The No Cloning Theorem

Cloning: If you want to make a copy of something, you need three things: the thing to be copied, The raw materials that you’re going to turn into the copy, and a procedure for transforming the raw materials into a semblance of the original thing. To copy a famous painting , you need a blank canvas, a brush, and the right colored paints, and then you carefully put paint on the blank canvas to match the original as closely as you can and hopefully sell it for a lot of money. But your painting isn’t exactly the same as the original – the red is a little too bright, that stroke is a little too heavy, there are a few too many atoms of carbon 14 in the new canvas, and so on – it’s a copy, but not a perfect one. Is a perfect copy, identical even at the subatomic level, even possible? Like, can you make a copy of my brain down to the neuron and beyond, so that even the position, momentum, and spin of every single sodium ion moving between neurons is exactly, indistinguishably, the same as in the original? Physicists call this kind of perfect copying “cloning”, even though it definitely isn’t the same thing as cloning in biology where two organisms share the same DNA but how they grow and develop can be very different – cloning in physics means a much perfecter copy, where the relative positions and momenta and energy levels of every particle and all of their bonds and interactions are exactly the same in the copy as the original, such that if you turned your back and randomly switched them, there literally would be no way of telling which was the original and which was the copy. Unfortunately, the universe is a party pooper, and perfect cloning is impossible. I don’t simply mean that we don’t know how, or that we haven’t succeeded yet because it’s really hard to do in practice; no, I mean that it has been mathematically proven that perfect cloning can’t be achieved even in principle. Here, now, is that proof, using as little math as possible. Everything in the universe is made up of elementary quantum particles and the forces by which they interact , so for the no-cloning proof we need to know what it means to clone a quantum particle, so first we’re going to need to know three important and fundamental properties shared by all quantum particles. Ok, quantum property number one: particles can be in several states at once. Like Schrödinger’s cat, stuck in a bunker with unstable gunpowder that has a 42% chance of exploding in any minute, but maybe it hasn’t yet, so that the gunpowder is in a superposition of “gunpowder has already exploded” and “gunpowder hasn’t exploded yet” . Or like a photon going through two slits at once to interfere with itself and make a nice pattern on the wall . Or an electron in an atomic orbital, its wavefunction occupying many points in space all at once. In summary: in quantum mechanics, the whole is equal to the sum (that is, superposition) of its different possible parts . Alright, property number two: multiple particles, when viewed together as one single “object” (like an atom, or entangled pair of photons, or the gunpowder together with Schrödinger’s cat, or whatever), are the product of their components, or, since it’s quantum mechanics, a superposition of products of their components, so the situation inside Schrödinger’s box could be described as a superposition of the product of “gunpowder has already exploded” and “the cat is dead” and the product of “gunpowder hasn’t exploded” and “the cat is alive” . In summary: composite quantum objects are multiplied together . And finally, quantum property number three: any change to a particle that’s in a superposition of states affects all of the states independently . Kind of like how if you go two miles to the right and one mile up and then rotate your map ninety degrees , that’s the same as first spinning each arrow individually 90° and then adding them together. Or if you have an electron in a superposition of “here” and “there” that’s moving to the right, that means that “electron in one second” will be in a superposition of “wherever ‘here’ is in one second” and “wherever ‘there’ is in one second”. In summary: when you have a superposition, aka, a sum of several parts , any change or transformation of the sum of the parts is equal to the sum of the transformations of the parts , whether that transformation is a rotation, a movement, or even an entire hypothetical cloning process. So let’s recap, for the no-cloning proof, we’ll use three of the properties that all fundamental particles in the universe obey: individual particles can be in superpositions, which looks like adding; groups or combinations of particles are products of their components (or sums of products of their components), which looks like multiplying; and any transformation of a particle or group of particles is the same as the sum of the transformation applied to the parts, which looks like distributing. Ok, now we can get into the meat of the proof! So in terms of the properties we just outlined, let’s talk about what it would mean to have a quantum cloning machine. We’d need the thing to be cloned , the materials to make a clone out of, and a procedure to transform the materials into an exact copy of the original . Our machine shouldn’t have to know in advance what the thing to be cloned is, otherwise it’s not really a machine for cloning things as much as a machine for building a known thing . So, if a cloning procedure were to exist, we should be able to “apply cloning” to any specimen we want , and end up with two copies of the specimen. The problem occurs, however, if the specimen we’re cloning is a superposition, like if it’s the gunpowder from inside Schrödinger’s cat’s box, in a superposition of “exploded” and “not exploded”. If we apply our hypothetical cloning to the whole gunpowder-inside-the-box-superposition, we get “exploded” plus “not exploded” times “exploded” plus “not exploded”. But since, in quantum mechanics, a procedure applied to the whole gets distributed through as the sum of the procedure applied to the parts, that means that we should get the same result by applying cloning to each part of the superposition , separately cloning “exploded” and “not exploded” and then adding them together. But, we don’t get the same thing, since exploded times exploded plus not exploded times not exploded is not the same as exploded times exploded plus exploded times not exploded plus not exploded times exploded plus not exploded times not exploded. There are these extra terms here that don’t match up. Basically, if both quantum mechanics and cloning are true, then A plus B, squared must be the same as A squared plus B squared. But A plus B, squared, is not the same as A squared plus B squared. And this contradiction means that either quantum mechanics is wrong (which would fly in the face of the most precise and accurate experimental tests in all of science ), or that a cloning procedure can’t exist. Spoiler alert: it ain’t looking so good for cloning. This, by the way, is an example of what’s called “proof by contradiction”, a logically sound (but not always pretty) kind of proof where you suppose that the opposite of what you’re trying to prove is true, is true, and show that such an assumption leads to a contradiction or other logical problems, so it can’t be true, and thus what you actually are trying to prove must be true instead. Like, to prove there’s no biggest even number, we’d first suppose there IS a biggest even number, call it E, which since it’s even it’s equal to two times some other number. But then if we add 1 to that other number and multiply by 2, we get an even number (since it has 2 as a factor), but this new number is bigger than E, which was supposed to be the biggest even number. This is a contradiction, so our supposition that there is a biggest even number can’t be right… so there is no biggest even number. Ok, but back to cloning. So to summarize the proof of no cloning theorem, we first suppose the cloning IS possible, then show that such cloning would logically results in the contradiction that a cloned whole would not be the same as the sum of its parts, and hence perfect cloning is not possible. I also want to point out that the proof of no cloning didn’t examine any specific apparatus or design for how cloning might be done – it just uses properties that we know any cloning apparatus would have to have. Like, it would have to exist in our physical universe, and it would have to be able to clone things. The proof proves that anything with both of these properties can’t exist. However, for those wanting to live in a sci-fi future, all is not lost. Even if perfect cloning isn’t possible, “pretty decent copies” cloning is. Like, it’s possible to clone a qubit with an average of 83% fidelity . And even more exciting: the no-cloning theorem is only about cloning; teleportation is still possible. That’s because teleportation consists of a subject, materials to make the teleported version out of, and a procedure to turn the teleported materials into the subject, leaving behind an empty machine. And a quick calculation shows that teleporting a superposition, or sum, is indeed equal to the superposition, or sum, of the individually teleported parts! What’s more, “no cloning” doesn’t mean you can’t have two or more copies of the same thing in the universe, it just means it’s not possible to take an existing thing that you don’t already know all the details about and make a perfect copy of it while leaving the original intact. You can build a machine to make multiple versions of things as long as you know in advance exactly what it is you’re making. So, is it possible to learn every single detail about something? Well, the Heisenberg uncertainty principle means that you can’t simultaneously measure all the relevant details of any one object, but if you have a number of objects that you know are the same, you can measure each of them in a different way to get the full picture. So the irony is that in quantum mechanics, you can’t perfectly clone a thing you have only one of, but if you already have a lot of copies of something , you can make more copies. However, as far as we know, there’s only one of each of us in the universe, so “no 100% perfect cloning” in quantum mechanics means “no 100% perfect cloning” in humans, either . While we may eventually be able to grow a child that’s genetically identical to you, we likely won’t ever be able to make a perfect clone of you that has all of your memories, thoughts, and loves. How close we can get, of course, depends on whether or not consciousness relies on quantum processes in the brain. But that’s a question for another day.

Channel: minutephysics

Views: 2,249,390

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Keywords: physics, minutephysics, science

Id: owPC60Ue0BE

Length: 10min 4sec (604 seconds)

Published: Tue Dec 27 2016

Reddit Comments

[u/Rodry2808]

It's the first time I see such a deep example of Linear Transformations

[u/N8CCRG]

First time I've heard "unstable gunpowder" in the box before.

[u/Zicon4]

I see, so that Linear Algebra class I took actually DOES do something worthwhile

[u/obewanjacobi]

Lol, "perfecter". This is why we do physics

[u/[deleted]]

(|a>+|b>)² = |a>² + |b>² when |a>.|b> = 0 = |b>.|a>. This is indeed zero because |a> and |b> are orthogonal (blowing up and not blowing up, dead and alive). So this proof is wrong? Am I missing something here?