What is Quantum Entanglement? - Detailed explanation

What is Quantum Entanglement?

Firstly: One reason I feel compelled to write this is that many people say something like this:

“Take two dissimilar objects that come in a pair (e.g. gloves), put them in two identical boxes, forget which box is which and send them to distant locations. Look inside one box and bam! you immediately (re)discover what’s in the other box! And that is just like entanglement!”

This is worse than useless as an analogy for quantum entanglement. Physicists call this sort of system a classically correlated mixed state, and it is totally different from entanglement. Obviously, there is nothing marvellous or extraordinary about classical correlation, and anyone who has ever listened to the terms in which physicists talk about entanglement must suspect that this is a cop-out. If you have ever heard this, please un-learn it right now, and read on.

Anyone who confuses entanglement with classical correlation deserves to be glove-slapped.

Now, to properly understand quantum entanglement, it’s crucial to talk about Bell's theorem. No need to go into the maths because fortunately, the basic concept is quite intuitive.

Imagine some quantum mechanical process (in my lab we use Spontaneous parametric down-conversion) produces a bunch of particle pairs. We take one particle from each pair, line them up in order and put them all in a box, which we give to an experimentalist called Alice. We similarly package the other particles and give them to Bob. These boxes each have two buttons (labelled 1 and 2), which when pressed perform a measurement on the next particle in line inside the box. The measurements have two possible outcomes, which show up as either a red light on the box, or a green light.

Alice and Bob go off to distant locations and every minute press either one of the buttons. Their choice of which button to press each minute is independent of all previous choices, and completely up to them. They each record which lights flash in response to each measurement. Later, Alice and Bob meet up and compare notes.

Immediately, and before looking at the other’s data, the first thing they agree is that there is no obvious pattern to which light flashes in response to which button is pressed. Whichever Alice presses on her box, either its red light or its green light flashes with apparently 50% probability. There seems to be no rhyme or reason to which one or why.

Shortly after, however, they realise there are strong correlations between the behaviour of the two boxes. For the particular settings of these boxes, Alice and Bob’s observations, which perfectly agree with the prediction of QM, are the following:

1. Where both Alice and Bob pressed button 1, the lights that came up on each of their boxes, whilst still apparently random, were the same as each other 100% of the time.
2. Where one of them pressed button 1 and the other pressed button 2, the lights that came up on each of their boxes were the same as each other 99% of the time, and different 1% of the time.
3. Where both Alice and Bob pressed button 2, the lights that came up on each of their boxes were the same as each other 96% of the time, and different 4% of the time.

Now initially this may seem arbitrary but not especially strange. Perhaps this set of results can be explained by something like the gloves-in-a-box situation? In other words, due to some process of which we are ignorant, each particle in each pair is allocated certain properties at their generation which give rise to these correlations. These properties, which stay with the particles after their generation right until their measurement, determine which light flashes. Since they are specific to each particle, they are called local, and since we can’t fully access them (the measurement outcomes seem random) they are called hidden: hence local hidden variables.

If that is your first thought, you aren’t alone. Einstein, Podolosky and Rosen thought exactly the same thing in their famous “EPR paradox". Their conclusion was that QM, which makes no mention of the local hidden variables, was incomplete: there must be a hidden layer of reality below the structures in QM, which if accessed gives precise and pre-determined answers to which outcome will occur in response to each measurement.

Einstein, Podolsky and Rosen were wrong about that. This is where the genius of Bell’s theorem comes in: it proves that the distribution of outcomes above is impossible with local hidden variables. You’ll need your seatbelts for this bit.

I’ll have you know God plays dice whenever he damn well chooses.

Assuming that these results are explained by local hidden variables requires that each particle knows nothing about what measurement is performed on any other particle. Hence, the only thing that the hidden variables can control is how each particle responds to a given measurement. There are therefore four predetermined lists of potential outcomes (i.e. lists where each entry is either RED or GREEN): the outcomes for Alice’s particles when subject to measurement 1 (list A1) or measurement 2 (list A2), and similarly, lists B1 and B2 for Bob’s particles.

Now we know from Alice and Bob’s comparison that performing measurement 1 on both particles always gives the same outcome: the lists A1 and B1 are identical.

We also know that A1 and B2 overlap 99%, as do A2 and B1. Now since A1 and B1 are the same, this means that the minimum overlap of A2 and B2 is 98%, which happens if all the discrepencies with B1 and A1 respectively happen on different pairs. BUT- this contradicts the final observation, which is that there is only 96% correlation between the A2 and the B2 measurements!

Hence there can be no predetermined lists A1, A2, B1 and B2. Local, predetermined outcomes, which are ubiquitous across the rest of physics and everyday experience, cannot explain quantum correlations! And yet, we see these correlations in our laboratories- I myself have measured them with my own hands. This is the central point of what quantum entanglement is- a real physical phenomenon that cannot be explained in the terms we normally fall back on.

Loss of Individuality!!!

Let me explain. All we need is a lil’ bit of Probability, some Ice Cream and Dank-memes!!!

Quantum entanglement has a very enigmatic persona in pop-culture and it’s quite infamous with it’s interpretations in physics community, especially during the early stages of development of Quantum-Mechanics.
Einstein use to call it, “Spooky action at a distance!”. But what is so ‘Spooky’ about quantum-entanglement, that even Einstein had a hard time wrapping his head around? We’ll see.

But first we need to understand a special kind of notation Physicists use to define Quantum-states, it’s called the Bra-Ket notation:
⟨S|Q|S⟩$⟨S|Q|S⟩$

Here quantity ‘Q’ is measured for a state ‘S’. The part left of ‘Q’ is called ”Bra” and the one on the right is called “Ket”. We can also define the probability of a state as:
A|S⟩$A|S⟩$

here A2$A^2$is the probability of state “S” happening, when the wave function collapses ( fancy way of saying, when the quantum system is observed).

So here’s how our story goes:

• Your friend Bablu comes over to meet you after a long time. He urges you to go out with him, as it’s been a long time since you guys hung-out.
  You tell him about this awesome Ice-cream shop near your house “Tillu-Ice-cream”.

• Both of you marched to the Ice cream shop. He asked for Strawberry, while you had Vanilla. But as you both were returning from the shop, his wife calls and asks him to come home as soon as possible. You guys planning to eat both the flavours.
  He asked you, “If there’s anything that we can do about it?”. And you being a Physicist, accepted his challenge.

• In your lab, you have made the Quantum Entangler-4000, a machine which can entangle any number of things together. So you insert both icecream buckets in the machine. And what you get are 2 buckets of “Entangled Flavour Ice Cream“.

Now that you have entangled both flavours, each bucket behaves as a quantum system of 2 superimposed states (here, flavours). So when you observe the system i.e. whenever you take scoop from the bucket and eat, then only you get to know what is the current flavour of your ice cream. The mathematical formulation of this process will look something like this:

Here the coefficients (A, B) would be such that A2+B2=1.$A^2+B^2=1.$ For example if both of the states (given in the kets) have equal probability then :|Vanilla⟩,|Strawberry⟩⟹12−−√|V,S⟩+12−−√|S,V⟩$|\text{Vanilla}⟩,|\text{Strawberry}⟩⟹\sqrt{\frac{1}{2}}|V,S⟩+\sqrt{\frac{1}{2}}|S,V⟩$

And if the states have different probabilities of occurrence, then their respective kets will have different coefficients, example:
|Vanilla⟩,|Strawberry⟩⟹14−−√|V,S⟩+34−−√|S,V⟩$|\text{Vanilla}⟩,|\text{Strawberry}⟩⟹\sqrt{\frac{1}{4}}|V,S⟩+\sqrt{\frac{3}{4}}|S,V⟩$

So if you get Vanilla in your scoop, Bablu will get Strawberry and vice-versa. You'll get one of the two flavour randomly with each scoop, but yours and Bablu’s ice-cream flavour will never be the same.

This is called Quantum-Entanglement, because here outcome of one bucket cannot be determined independently without simultaneously getting to know the state of the other bucket.
(“Hmmmmm, ok so this Quantum-entanglement is a little weird”, you say “but it’s not mindblowing or spooky or anything like that…. huh?”)

BUT WAIT! There is more…

• So Bablu happily goes to his home, now that he can eat both flavours.
  15 minutes later, your girlfriend arrives and says, “Hey! Guess what? I brought the Chocolate flavour from Tillu’s Ice cream, and looks like you already have one bucket. We can entangle them and then eat, just like we did the last time.”

• So now you again put your buckets in the Quantum Entangler-4000, what comes out is 2 buckets of 3-flavour entangled ice cream. (Vanilla-Strawberry-Chocolate)

• So you both start eating and in the first bite you find, it’s Vanilla. You ask your girlfriend, what did she get? “Strawberry!!!”,she says.
  Then she asks, “Who has the Chocolate flavour?”

• At the Same-Time, somewhere far far away….

It doesn’t matter where Bablu lives, it can be 10 Km away or 1000 Km away or 100 Light-years away. Superposition of states will collapse simultaneously. Due to quantum-entanglement 3 buckets have lost their independent identity and have started behaving like a single entity.