefully this is News’ very colorful commentary on UD’s color commentator’s comments in some other post? For example…

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.

Huh? Apparently, by “fixed” you mean already serves a purpose, and that cannot be used to store other information at a more fundamental level. But that’s simply not true either.

You can store different information in a JPEG file despite the fact that JPEG files were designed to store images, not other information.

Quantum information is distinguished from classical information by the fact that some physical tasks are not possible at the small scale. At which point Shannon’s theory become an approximation as it cannot explain information at the quantum scale.

This is in the same sense that Newton’s laws are an approximation. While it can explain how we can launch rockets into space, it cannot explain how to keep a GPS satellite in sync with GPS ground receivers. Generally reality can explain both the rocket launch and GPS satellite. Newton’s laws cannot.

15. 15
EricMH says:

@CR, I believe the basic point Dr. Sheldon is making is that classical information counts individual items of a certain type, while Quantum information counts groups.

If we a have a bitstring of length 10, then the first kind of information would count how many different amounts of 0s and 1s can be represented with the bitstring, irrespective of order, which trivially is 10. I.e. there can be a bitstring with zero 1s, a bitstring with one 1, etc. all the way up to a bitstring with ten 1s. Dr. Sheldon calls this classical information.

The second type of information counts how many different groupings of bits are possible, which is 2^10. Dr. Sheldon calls this quantum information, since entangled particles form groups.

I believe Dr. Sheldon is claiming that if groupings of bits matters, then Lloyd’s 10^90 bits becomes 2^(10^90) available groupings for Quantum information.

So, in response to my previous query, Dr. Sheldon is not claiming quantum information increases the number of available trials. He is claiming quantum information dramatically increases the availability of encodings in the universe, which is true.

However, it is less clear how this increases the amount of information that can be embedded in the universe. What I think is being assumed is that a single particle can be a member of different entangled groups. If that is true, and the number of possible groups per particle is unlimited, then 10^90 particles provides 2^(10^90) bits of quantum information. The quantum information bits are set according to which entangled groups are instantiated in our universe.

This is conceptually plausible, and does provide a source for the incredible amount of classical information in DNA, since the classical information cannot come from random mutation and selection.

16. 16
critical rationalist says:

@CR, I believe the basic point Dr. Sheldon is making is that classical information counts individual items of a certain type, while Quantum information counts groups.

As I pointed out, classical information can “count groups” as well. As such, it’s unclear how that can be a key difference between classical and quantum information.

If we a have a bitstring of length 10, then the first kind of information would count how many different amounts of 0s and 1s can be represented with the bitstring, irrespective of order, which trivially is 10

Why can’t you take the order into account in the case of classical information?

The second type of information counts how many different groupings of bits are possible, which is 2^10. Dr. Sheldon calls this quantum information, since entangled particles form groups.

Yet, “forming groups” is possible in classical systems.

I believe Dr. Sheldon is claiming that if groupings of bits matters, then Lloyd’s 10^90 bits becomes 2^(10^90) available groupings for Quantum information.

“grouping of bits” isn’t what make quantum information “quantum”. What makes it quantum is that classical information theory simply isn’t capable of explaining it. This is because specific physical tasks cannot be performed on it in respect to measurements.

So, in response to my previous query, Dr. Sheldon is not claiming quantum information increases the number of available trials. He is claiming quantum information dramatically increases the availability of encodings in the universe, which is true.

Classical information is to quantum information as Newton’s laws of motion is to Einstein’s general relativity.

We can use Newton’s laws of motion to launch rockets into space, but it is only an approximation. Only Einstein’s general relativity can be used to explain GPS satellites and falling apples. Newton’s laws are scale dependent in the case of velocity. In the same sense, we can only use classical information in the case of classical systems, like classical computers. However, that is an approximation. Only quantum information can be used to define information at both the very small scale and the large scale. Classical information is scale dependent in the case of size.

Furthermore, all the rest of those “beans” are not in our classical universe. They are in other classical universes that are nearly identical to ours. They differer in respect to the states of those same “beans.”, but are otherwise duplicate classical universes. So, quantum information refers to information that is defined across multiple classical universes across the multiverse. This is why quantum computers can perform computations requiring more resources than we have particles in our universe, such as factoring numbers that would take classical computers exponentially more time.

This is conceptually plausible, and does provide a source for the incredible amount of classical information in DNA, since the classical information cannot come from random mutation and selection.

Quantum information is not a source of classical information. Classical information is an approximation of information at a specific scale. That would be like saying General Relatively is the source of Newton’s laws of motion. Both ideas would represent a category error.

17. 17
EricMH says:

@CR yes, at some point this ends up being a semantics issue. You are right, we can talk about grouping and such in classical information theory. In this case, the terms are just being used to distinguish two different ways information can be embedded in the universe, one giving 2^90 bits and the other giving 2^(2^90) bits, as in your quantum computer example. To avoid semantic discussions, the first kind of information without entanglement is info1 and the second with entanglement is info2. So, if a given DNA strand is made from 256 info1 bits, we only need 8 info2 bits to encode the strand. If ordering is taken into account for info2, then even fewer info2 bits are needed.