The CHSH Game: How Quantum Beats the Classical Limit

The setup

Alice and Bob are siblings who love matching colors. Each morning, they each put on either red or blue.

Except: they're pretty moody. Each has a 50% chance of waking up happy 😊 or grumpy 😠.

The game is simple: when they wake up, each sibling has to figure out what to wear without communicating with the other. The objectives are the following:

Both happy : they want to match colors.

Both grumpy : they definitely do not want to match.

One happy, one grumpy : the happy one will cheer the other up, and they'll want to match.

To summarize:

Alice

😊

😠

Bob😊

Match

😠

Match

Don't match

The four mood combinations, and what counts as a win in each.

Each of the four mood combinations is equally likely (25% each). Each sibling only knows their own mood, not the other's.

The night before, Alice and Bob can strategize together, but once they wake up they can't communicate at all. There's no way to predict if either will be happy or grumpy.

So the question is:

If a sibling wakes up grumpy, what should they wear?
If a sibling wakes up happy, what should they wear?

How to Almost win

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

Always wear blue.

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

Win rates · “Always wear blue”

Alice 😊 happy

Alice 😠 grumpy

Bob 😊happy

should match

100%

should match

100%

Bob 😠grumpy

should match

100%

should not match

Three of four combos win every time. One combo (both grumpy) loses every time. Average win rate: 75%.

Can we do better?

Not without quantum. 75% is the best any classical strategy can achieve. Any agreed-upon plan, any private information, any randomness shared ahead of time, none of it gets them above 75%.

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

How to win

What if Alice and Bob shared an entangled pair of qubits? It turns out they 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 Alice and Bob 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.

Let's say that Alice and Bob each hold an entangled qubit. The entanglement is simple: if either Alice or Bob measures their qubit to be pointing a certain direction, the other qubit will collapse to the opposite direction. This is the only thing they have.

How can they use this behavior to beat the game?

In the visualization below, we've labeled different points on the sphere with colors. If Alice or Bob measure their qubits and it's pointing toward the color red, they wear red, and vice versa for blue. Alice and Bob measure on different axes depending on whether they're grumpy or happy. Notice that Bob's axes are rotated 45° from Alice's.

If they follow this procedure, with axes as defined below, they will actually win 85% of the time, regardless of who's happy or grumpy.

Alice and Bob each have two measurement axes (one for each mood). Bob's axes are rotated 45° from Alice's, and his colors are flipped.

How to play

Alice and Bob each hold one half of an entangled pair.

You get to choose Alice and Bob's moods, and see if they get it right! Go ahead and measure Bob and Alice's qubit in the appropriate directions by pressing the buttons. Then, repeat that experiment many more times by pressing the button that appears! The results are tracked below in the grid. Try all four combinations of moods, and see what accuracies you get!

Alice

Alice's measurements

No measurements yet.

entangledmeasured

Bob

Bob's measurements

No measurements yet.

Alice and Bob can each pick either measurement. Bob's axes are rotated 45° from Alice's, and his colors are flipped.

Why this works

This works because of entanglement, and the way we have chosen the measurement axes.

Here's a summary, which we will dig into below:

We begin with our qubits in the “singlet state”, which is perfectly anti-correlated. Notably, when one person measures their qubit, the second qubit collapses to the exact opposite spot.
After the second qubit collapses, we've positioned the red/blue labels around the sphere such that the correct color is always only 45° away from the collapsed state. This way, it's 85% likely to get the proper color!

Recall the figure for single-qubit collapse probabilities:

If a qubit is pointing at the top of the sphere, the probability of collapsing to the +outcome of an axis at angle θ is cos²(θ/2).

Now we can look at all the options for Alice, who we'll assume measures first. (It would be the same if Bob measured first.) For each option of Alice's mood and her result (on the left), we show the probabilities of Bob getting each color if he's grumpy or happy. Notice that the correct color is always just 45° away for him.

All four cases of Alice's measurement. In each row, Alice's qubit has collapsed to one of her axis ends; Bob's anti-correlated qubit then sits at the opposite spot, and each of his four axis ends has a cos²(θ/2) probability of being measured. The “match” outcome lands at 85% in every row.

Mapping it to the math

The math is short, and you don't need much background to follow it. Three ideas:

1. The singlet, written down

Physicists write the singlet, the special pair Alice and Bob share, like this:

∣ ψ^{-} ⟩ = \frac{1}{2} (∣01 ⟩ - ∣10 ⟩)

Read it like this. $∣01 ⟩$ means “Alice's qubit is 0 and Bob's is 1.” $∣10 ⟩$ means the opposite. The singlet is a 50/50 mix of those two, but a special mix.

The defining property: measure both qubits along the same axis and you always get opposite outcomes. That's the “perfectly anti-correlated” rule we've been using. The thing worth knowing is that this works for any axis they pick, not just one specific direction.

2. Computing the probabilities

Quantum mechanics gives one rule for the joint outcome. Say Alice's axis points along the unit vector $a$ (so her $+ 1$ outcome is the qubit state $∣ a ⟩$ ) and Bob's along $b$ . The probability that Alice measures her qubit into $∣ a ⟩$ and Bob measures his into $∣ b ⟩$ , starting from the singlet, is the squared inner product of that joint outcome state with the singlet:

P (∣ a ⟩, ∣ b ⟩) = ⟨ a ∣ \otimes ⟨ b ∣ ∣ ψ^{-} ⟩^{2}

This is the Born rule applied to the joint state. Plug in the singlet, expand the algebra, and everything collapses to a clean closed form:

⟨ a ∣ \otimes ⟨ b ∣ ∣ ψ^{-} ⟩^{2} = \frac{1}{2} sin^{2} (\frac{θ _{ab}}{2})

where $θ_{ab}$ is the angle between Alice's and Bob's measurement axes. Two quick sanity checks: same axis ( $θ = 0$ ) gives zero, which confirms the anti-correlation rule (both can never get $+ 1$ ). Opposite axes ( $θ = 180°$ ) give 1/2, which says they always agree when measuring opposite directions.

3. Test it at 45°

In our setup, Alice's axes are vertical and horizontal; Bob's are rotated 45° from hers. So three of the four mood combinations have $θ_{ab} = 45°$ . Plugging in:

P (Alice + 1, Bob + 1) = \frac{1}{2} sin^{2} (22.5°) \approx 0.073

That's the chance both Alice and Bob get the $+ 1$ outcome. Because Bob's colors are flipped relative to Alice's, same measurement value means different color — a no-match. The matching cases are when Alice and Bob get opposite measurement values, and summing those two joint probabilities gives:

P (match) = cos^{2} (22.5°) \approx 0.854

That's the 85% in every row of the figure above. The fourth case “both grumpy” works out the same by symmetry, since the game flips the win condition there.

Hold them

Two qubits you can hold in your hands.

Qubi is a model qubit. Pair them up, run the gates, build the intuition that this guide just laid out, by touch.

Get your model qubits Read: What is quantum entanglement?

The Bell Paradox

Two qubits you can hold in your hands.