If you know one thing about quantum mechanics, it’s that Schrödinger’s cat is both dead

and alive.

This is what physicists call a “superposition”.

But what does this really mean?

And what does it have to do with entanglement?

This is what we will talk about today.

The key to understanding superpositions is to have a look at how quantum mechanics works.

In quantum mechanics, there are no particles and no waves and no cats either.

Everything is described by a wave-function, usually denoted with the Greek letter Psi.

Psi is a complex valued function and from its absolute square you calculate the probability

of a measurement outcome, for example, whether the cat is dead or whether the particle went

into the left detector, and so on.

But how do you know what the wave-function does?

We have an equation for this, which is the so-called Schrödinger equation.

Exactly how this equation looks like is not so important.

The important thing is that the solutions to this equation are the possible things that

the system can do.

And the Schrödinger equation has a very important property.

If you have two solutions to the equation, then any sum of those two solutions with arbitrary

pre-factors is also a solution.

And that’s what is called a “superposition”.

It’s a sum with arbitrary pre-factors.

It really sounds more mysterious than it is.

It is relevant because this means if you have two solutions of the Schroedinger equation

that reasonably correspond to realistic situations, then any superposition of them also reasonably

corresponds to a realistic situation.

This is where the idea comes from that if the cat can be dead and the cat can be alive,

then the cat can also be in a superposition of dead and alive.

Which some people interpret to means, it’s neither dead nor alive but somehow, both,

until you measure it.

Personally, I am an instrumentalist and I don’t assign any particular meaning to such

a superposition.

It’s merely a mathematical tool to make a prediction for a measurement outcome.

Having said that, talking about superpositions is not particularly useful, because “superposition”

is not an absolute term.

It only makes sense to talk about superpostions of something.

A wave-function can be a superposition of, say, two different locations.

But it makes no sense to say it is a superposition, period.

To see why, let us stick with the simple example of just two solutions, Psi 1 and Psi 2.

Now let us create two superpositions, that are a sum and a difference of the two original

solutions, Psi 1 and Psi 2.

Then you have two new solutions, let us call them Psi 3 and Psi 4.

But now you can write the original Psi 1 and Psi 2 as a superposition of Psi 3 and Psi 4.

So which one is a superposition?

Well, there is no answer to this.

Superposition is just not an absolute term.

It depends on your choice of a specific set of solutions.

You could say, for example, that Schrodinger’s cat is not in a superposition of dead and

alive, but that it is instead in the not-superposed state dead-and-alive.

And that’s mathematically just as good.

So, superpositions are sums with prefactors, and it only makes sense to speak about superpositions

of something.

In some sense, I have to say, superpositions are really not terribly interesting.

Much more interesting is entanglement, which is where the quantum-ness of quantum mechanics

really shines.

To understand entanglement, let us look at a simple example.

Suppose you have a particle that decays but that has some conserved quantity.

It doesn’t really matter what it is, but let’s say it’s the spin.

The particle has spin zero, and the spin is conserved.

This particle decays into two other particles, one flies to the left and one to the right.

But now let us assume that each of the new particles can have only spin plus or minus 1,

This means that either the particle going left had spin plus 1 and the particle going

left had spin minus one.

Or it’s the other way round, the particle going left had spin minus one, and the particle

going right had spin plus one.

In this case, quantum mechanics tells you that the state is in a superposition of the

two possible outcomes of the decay.

But, and here is the relevant point, now the solutions that you take a superposition of

each contain two particles.

Mathematically this means you have a sum of products of wave-functions.

And in such a case we say that the two particles are “entangled”.

If you measure the spin of the one particle, this tells you something about the spin of

the other particle.

The two are correlated.

This looks like it’s not quite local, but we will talk about just how quantum mechanics

is local or not some other time.

For today, the relevant point is that entanglement does not depend on the way that you select

solutions to the Schroedinger equation.

A state is either entangled or it is not.

And while entanglement is a type of superposition, not every superposition is also entangled.

A curious property of quantum mechanics is that superpositions of macroscopic non-quantum

states, like the dead and alive cat, quickly become entangled with their environment, which

makes the quantum properties disappear in a process called “decoherence”.

We will talk about this some other time, so stay tuned.

Thanks for watching, see you next week.

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