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  • Cloning: If you want to make a copy of something, you need three things: the thing to be copied,

  • The raw materials that youre 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 originalthe 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 onit’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 copyingcloning”, 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 differentcloning 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 were 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 onceLike 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 ofgunpowder

  • has already explodedandgunpowder 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 singleobject” (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 ofgunpowder has already explodedandthe cat is deadand the product

  • ofgunpowder hasn’t explodedandthe 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

  • hereandtherethat’s moving to the right, that means thatelectron

  • in one secondwill be in a superposition ofwhereverhereis in one second

  • andwhereverthereis 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, well 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 toapply cloning

  • to any specimen we want , and end up with two copies of the specimen.

  • The problem occurs, however, if the specimen were cloning is a superposition, like if

  • it’s the gunpowder from inside Schrödinger’s cat’s box, in a superposition ofexploded

  • andnot exploded”. If we apply our hypothetical cloning to the whole gunpowder-inside-the-box-superposition,

  • we getexplodedplusnot explodedtimesexplodedplusnot 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 cloningexploded

  • andnot explodedand 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 calledproof by contradiction”, a logically

  • sound (but not always pretty) kind of proof where you suppose that the opposite of what

  • youre 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 rightso 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 copiescloning 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 cloningdoesn’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 youre 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, sono

  • 100% perfect cloningin quantum mechanics meansno 100% perfect cloningin 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.

Cloning: If you want to make a copy of something, you need three things: the thing to be copied,

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B2 cloning exploded superposition quantum gunpowder sum

The No Cloning Theorem

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    林宜悉 posted on 2020/03/28
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