# Quantum Entanglement

Alice and Bob are very anti-social electrons. For financial reasons, they have to share a house (a helium atom). Due to their antisocial nature, they come to a condition where both of them never are in the same spot (state) at the same time. We’re interested in their spin, (political spin). If we ask one of them what their spin is, a question that technically would be phrased like “is your spin aligned with the y-axis?”, the answer we must get is “with” or “against”. That is all the answer they can give. And if we then immediately go and ask the other the same question, we will have to get the opposite answer. If Alice answered “with”, then Bob will answer “against”.

This all assumes that Alice and Bob are in their “ground state”, which is the worst financial condition (lowest energy) possible. If one of they hae a little cash, they could be in an “excited state” and they could end up with the same spin. But in that case, they would still have to be found in different positions (and with different energies). For example, Alice could be downstairs eating while Bob is upstairs sleeping.

The physicists say that these sort of living together difficulties arise because electrons are fermions: anti-social quantum creatures in that two of them are never found in precisely the same quantum state. This is called Fermi-Dirac statistics, or the Pauli exclusion principle. Fermi, Dirac and Pauli are three physicists. “Statistics” from the fact that when you make computations using “statistical mechanics,” you have to count up the number of ways a certain situation can be achieved, and if the particles can’t fit into the same state it reduces the number of ways. Fewer ways makes that situation less likely. Statistical mechanics forms the foundation of thermodynamics, the science of temperature, pressure, volume and all that.

So electrons are fermions. This is a good thing. The reason that gravity doesn’t pull you down to the center of the earth is because the electrons in your shoes can’t fit into the same quantum states as the electrons in the stuff you’re walking on. The same principle keeps neutron stars from collapsing into black holes. Hooray for fermions!!!

The helium atom has two electrons. While two electrons cannot be in the same state, all electrons are identical in that they all have negative charge and are attracted to positively charged things. In the case of the helium atom, the nucleus is positively charged and the two electrons are attracted to it. They can’t escape, but they can’t be in the same state either.

Ground States of Helium

You can think of a quantum state of an electron as being a description of where it is and what its spin is. For the case of the helium atom, different quantum states have different distances from the nucleus, or they have different spin, or both. There are two states that are as close to the nucleus as possible, these are the most attractive for the electrons and they’d both like to be there. Those two states are called the ground states. (By the way, when even a wikipedia physics article admits that it “contains too much jargon” you know it’s reached the level where it is a self-parody.)

In bigger atoms, like oxygen or uranium, there’s more than 2 electrons and so not all of them can fit into the two ground states. But with helium, there are exactly enough. It’s just right. And that makes it a good place to begin talking about entanglement.

Those two ground states differ in their spin. For example, if you measured one of the electrons and found that it had spin “with” the x-axis, the other electron would have to have spin “against” the x-axis. They have to be opposites. And this applies to no matter what direction you measure spin in. If you measure the spin of one electron in the y-axis, and get “against”, the other will measure out as “with”.

Entanglement

It gets even more interesting when the two electrons from a helium atom are separated. If the separation is done in such a way that the spins are not changed, then the two electrons will still have opposite spins even though they are a long distance from each other. This is called entanglement.

To put it in human terms, it’s as if two people who once lived together ended up with opposite opinions on everything. They even have opposite opinions on things that neither of them have ever thought of before. If you ask them one of these brand new questions, either one of them could go either way, but in a “spooky action at a distance” it turns out that whatever answer one of them gives you, the other will give the opposite.

Classically, you could think of this as the first electron having a definite value of spin “with” the x-axis, and “against” the y-axis, while the other electron has spin “against” the x-axis and “with” the y-axis. But you would have to do this for all possible directions; you would have to assume that the first electron had a definite spin with respect to all other possible axes. For example, the first electron would have to have defined spin with respect to the axis that points towards the missing brick on the chimney on the house of the neighbor down the street, you know, the one who doesn’t weed his yard. That’s a lot of directions, but it’s possible, at least classically.

However, in quantum mechanics, it is required that those electrons not have definite values of spin until they are actually measured, at which time one of the electrons ends up having spin “with” and the other ends up having spin “against”. Which one ends up with which is a matter of probability (luck). The argument for why this is the case is too complicated for this blog post though it is simple enough to talk about in a single post devoted to that (interesting) subject. Einstein, along with two other physicists, Podolsky and Rosen, famously thought this was too bizarre to be real (the idea that the spin wasn’t decided until it was measured). Einstein called it “spooky action at a distance,” but quantum mechanics was right and Einstein was wrong. The effect is known as the Einstein Podolsky Rosen (EPR) Paradox.

Because the two particles are entangled, quantum mechanics has to assign probabilities to the various cases for what could happen when you measure them, individually or apart. For example, if the two measurements you are considering are spin in the x-direction (with or against), then quantum mechanics will give you four probabilities:

Alice with, Bob with;
Alice with, Bob against;
Alice against, Bob with;
Alice against, Bob against.

But we know that Alice and Bob can’t be in the same states so we can reduce this to just:

Alice with, Bob against;
Alice against, Bob with.

And since Alice and Bob are electrons and all electrons are identical and we really can’t tell which is which, these two probabilities have to be equal. They are both 50%. It gets more complicated when we look at measurements that involve interfering the two particles with each other. These correspond to more complicated questions which we will discuss in the next post.

Technical commentary for those who really didn’t have any reason to read all this

If that’s all there was to it, 50% probabilities, it would be a pretty dull story. However, in quantum mechanics, there is another measurement that you can always make of a particle, the phase. (Actually “relative phase” is the observable; phase itself cannot be observed because all quantum systems are symmetric with respect to global changes to phase.) And associated with this, you can mix two particles up by linear superposition. This complicates things beautifully, and I think it should be the topic of another post. And after THAT post, I think we can talk about quantum teleportation. Oh, maybe we have to do “unitary” transformations in between.