The CHSH game

You and your twin love matching colors. You each wake up in the morning and put on either red or blue.

Except: you guys are pretty moody. You each have a 50% chance of waking up happy 😊 or grumpy 😠.

The game is simple: when you wake up in the morning, each of you have to figure out what to wear without communicating with each other. The objectives are the following:

Each of the four mood combinations is equally likely (25% each). You wake up and only know your own mood, not your twin's.

The night before, you can strategize with your twin, but when you wake up, you can't communicate at all. There's no way for you to predict if anyone will be happy or grumpy.

So the question is:

It turns out that a very simple strategy already lets you win 75% of the time:

Always wear blue.

Think through it. Three of the four mood combinations want you to match, and if you both always wear blue, you always match. Only the “both grumpy” case (which wants you to not match) loses. That's 3 out of 4. 75%.

No. Since you can't know if your twin is happy or grumpy, the “both grumpy” case is a 50/50 guess at best, and the math works out so that 75% is the best any classical strategy can achieve. Any agreed-upon plan, any private information, any randomness you share ahead of time, none of it gets you above 75%.

That's an upper bound. It's been proven impossible.

What if you and your twin shared an entangled pair of qubits? It turns out you can win ~85% of the time.

That's shocking. We know no information can be sent over entanglement. Yet entangled qubits beat the classical limit by 10 percentage points. Something real is being shared between you and your twin that can't be reduced to any pre-shared classical strategy.

That gap, between 75% and 85%, is what John Clauser, Alain Aspect, and Anton Zeilinger spent their careers measuring. They showed that real entangled qubits really do beat the classical bound. That's what won them the 2022 Nobel Prize in Physics.

Let's see how it works.