# the chsh game

The CHSH game, and the inequality that is central to it, demonstrates the fundamental difference between classical and quantum mechanics. More precisely it demonstrates that it is not possible that quantum mechanics is a classical theory with local hidden variables.

This article details the rules of the game, accompanied by some hefty flavour text. You can check these other articles for the best classical strategy, the quantum strategy and the conclusion of this game.

## the premise

Imagine the following situation: you are a secret spy tasked with the assasination of a certain public figure. Together with your accomplice you will infiltrate a ceremony at which your target will be giving a toast ā there, you'll have the chance to poison the target's glass of champagne. However, there is a catch. The target has three doppelgangers ā therefore there is a total of four people who could give a toast at the ceremony, only one of which is your target.

## the target

Luckily, there are two pieces of information that can help you tell the target apart. First, the target and one of their doppelgangers is left-handed, while the other two are right-handed. Second, you know the exact dimensions of the target's suit ā which accidentally match the dimensions of one of the right-handed doppelganger's suit, but don't match the others.

left-handed right-handed
matching suit size ā target šÆ doppelganger
suit size doesn't match doppelganger doppelganger

Therefore, you have to have know both pieces of information in order to tell the target reliably. The two of you decide to infiltrate the ceremony, one disguised as a cloakroom attendant, the other as a waiter. The target is well-known for issuing a cheque beforehand, so the waiter can see which hand they use to sign it. Meanwhile, the cloakroom attendant can measure the dimensions of the clothes left there. Due to the nature of your disguises, you won't be able to communicate with each other during the operation ā each of you will be on your own.

## the weapon

Since you won't be able to meet and collectively decide whether to poison the drink, one of you will have an ampoule with poison, and the other will have an ampoule with the antidote. The antidote has the capacity to neutralize the poison, but digested on its own it is just as deadly.

poison not used poison used
antidote not used š° safe ā ļø lethal
antidote used ā ļø lethal š° safe

Each of you will have a chance to pour the contents of your ampoule into the drink at different times. Since both the poison and the antidote are designed to be undetectable, neither of you will be able to tell whether the other one did or did not use their ampoule.

## mission statement

It is becoming increasingly clear that you're playing a game of chance. Not only that you can't tell if the target will be there, even during the mission you'll only have incomplete information that you cannot share with each other. Therefore, you should formulate a strategy that maximizes the probability of mission success.

Success of your mission is defined as either poisoning the target ā if present ā or not poisoning the doppelgangers if the target is not present. If you either poison a doppelganger, or fail to poison the present target, the mission is deemed a failure.

target present šÆ target not present š¤µ
poisoned ā ļø ā success ā failure
not poisoned š° ā failure ā success

What is the highest probability of success you can achieve with the perfect strategy? That will depend on whether you live in a universe where quantum mechanics apply, or a completely classical universe.

See the derivation of the best classical strategy, then check whether the this quantum strategy beats it. Finally, you can read the conclusion of this little thought experiment.