# CHSH

CHSH is a collaborative game for two players. We'll call them Alice and Bob.

• Each player flips a coin.
• Each player then chooses to output one of two symbols.We'll use ▲ and ●, but they could be any two symbols.

Click anywhere in the figure to flip some coins.

A judge looks at Alice and Bob's output, and decides whether they collectively win or lose according to the following rules:

• If either Alice or Bob flip heads (HH or HT or TH), they should output the same symbol. So, ▲ ▲ and ● ● are winning moves.
• If Alice and Bob both flip tails (TT), they should output different symbols. So, ▲ ● and ● ▲ are winning moves.

The aim of the game is for Alice and Bob to collaborate, so as to win as often as possible. We can summarize the rules with a symbol:

Alice and Bob are playing a perfect game. Since their coins are not hidden, they can easily negotiate a strategy such that they win 100% of the time. The game is too easy. Boring.

Let's make things more interesting. We'll allow Alice and Bob to meet and discuss a strategy before the coin flip. This strategy might be exchanged verbally, or written on a piece of paper, or stored in a computer or whatever — Alice and Bob can collect some arbitrary stuff from a meeting point, have a chat and a cup of tea, whatever they like. However, once the coins have flipped, they are forbidden from communicating . There is no way that Alice can learn the outcome of Bob's coin flip, and vice-versa. No phone calls, no texting, no wifi.

Previously, Alice and Bob won 100% of the time. Now that they can't communicate the outcome of their coin flips, what is the best possible strategy? How often can they win? I'll give you a clue: they can't win 100% of the time any more.

It might help to think about some examples.

• What if Alice and Bob independently output ● or ▲ at random?
• What if each party outputs ● when they flip H and ▲ when they flip T?
• What if they write down a really complicated plan in advance?
• What if they each have a supercomputer, or a submarine, or a puppy?
• What if they simply always output ●?

Alice and Bob are using a really simple strategy. They're just saying "●" every time. They win three games out of four on average. They only lose when they flip TT. Is there a more complicated strategy which does better? Nope — it can be proven that no classical strategy outperforms this simple method.

I suggest that you do not progress to the next section until you are convinced (intuitively, or by finding a proof) that 75% is a hard limit. You should be able to see that no matter what stuff Alice and Bob exchange during their meeting — computers, messages, whatever — nothing helps beat 75%.

In the figure below, the object on the left is a photon source. A laser zaps a crystal, generating two photons — tiny particles of light. We give one photon to Alice, and the other to Bob. These photons are going to play the role of the pieces of paper or mental plans which Alice and Bob previously exchanged. We're not cheating. Just as before, Alice and Bob are allowed to collect some objects from a meeting place before they flip their coins.

Having collected their photons, Alice and Bob flip their coins. Alice has some optical apparatus: a piece of glass and a photon detector. She configures her device, twisting a piece of glass to the left if she flips H, or to the right if she flips T. The photon goes through the optical apparatus, and into the detector. The detector has two possible outputs. Alice has labelled them, "●" and "▲". She then simply shouts out the output of her detector.

At the same time, Bob follows the same procedure. He flips his coin, sets his apparatus, and dutifully chooses "●" or "▲" depending on the output of his detector. Let's play the game:

Now Alice and Bob win 85.36% of the time on average. What is going on? How is this possible?

Maybe Alice and Bob are cheating, they're calling each other up on the phone? Well, let's make sure. We'll give them their photons, and then put them in sealed boxes, with a Faraday Cage each, on different sides of the solar system. Now there's no way they can communicate with each other. What happens? They still win with the same probability!

This effect has been tested and independently confirmed in hundreds of experiments, performed by respected physicists across the world. It's been implemented using photons, electrons, atoms, and even superconducting wires. In order to rule out cheating by classical communication, one group even placed Alice and Bob on different islands in Tenerife!

How is it that the game can be beaten with a couple of little photons, but not by humans with computers and JCBs and brains?

The short answer is that small things, quantum mechanical things like photons, electrons, and atoms do not play according to common-sense rules. Quantum mechanics really is very alien, very far removed from our macroscopic classical experience. This has real practical concequences. We've seen here a specific example where a quantum machine can outperform all possible classical machines — albeit in a rather contrived task. However, we believe that we can exploit counterintuitive quantum effects to build really useful machines. These include ultra-secure communication links, and ultra-fast quantum computers.

The long answer involves learning more quantum mechanics.