Quantum Entanglement in a nutshell

As well as the article below, check this...
https://www.sciencenews.org/blog/context/entanglement-spooky-not-action-distance
Part 2 here https://www.sciencenews.org/blog/context/quantum-spookiness-survives-its-toughest-tests?mode=blog&context=117

Plus this paper on string theory https://arxiv.org/pdf/1512.02477v5.pdf

An excellent overview of Quantum Entanglement by Barak Shoshany, Graduate Student at Perimeter Institute for Theoretical Physics

For my next trick I will explain quantum entanglement in 5 minutes to anyone with basic knowledge of linear algebra (no prior knowledge of physics or quantum mechanics necessary), as I promised elsewhere on Quora.

Let's say I have a physical system (a particle, for example). This system has some properties (position, momentum, spin and so on). In quantum mechanics we write the quantum state of a system as |ψ⟩$|\psi ⟩$. This is just a fancy way of writing a vector. I could have just written ψ⃗$\overrightarrow{\psi }$ but physicists like to write things in a fancy way.

The thing inside the |⟩$|⟩$ can be anything; the letter ψ$\psi$(psi) is commonly used for historical purposes, but |cat\ is\ alive⟩$|\text{cat is alive}⟩$ is also a perfectly good quantum state.

These quantum states live in a vector space. We call this a Hilbert space and we say that all the possible states of the system are vectors in this space. Now, as you know, if you have some vectors in a vector space you can always write a linear combination of them. For example, |A⟩+|B⟩$|A⟩+|B⟩$ is the linear combination of the states |A⟩$|A⟩$ and |B⟩$|B⟩$. (Again, in quantum mechanics we like to use fancy language so we call this a superposition of states. But it's just a linear combination of vectors.)

In quantum mechanics the coefficients of each state in the superposition are called probability amplitudes, and in general they are complex numbers (since our Hilbert space is a complex vector space).

Roughly speaking, if we perform a measurement on the superposition (assuming that it's in the right basis, etc., but I don't want to get into too much detail since we only have 5 minutes) we will measure only one of the states in the superposition, with the probability given by the absolute value squared of the amplitude.

Here's an example. If my superposition is
15‾‾√|A⟩+25‾‾√|B⟩+25‾‾√|C⟩$\sqrt{\frac{1}{5}}|A⟩+\sqrt{\frac{2}{5}}|B⟩+\sqrt{\frac{2}{5}}|C⟩$
Then I will measure A with probability 1/5, B with probability 2/5 or C with probability 2/5. (Again, the probabilities are the squares of the coefficients.)

(For more information about quantum states and measurements see for example my answer to In layman's term, what is a quantum state? and my answer to What are the postulates of quantum mechanics?)

Are you still with me? Great! Now, every 7 year old can tell you that if you want quantum entanglement you need to have more than one particle, right? So let's say I have two particles. Now my system is a so-called composite system of two separate systems. It's still a vector space, just a bigger one.

In this bigger Hilbert space, I can write a quantum state like so: |A⟩|B⟩$|A⟩|B⟩$. This is just a fancy way of saying that particle 1 is in the state |A⟩$|A⟩$ and particle 2 is in the state |B⟩$|B⟩$. (The order matters! The state on the left or right is always that of particle 1 or 2 respectively.)

So, we had 1-particle states, then we had superpositions of them, then we had 2-particle states. The next step is, of course, superpositions of 2-particle states. And this is where quantum entanglement happens.

Let's say I have the following superposition:

12‾‾√|↑⟩|↑⟩+12‾‾√|↓⟩|↓⟩$\sqrt{\frac{1}{2}}|\uparrow ⟩|\uparrow ⟩+\sqrt{\frac{1}{2}}|\downarrow ⟩|\downarrow ⟩$

The arrows are meant to represent spin up |↑⟩$|\uparrow ⟩$ and spin down |↓⟩$|\downarrow ⟩$. But the states can be anything, they don't have to be spin states.

The above superposition means that, if I measure the spin of the 2-particle system, I have a probability of 1/2 to get |↑⟩|↑⟩$|\uparrow ⟩|\uparrow ⟩$ (i.e. both particles have spin up) and a probability of 1/2 to get |↓⟩|↓⟩$|\downarrow ⟩|\downarrow ⟩$ (i.e. both particles have "spin down").

And that, folks, is quantum entanglement! (Or at least, a very simple example of it.)

Reader: "Wait, what? That's it?"

Me: "Yes, that's it."

Reader: "What do you mean, that's it? Where is the faster-than-light communication?"

Me: "It's nowhere. There is no such thing. It's just a common misconception. There is no communication of any kind taking place, not in any speed and certainly not faster than light."

Reader: "What about time travel? And parallel universes? And wormholes?"

Me: "Completely unrelated. You've been watching too many sci-fi movies."

Reader: "Well, that is disappointing."

Me: "I'm sorry. But hey, at least now you know how quantum entanglement works!"

Reader (after some time): "Aha! Wait a minute! What if I take one particle to the Andromeda galaxy, 2.5 million light years from Earth, and then measure its spin? How will the particle on Earth instantly know to have the same spin? They must communicate somehow. Surely something fishy is going on here!"

Me: "Nope. The entangled state I defined above simply says that the measurements of the spins of the two particles are correlated. It doesn't matter what the distance between them is. If one is measured to have spin up, then the other will also have spin up simply because they were entangled in such a way that their spins are correlated."

Reader (after reviewing the math): "Okay, I get that this is what the math says, but it still doesn't make sense to me."

Me: "Congratulations; you're in good company. Great minds such as Einstein also thought it doesn't make sense. Physicists and philosophers of physics have been debating the meaning of quantum mechanics in general, and quantum entanglement in particular, ever since quantum mechanics was first formulated, and are still debating it today, almost 100 years later."

Reader: "Surely they have reached some conclusions after 100 years..."

Me: "Sort of. They have come up with a very long list of interpretations of quantum mechanics which attempt to make sense of weird quantum phenomena, such as entanglement. An interpretation of quantum mechanics is an attempt to explain what is "really" going on behind the math."

Reader (after some time): "I clicked on the link. There are so many different interpretations... Which is the correct one?"

Me: "Unfortunately, since the experimental predictions are unchanged by the interpretations, it's impossible to determine which interpretation is the correct one, or if there even is a correct one! The only thing we know for sure is that all experiments ever performed have supported the validity of quantum mechanics. It's true independently of how you interpret it."