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.