# Chapter 2: Entanglement

Welcome to Chapter 2 of our Quantum Explainers Series! If you are inspired dive deeper into this topic, check out the Quantum Shorts Contest. You can take inspiration from this article (we will even hint at potential topics that you can make your video about) and provide your own take on entanglement for the contest!

By Yi Zhu

April 14, 2024

Let’s return to the example of baking cakes as described in Chapter 1. Suppose we modify the recipe from before and again bake two cakes. After baking, when we open the oven we find that both cakes are raw. If our new recipe still prepares each cake in an even superposition state (as in Chapter 1), then each cake would have 1/2 probability to be raw, and so we have probability ½ * ½ = ¼ to find both cakes to be raw.

Our predictions for when we bake 2 cakes 100 times (each blue dot is one time we baked 2 cakes). If both cakes are in an equal superposition state, we expect to get each combination ¼ of the time.

Mathematically, we write the state of two cakes, each in an even superposition state as:

Here the subscripts A and B indicate which cake we are describing. For example,

means that there is a ¼ probability that we measure cake A to be raw and cake B to be burnt.

If our theory above is correct, then we expect to measure all four combinations of raw/burnt cakes with equal probability. So we repeat this experiment 100 times: each time we bake two cakes at a time and check if they are raw or burnt. 200 cakes later, we come to an interesting conclusion: for every iteration of the experiment, we find the two cakes to be either both raw (approximately half the time) or both burnt. This does not correspond at all to our predictions!

What we actually see when we bake 2 cakes 100 times (each blue dot is one time we baked 2 cakes). We see that half the time both cakes are burnt and half the time both cakes are raw!

Since our steps for preparing the two cakes were identical each time, we know that we must have prepared a pure quantum state and not a statistical mixture of various different quantum states. So how do we describe this new quantum state that we’ve prepared?

We know that half the time we measure the cakes to be

and half the time we measure the cakes to be

Following the mathematical notation we’ve developed, we can write this state as:

This state is called an “entangled state” because the measurement results of A and B are correlated. If we measure cake A to be raw, then we have collapsed the entangled state to

and so we must measure cake B to be raw as well.

Suppose we prepare this entangled cake state, then box up each cake individually without measuring the state (e.g. not checking if it’s raw or burnt). We give our friends Alice and Bob each a cake, and have them open the box when they get home. If Alice opens her cake box first and finds her cake to be raw, only then has her measurement collapsed the entangled state. Now when Bob opens his cake box at a later time, he must also find the cake to be raw.

This behavior seems indeed quite strange because Alice’s action of measuring her cake has somehow affected the state of Bob's cake, which could be miles away. This is why entanglement is often described as “spooky action at a distance.”

In our everyday lives, we never encounter this “spooky action” because it is challenging — impossibly challenging — to entangle objects on a human scale. However, research have demonstrated entanglement of quantum-scale objects over distances as far as 240,000 miles (the distance between Earth and the moon [1]).

The phenomenon of entanglement is consistent with researchers’ experimental observations, but is it consistent with the fundamental laws of physics? One such fundamental law is that no object or information can travel faster than the speed of light. Could we use entangled particles to communicate faster than the speed of light? Luckily for our quantum theory, the answer is no: we can not use entanglement to communicate faster than the speed of light. Why not?

[1] Cao, Yuan, et al. "Bell test over extremely high-loss channels: towards distributing entangled photon pairs between earth and the moon." Physical review letters 120.14 (2018): 140405.