No cloning theorem: the double edge sword

Essentially, there are 2 types of cloning…

Biological cloning leading to clones such as identical twins who share exactly the same DNA.

Or artificial, genetically engineered cloning leading to such clones as plants whose DNA is also identical.

But there exists a more precise kind of cloning in physics that reaches all the way to the subatomic level of particles. Everything in the universe is made up of elementary quantum particles and the forces by which they interact, including DNA and us. This kind of cloning is more detailed because it involves the superposition of subatomic particles; their relative positions (particles can be in more than 1 position or state at the time), momenta and energy levels of every particle and all of their bonds and interactions are exactly the same in the copy (clone) as the original.

This kind of perfect cloning is impossible. It has been proven mathematically and formulated into the no cloning theorem, which states:

“In physics, the no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state.”

How those two types of cloning apply to life systems, such as us, our DNA and so forth?

As the example of identical twins or genetically cloned plants or animals indicate, biological clones are identical as far as their DNA, but they are not identical on subatomic level, where their quantum correlations, relative positions, momenta and energy levels their bonds and interactions are NOT exactly the same in the copy (clone) as in the original.This means that each time the cell divides, making the exact copy of itself, including DNA, the new copy could be identical biologically (it doesn’t have to be) but it is definitely not identical on subatomic level due to no cloning theorem, that prohibits such identical cloning.

So, what does it all mean?

There are many implications due to these very facts but I would like to focus on one in this OP – evolution.

The Darwinian theory of evolution relies on the premise that for evolutionary processes to accomplish their “goals” of building new structures, organs and life forms, the information in the life system undergoing evolution has to continue to increase. However, due to the no cloning theorem, even an identical copy of the a cell and it’s DNA can’t be made, not to mention a copy of a cell or DNA with an increased quantum information. So, how could evolution progress then if the quantum information cannot increase?

How about mutations?

Darwinists claim that mutations, the errors in the biological processes of the cell attempting to make an identical copy of itself, including DNA, lead to some genetic variation, which lead to an increase of information.

Actually, it’s the opposite.

Scientists have put this idea to the experimental test. It was discovered that it’s possible to “clone” a qubit of information with an average of 83% fidelity. This means that on average 17% of quantum information was lost each time a qubit of information was “cloned”…

I just don’t know where the additional information would come from for Darwinism to be even close to its abilities to increase information, if each time a qubit of information is “cloned”, such as in DNA, there occurs a loss of information and not an increase…

The only evidence I see in all this is proof for devolution and not evolution, just as Behe observed it, and wrote about it in his new book Darwin Devolves.

It follows that the loss of quantum information with each cell and DNA replication is gradually leading to more mutations and more damaging ones, especially if they are artificially selected for, just like in dog breeding in the devolution of wolf to chihuahua…

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32 thoughts on “No cloning theorem: the double edge sword

  1. “In physics, the no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state.”

    As best I can tell, this has no consequences at all for biology.

    The Darwinian theory of evolution relies on the premise that for evolutionary processes to accomplish their “goals” of building new structures, organs and life forms, the information in the life system undergoing evolution has to continue to increase.

    That is surely mistaken.

    Even if we accept the dubious view of information used by ID proponents, that at most requires some new information. It does not deny that some existing information could be lost in the transition.

    The only evidence I see in all this is proof for devolution and not evolution, just as Behe observed it, and wrote about it in his new book Darwin Devolves.

    Then I guess we must all be devolved bacteria. Viva la devolution.

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  2. I’d like to take this opportunity to thank Gordon Davisson for bringing up this subject…
    I have always wanted to do this OP but I was lacking the motivation to research the details, so that those who lack the fundamental knowledge of quantum mechanics could understand the issue…
    It’s easy for me to picture things in quantum world, such as no time factor, effect before cause or quantum shadow information in doublehelix.. Explaining it in easy terms is a challenge…

    For those who are fascinated with quantum mechanics like I am, although making identical copies of ourselves is not possible, the good news is quantum teleportation via entanglement still is a possibility… 😉

    Some may remember my old OP on “How Did The Designer/God do it?”

    http://theskepticalzone.com/wp/how-did-the-designergod-do-it/

    I have learned few things since then 😁

    While quantum creation still looks like the best mechanisms known, there are other issues that have come to light that may only be explained by some animating matter energy, like dark energy…

    Unfortunately, I’m only planning to do one more OP and I will have to take a break from blogging…I’m tempted to do a newer version of How God Did It plus quantum resurrection/quantum consciousness, but I won’t have the time to research it…

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  3. J-Mac:

    I’d like to take this opportunity to thank Gordon Davisson for bringing up this subject…

    Why not show your appreciation by answering his questions instead of continuing to dodge them?

    He asked you:

    Yes, I do know what a quantum state is. I took a year of graduate-level QM while I was in college. I don’t claim to be an expert (and I’m certainly rusty), but I’m familiar with the basics and relatively used to thinking in quantum terms.

    Do you understand QM at even a basic level? Can you actually say anything more specific than “it’s quantum information!” like some sort of magic incantation, and quote other people on the subject? Do you know how to think in quantum terms?

    Here’s a quick quiz about QM, quantum information, and the relevance of the no cloning theorem (in a simple case). It doesn’t require any particularly deep understanding, just a basic knowledge of the notation, jargon, and some algebra. If you actually understand QM as well as you seem to think, this should be easy for you.

    (Ok, there’s a kind of annoying amount of algebra, to the point where I can’t guarantee I didn’t lose track of the sign of any terms when I worked through it. My goal isn’t to make you do the algebra, but to make you prove you know what algebra to do, and needing to work through it is just part of proving that. ‘Course, if you’re a real expert, you’ll probably just set up the relevant transformation matrix and let Mathematica do everything for you…)

    Suppose we have a a qbit, with the usual basis states |0⟩ and |1⟩. Suppose we have a qbit “copier” that, given a gbit in the state |0⟩ produces a pair of qbits in the state |0⟩|0⟩, and given one in the state |1⟩, produces a pair in the state |1⟩|1⟩. Straightforward so far, right?

    (I did think about doing this with a 4-state system with {|A⟩, |C⟩, |G⟩, |T⟩} as its basis, but it just made the algebra sloggier.)

    Now, let’s define two other states for our qbits, |+⟩ = (|0⟩ + |1⟩)/sqrt(2) and |-⟩ = (|0⟩ – |1⟩)/sqrt(2). These are orthogonal and unitary, so they form another basis for our qbit’s state space.

    Question 1: Suppose we fed a qbit in the state |+⟩ into the “copier”. What state would the pair of qbits it produces be in?

    Question 2: Suppose, given two qbits in the state from question 1, we measured each of them in the {|+⟩, |-⟩} basis. What’re the possible combinations of results we might get and their probabilities?

    Question 3: Suppose we lost the copy (i.e. the second qbit). Is there any test we can perform on the first qbit that’ll distinguish it from being in a random state? If so, what is it?

    So there’s your challenge. Either answer the questions, or admit that you don’t understand QM well enough to.

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  4. Neil Rickert:
    “In physics, the no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state.”

    As best I can tell, this has no consequences at all for biology.

    Are you suggesting that biological life systems are exempt from the no cloning theorem?

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  5. Neil Rickert:
    The Darwinian theory of evolution relies on the premise that for evolutionary processes to accomplish their “goals” of building new structures, organs and life forms, the information in the life system undergoing evolution has to continue to increase.

    That is surely mistaken.

    Wishful thinking instead?

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  6. J-Mac, in the OP:

    I just don’t know where the additional information would come from for Darwinism to be even close to its abilities to increase information, if each time a qubit of information is “cloned”, such as in DNA, there occurs a loss of information and not an increase…

    Dude, where did you get the (batshit) idea that DNA replication was equivalent to quantum cloning?

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  7. Neil Rickert:
    The Darwinian theory of evolution relies on the premise that for evolutionary processes to accomplish their “goals” of building new structures, organs and life forms, the information in the life system undergoing evolution has to continue to increase.

    That is surely mistaken.

    Wishful thinking instead?

    keiths: Suppose we have a a qbit, with the usual basis states |0⟩ and |1⟩. Suppose we have a qbit “copier” that, given a gbit in the state |0⟩ produces a pair of qbits in the state |0⟩|0⟩, and given one in the state |1⟩, produces a pair in the state |1⟩|1⟩. Straightforward so far, right?

    Lol
    This is all suppositions based on no experimental tests..
    The perfect example that supposition and tests are nonsense is that when you make a copy of a qubit, as perfect clone as possible , you will any achieve 83% fidelity…

    I linked the experiment in the OP…
    Stop embarrassing yourself!

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  8. Here we go
    The ability to copy information is fundamental for
    many processes in distributing and dealing with data.
    While classical information can often be copied many
    times without any significant loss, the cloning of quantum
    information is seriously restricted by the laws of quan-
    tum mechanics. This limitation is known today as the
    non-cloning theorem [1] and has found its applications
    in quite different fields of quantum information theory,
    such as quantum computation [2, 3] and quantum cryp-
    tography [3, 4].
    In processing quantum information, formally, any de-
    vice that provides for M unknown input states of a given
    system N (> M) output states (of the same or some
    analogue systems) is called a M → N quantum cloning
    machine (QCM). To ensure a proper quantum behaviour,
    such machines are usually represented as unitary trans-
    formations; they are called symmetric if the N output
    states are identical to each other, and are said to be
    nonsymmetric otherwise. In this work, we shall consider
    symmetric 1 → 2 QCM that provides for a (pure) input
    state |sai (of qubit a) the two qubits a and b in the
    same output state ρ
    out
    a = ρ
    out
    b
    , and where ρ
    out denotes
    the (reduced) density matrix of the corresponding qubit.
    For an ideal transformation, of course, we might expect
    |sia
    |0i
    b
    |Qi
    c −→ |sia
    |si
    b

    c

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  9. Several QCM’s have been discussed in the literature
    before [4–7]; Buˇzek and Hillery [5], for example, worked
    out a symmetric 1 → 2 QCM that provides copies with
    the fidelity F = 5/6 ≈ 0.83 independent of the given
    input state. This state-independent transformation,
    which is now known as universal QCM, has found re-
    cent attention and application in quantum cryptography,
    namely, in optimal eavesdropping attack [8] within the
    six-state protocol [9]. For some protocols in quantum
    cryptography, however, only few states {|sai} are (pre-)
    selected from the Bloch sphere and, hence, one may wish
    to find a state-dependent QCM that provides copies with
    higher fidelity than the universal Buˇzek-Hillery machine
    for this particular set of states

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  10. Neil Rickert: Even if we accept the dubious view of information used by ID proponents, that at most requires some new information. It does not deny that some existing information could be lost in the transition.

    Both classical and quantum information get lost in translation…
    Proving that new information can be created by blind, random processes is fundamental.
    No cloning theorem poses one of those changes for Darwinism…

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  11. J-mac :
    The only evidence I see in all this is proof for devolution and not evolution, just as Behe observed it, and wrote about it in his new book Darwin Devolves.

    Neil Rickert
    Then I guess we must all be devolved bacteria. Viva la devolution.

    This not only a problem for Darwinism. It is a problem for those from ID, like Behe who subscribe to common descent.
    I don’t have that problem. The know cloning theorem says common descent is impossible…unless someone can explain how biological systems break that law…

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  12. J-Mac,

    You’re not even reading your own copy-pasta, which clearly states

    While classical information can often be copied many times without any significant loss, the cloning of quantum information is seriously restricted by the laws of quantum mechanics.

    DNA replication is not equivalent to the cloning of quantum states.

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  13. The answer is simple and obvious: quantum information is not copied during cellular division (including DNA replication); only classical information is copied. All this stuff about quantum information is your own personal delusion, with no connection to reality.

    If replication of quantum information were needed, it wouldn’t just be a problem for evolution, it’d be a huge problem for the development of multicellular organisms. Making all of the cells that make up a large metazoan like a human requires a huge number of replications in each cell lineage; if there’s a significant information loss with each cell division, there’d be basically no information left in the cells that make up an adult organism.

    And of course, the same is true between generations of organisms. Each egg and sperm cell is the result of a number of divisions, so any quantum information will be basically gone in a single generation. Not degraded, gone.

    But of course there’s no similar problem with classical information.

    BTW, the paper you cited about copying qbits with 83% fidelity only works that well with qbits in the “Eastern meridian” — the line on the Bloch sphere running from |0⟩ through |+⟩ to |1⟩. It won’t work nearly as well for other states, like |-⟩ or |+i⟩ or |-i⟩. This is like making a GPS system that only works semi-well for longitude=90° East, but fails in both East and West of that meridian. And I see no sign that they “put this idea to the experimental test”, just that they analyzed it theoretically.

    The reason copying classical info isn’t problematic is that it only requires the equivalent of copying the |0⟩ and |1⟩ states — not |+⟩, not |-⟩, no meridians or anything. Copying those states can be done with arbitrarily close to 100% fidelity.

    Some other random points while I’m at it: You said

    The Darwinian theory of evolution relies on the premise that for evolutionary processes to accomplish their “goals” of building new structures, organs and life forms, the information in the life system undergoing evolution has to continue to increase.

    This is sort-of-right, but garbled to the point of actually being wrong. First, evolution doesn’t have a direction. Evolution is change. Changes that increase information (by whatever measure you happen to like), changes that decrease information, changes that leave the amount of information the same, whatever. Now, if you look at the entire lineage of a particular organism, you’ll find that for many definitions/measures of information there’s been a net increase over that entire history, but that doesn’t mean it’s continuous or uniform, and it’s certainly not an inherent part of the definition of evolution.

    Also, to the extent that evolution does need to add additional information, that has nothing to do with accurate copying of old information. Copying existing information doesn’t add information. Adding new information adds new information. It’s a different thing.

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  14. J-Mac: Are you suggesting that biological life systems are exempt from the no cloning theorem?

    I’m suggesting that biological systems are not doing anything that depends on quantum cloning.

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  15. J-Mac: Proving that new information can be created by blind, random processes is fundamental.

    Yes, it is fundamental — but only to the misunderstanding of biology by ID proponents.

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  16. J-Mac: [quotes keiths quoting my question about a qbit “copier” that works for the |0⟩ and |1⟩ states]

    Lol
    This is all suppositions based on no experimental tests..
    The perfect example that supposition and tests are nonsense is that when you make a copy of a qubit, as perfect clone as possible , you will any achieve 83% fidelity…

    I linked the experiment in the OP…
    Stop embarrassing yourself!

    Go look at the paper you linked. Look at the section about the Wootters-Zurek transformation (page 3, middle of the left column). Compare it to the copier I assumed.

    Hint: they’re the same, other than them using more explicit notation.

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  17. Gordon Davisson:
    The answer is simple and obvious: quantum information is not copied during cellular division (including DNA replication); only classical information is copied. All this stuff about quantum information is your own personal delusion, with no connection to reality.

    If replication of quantum information were needed, it wouldn’t just be a problem for evolution, it’d be a huge problem for the development of multicellular organisms. Making all of the cells that make up a large metazoan like a human requires a huge number of replications in each cell lineage; if there’s a significant information loss with each cell division, there’d be basically no information left in the cells that make up an adult organism.

    And of course, the same is true between generations of organisms. Each egg and sperm cell is the result of a number of divisions, so any quantum information will be basically gone in a single generation. Not degraded, gone.

    But of course there’s no similar problem with classical information.

    BTW, the paper you cited about copying qbits with 83% fidelity only works that well with qbits in the “Eastern meridian” — the line on the Bloch sphere running from |0⟩ through |+⟩ to |1⟩. It won’t work nearly as well for other states, like |-⟩ or |+i⟩ or |-i⟩. This is like making a GPS system that only works semi-well for longitude=90° East, but fails in both East and West of that meridian. And I see no sign that they “put this idea to the experimental test”, just that they analyzed it theoretically.

    The reason copying classical info isn’t problematic is that it only requires the equivalent of copying the |0⟩ and |1⟩ states — not |+⟩, not |-⟩, no meridians or anything. Copying those states can be done with arbitrarily close to 100% fidelity.

    Some other random points while I’m at it: You said

    This is sort-of-right, but garbled to the point of actually being wrong. First, evolution doesn’t have a direction. Evolution is change. Changes that increase information (by whatever measure you happen to like), changes that decrease information, changes that leave the amount of information the same, whatever. Now, if you look at the entire lineage of a particular organism, you’ll find that for many definitions/measures of information there’s been a net increase over that entire history, but that doesn’t mean it’s continuous or uniform, and it’s certainly not an inherent part of the definition of evolution.

    Also, to the extent that evolution does need to add additional information, that has nothing to do with accurate copying of old information. Copying existing information doesn’t add information. Adding new information adds new information. It’s a different thing.

    Thanks Gordon.

    If I may ask a very elementary question

    From Wiki
    https://en.wikipedia.org/wiki/Quantum_information

    In physics and computer science, quantum information is the information of the state of a quantum system.

    What is a quantum system as it is discussed here. Going back to the pedagogical example of a particle in a box

    https://en.wikipedia.org/wiki/Particle_in_a_box

    What is the quantum information here? I’m sorry for the elementary question, but it’s one of those that I should know, but didn’t learn!

    Thanks in advance!

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  18. Gordon,

    If I can give my likely errant take on some of the changes going form quantum to classical, when I apply a Hermitian operator to a quantum system like a particle in a box, I extract observable data — like the energy of the system.

    There may be two boxes with a particle in each box, and when I apply the Hermitian Operator like the Energy operator, I can ascertain the energy of each particle in each box.

    Hypothetically we can have a “copy” the process of making several boxes each with the same kind of particle at the same kind of energy level as determined by the Hermitian operator. However something like exact position and momentum can’t be copied for each particle.

    We can only get them in the same state that can be observed through the Hermitian operator.

    Is that sort of right?

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  19. The particle in a box wiki entry gives one biological exmaple regarding the Beta-Carotene protein.

    https://en.wikipedia.org/wiki/Particle_in_a_box

    The way I understand it is that what can be “copied” are things like energy levels, which can be extracted mathematically from a quantum system using things like a Hermitian Operator, and then practically and simply by measuring instruments that measure the wavelength of a photon going in and out of a system.

    So hypothecially I can “copy” the energy states of two independent Beta Carotene molecules simply by shooting light of the same wavelength into them (presuming the Beta Carotenes started at the same ground state).

    I cannot however copy the position and momentum of the electrons relative to their respective nuclei from one Beta Carotene to another. This would violate Heisenberg Uncertainty.

    How this relates to the no-cloning theorem, I don’t know, but I suspect it is related.

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  20. stcordova: What is a quantum system as it is discussed here. Going back to the pedagogical example of a particle in a box

    https://en.wikipedia.org/wiki/Particle_in_a_box

    What is the quantum information here? I’m sorry for the elementary question, but it’s one of those that I should know, but didn’t learn!

    Hmm, lemmie try to explain it via a sort of history of information theory. In the beginning, there was combinatorial information theory, which measured information in terms of the number of possibilities. For instance English has 26 letters, so it can encode n^26 different words with n letters.

    Then came Claude Shannon and statistical information theory, which refined this by taking into account not only how many possibilities there were but what their probabilities were. For example, English has 26 letters but some are hardly over used (Q, Z, …), and some combinations are more common than others etc… and this decreases the amount of information it carries. This theory was originally intended to be applied to messages in a communication system, but it turned out that it could be applied to pretty much anything that involved random variables and probability distributions.

    Note that in both of these theories, what happens isn’t as important as what else could’ve happened. You can’t talk about the information in a particular message (or state, or whatever) unless you also talk about all of the other messages (/states/etc) that were also possible.

    As I said, you can apply statistical information theory to anything you have a probability distribution of, but if you apply it to a distribution of quantum states things get weird. A single specific quantum state (“pure state”) can include a superposition of multiple basis states. When we add a probability distribution, we’re talking about a distribution that can include multiple superpositions (with different probabilities), each of which can include multiple basis states (with different probability amplitudes). Essentially, there are two different levels of multiple-possibilities here, a classical one (the probability distribution) on top of a quantum one (superpositions). And these two levels don’t always stay cleanly separate.

    The standard way to represent one of these “mixed” states — states that have both classical and quantum uncertainty — is with a density matrix. These are the bastard offspring of probability distributions and quantum state vectors, and while I understand both distributions and state vectors, I’ve never properly wrapped my head around density matricies.

    Anyway, to answer your question about the particle in a box: if the particle is in a specific pure state (even if it’s a complicated superposition), there’s no information there. In order for there to be information, there has to be some classical-style uncertainty as well as quantum uncertainty. But if it might be in several different pure states (say, B, C, D, E, or F in the Wikipedia figure) — whether you’re using the states to encode meaningful information or it’s just random — then you have information stored there.

    Concerning copying quantum information via observations: what you’re describing there is essentially destroying (collapsing or decohering) the the quantum-level uncertainty in a system, extracting the classical-level information, and copying that. You can certainly do that, but you’re losing the quantumness in the process. The really interesting interactions let quantum systems interact without collapsing anything (and the list of possibilities here does not include copying).

    If you want to get an idea what operations you can do with quantum information, I’d try to find a good intro to quantum computing, and quantum logic “gates” in particular. IBM Q used to have a good tutorial on the standard gates, but I can’t find it anymore (though they still have a good circuit simulator you can play around with). Wikipedia has a list of common logic gates, but it’s not really … introductory. The detail article on the controlled NOT gate might give you an idea of the sorts of things you can do.

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  21. J-Mac:
    Proving that new information can be created by blind, random processes is fundamental.

    There is no need to prove that, because nature isn’t blind nor is it random (at least, not totally).

    As an Earth Scientist I know for fact that natural (geological) processes create (and destroy) information all the time. All because of the laws of nature plus the vagaries of the ordinary variation in the boundary conditions. For all the babble about ‘quantum cloning’ I see absolutely no reason why the same can’t happen in biology.

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  22. Neil Rickert: Even if we accept the dubious view of information used by ID proponents

    Is there a single view of information used by ID?

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  23. BruceS: Is there a single view of information used by ID?

    No, and the ID community should stop using or de-emphasizing the term.

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  24. stcordova: From Wiki
    https://en.wikipedia.org/wiki/Quantum_information

    In physics and computer science, quantum information is the information of the state of a quantum system.

    What is a quantum system as it is discussed here. Going back to the pedagogical example of a particle in a box

    https://en.wikipedia.org/wiki/Particle_in_a_box

    What is the quantum information here? I’m sorry for the elementary question, but it’s one of those that I should know, but didn’t learn!

    Hmm, lemmie try to explain it via a sort of history of information theory. In the beginning, there was combinatorial information theory, which measured information in terms of the number of possibilities. For instance English has 26 letters, so it can encode n^26 different words with n letters.

    Then came Claude Shannon and statistical information theory, which refined this by taking into account not only how many possibilities there were but what their probabilities were. For example, English has 26 letters but some are hardly over used (Q, Z, …), and some combinations are more common than others etc… and this decreases the amount of information it carries. This theory was originally intended to be applied to messages in a communication system, but it turned out that it could be applied to pretty much anything that involved random variables and probability distributions.

    Note that in both of these theories, what happens isn’t as important as what else could’ve happened. You can’t talk about the information in a particular message (or state, or whatever) unless you also talk about all of the other messages (/states/etc) that were also possible.

    As I said, you can apply statistical information theory to anything you have a probability distribution of, but if you apply it to a distribution of quantum states things get weird. A single specific quantum state (“pure state”) can include a superposition of multiple basis states. When we add a probability distribution, we’re talking about a distribution that can include multiple superpositions (with different probabilities), each of which can include multiple basis states (with different probability amplitudes). Essentially, there are two different levels of multiple-possibilities here, a classical one (the probability distribution) on top of a quantum one (superpositions). And these two levels don’t always stay cleanly separate.

    The standard way to represent one of these “mixed” states — states that have both classical and quantum uncertainty — is with a density matrix. These are the bastard offspring of probability distributions and quantum state vectors, and while I understand both distributions and state vectors fairly well, I’ve never properly wrapped my head around density matricies.

    The way I’d think of it is that quantum information is a generalization of classical (Shannon-style) statistical information, where the classical probability distribution might get tangled up with quantum uncertainty. Conversely, classical information can be considered a special case of quantum information, where the classical level of uncertainty doesn’t mix with a quantum level.

    Anyway, to answer your question about the particle in a box: if the particle is in a specific “pure” state (even if it’s a complicated superposition), there’s no information there. The “information” is in the classical probability distribution level, so in order for there to be information, there has to be some classical-style uncertainty as well as quantum uncertainty. But if it might be in several different pure states (say, B, C, D, E, or F in the Wikipedia figure) — whether you’re using the states to encode meaningful information or it’s just random — then you have information stored there.

    Concerning copying quantum information via observations: what you’re describing there is essentially destroying (collapsing or decohering) the the quantum-level uncertainty in a system, extracting the classical-level information, and copying that. You can certainly do that, but you’re losing the quantumness in the process. The really interesting interactions let quantum systems interact without collapsing or decohering anything (and the list of possibilities here does not include copying).

    If you want to get an idea what operations you can do with quantum information, I’d try to find a good intro to quantum computing, and quantum logic “gates” in particular. IBM Q used to have a good tutorial on the standard gates, but I can’t find it anymore (though they still have a good circuit simulator you can play around with). Wikipedia has a list of common logic gates, but it’s not really … introductory. The detail article on the controlled NOT gate is worth a read-through.

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  25. Thank you Gordon! I’m realizing I have so much re-learning of QM as I read your response.

    It’s nice seeing you visit here again.

    Anyway, to answer your question about the particle in a box: if the particle is in a specific “pure” state (even if it’s a complicated superposition), there’s no information there.

    If I’m understanding this, the Beta-Carotene protein (a real world approximation of the particle in a box) might hypothetically be at some reference state but then absorbs a photon of specific wavelength, we would assume the system went from one pure state to another and hence no quantum information there — although classically we might consider this flipping the state of a bit and thus changing the classical information.

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  26. I just realized something looking through my textbook by Griffiths, pretty much all of the examples were in pure states! I guess the math was hard enough to do when just dealing with pure states.

    I looked at the index and at least some of the terminology in this discussion wasn’t there like “Pure State” or even the word “Superposition”. My prof was really into the math, and also, I’m realizing now just how elementary and introductory Griffith’s book was!!

    The chapter’s were:

    1. The Wave Function (Schrodinger Equation, Statistical Interpretation, Probability, Normalization, Momentum, The Uncertainty Principle)

    2. Time-Independent Schrodinger Equation (Stationary State, The Infinite Square Well, The Harmonic Oscillator, the Free Particle, the delta-function potential, the finite square well)

    3. Formalism (Hilbert Space, Observables, Eigenfunctions of a Hermitian Operator, Generalized Statistical Interpretation, The Uncertainty Principle, Dirac Notation)

    4. Quantum Mechanics in Three Dimensions (The Schrodinger Equation in Spherical Coordinates, The Hydrogen Atom, Angular Momentum, Spin)

    5. Identical Particles (Two Particle Systems, Atoms, Solids, Quantum Statistical Mechanics)

    6. Time-Independent Perturbation Theory (Nondegenerate Perturbation Theory, Degenerate Perturbation Theory, The Fine Structure of Hydrogen, the Zeeman Effect, Hyperfine Splitting)

    7. The Variational Prinicple (Theory, The Ground State of Helium, The Hydrogen Molecule Ion)

    8. The WKB Approximation (The “Classical” Region, Tunneling, The Connection Formulas)

    There were more chapters but the class didn’t get beyond these listed.

    Now it’s coming back to me. It’s been a awhile!

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  27. Thanks, Sal. I agree, it’s good to go through some of this again — I think about QM from time to time, but don’t actually go through the math often enough to keep it fresh in my mind. And I think you’re right about standard treatments of QM not getting into quantum information theory — it’s a whole ‘nother layer of complication that’s not at all necessary to understand how QM applies to pure states. Mind you, I haven’t studied quantum statistical mechanics at all, but I’m pretty sure there’s a lot of overlap between that and quantum information theory, so I think that’d be the normal route to run into it.

    As for the Beta-Carotene protein example you brought up: whether there’s any information there depends on whether there’s multiple options about what might happen. If a molecule starts in its ground state and there’s nothing like light of the right wavelength hitting it, it’ll necessarily stay in the ground state so there’s no information. Similarly, if you deterministically drove it into the excited state, again there’s only one possible resulting state and no information.

    But if, say, you flipped a coin and based on that decided whether to leave it in the ground state or drive it into the excited state, then you’d be putting information into the state of the Beta-Carotene protein. Since Beta-Carotene (like all real physical things) is a quantum system this would technically be quantum information. But since the fact that the information isn’t really doing anything interesting at the quantum level, we could get away with treating it as classical information just fine.

    A more interesting variation occurs to me, though: suppose we (deterministically) fired a photon with the right energy at a Beta-Carotene protein. The protein will have some cross section for absorbing the photon (and switching into its excited state). Classically, that’d correspond to a probability that the photon will get absorbed (vs. just passing by without interacting), but in QM it’s going to deterministically produce a superposition of the absorbed and passed-by states. If a and b were the probability amplitudes for the two outcomes, the resulting state would be something like:

    a * |protein in excited state⟩|photon gone⟩ + b * |protein in ground state⟩|photon still going⟩

    Note that this is a pure state, so there’s no quantum information here. More technically, if you constructed the corresponding density matrix, its Von Neumann entropy (which is basically a generalization of Shannon entropy to quantum systems) will be zero.

    But if you consider the two pieces — the protein and the photon (/EM field where the photon would be) — separately, the entangled superposition turns into classical-style uncertainty about the specific states. The protein will be in |protein in excited state⟩ with probability |a|^2, and |protein in ground state⟩ with probability |b|^2, and will have a Von Neumann entropy entropy of – |a|^2 * log2(|a|^2) – |b|^2 * log2(|b|^2), which is greater than zero. The photon (/space) will have the same entropy.

    The entanglement has turned into information!

    This is actually one of the general features that distinguishes quantum information from classical information. In the Shannon theory, if you look at the joint entropy of two things (that is, the entropy of them taken together), it’ll be greater than or equal to the higher of their individual entropies, and less than or equal to the sum of their individual entropies. The joint entropy is equal to the sum when the two things are statistically independent, and equal to the larger of the two when they are maximally correlated. Essentially, if there’s information duplicated between the two, then it only counts once when you look them together.

    But with quantum information, the joint entropy of two things can be less than their individual entropies; the minimum it can be is the absolute value of the difference between their entropies. This is because entanglement between the two decreases the joint information even more than classical correlation does.

    A quick example: suppose you have two messages (or states or whatever), with entropies of 7 and 10 bits. With both classical and quantum information, the maximum their joint entropy might be is 17 bits. With classical information, the minimum the joint entropy could be is 10 bits (if the 7-bit message completely duplicates information in the 10-bit message). But with quantum information, the minimum their joint entropy could be is only 3 bits, if they’re maximally entangled.

    In the case of the Beta-Carotene protein, the protein and the photon (/space) have equal individual entropies, so the minimum possible joint entropy is zero… and that’s what their actual joint entropy turns out to be.

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  28. Gordon Davisson: But with quantum information, the joint entropy of two things can be less than their individual entropies;

    I find this situation more intuitive if I think of entropy as missing. information.

    So if we have two maximally entangled n-state subsystems, the reduced density matrix for measurements of each subsystem is 1/n on the diagonal (assuming the appropriate basis). That matrix means maximal entropy for each subsystem.

    But the overall entropy is lower because it includes the information in the entangled system, taken as a whole.

    That’s the reason that someone measuring one subsystem A of maximally entangled systems A, B always sees probabilities 1/n, regardless of whether the other system B has already been measured (which means that measurer of B knows the results A will get).

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  29. Gordon Davisson,

    Thank you, that was exceptionally clear, especially the part about the photon fired at an angle into the Beta-Carotene protein and the uncertainty it creates in the system separated from the whole. This was the best comment I’ve seen in the last year or more. Extremely informative and educational. I went back and started reviewing the Bra-Ket notation and associated conventions (something that Griffiths almost unbelievably only spent like 2 pages on) and which I never had mastery of.

    Thank you again, it re-kindled my interest in all this especially since chemistry is so reliant on Quantum Mechanics — its really closer to the topics I’m researching and reporting on right now.

    General Relativity and Plasma Physics were OK, Classical Mechanics almost buried me, but something about QM is almost magical both in its behavior and math.

    And if you’ll permit me, I’d like pay tribute again to your grandpa who was a pioneer of QM:

    https://en.wikipedia.org/wiki/Clinton_Davisson

    https://en.wikipedia.org/wiki/Davisson%E2%80%93Germer_experiment

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