In this post: I try to explain Bell Inequalities and the NoCommunication Theorem, while you try to write short javascript strategies that violate them.
An Apparent Paradox
Quantum mechanics is frustratingly hard to explain. There are a lot of reasons for that. One of the main reasons, and the one we'll be touching on throughout this post, is QM's complicated relationship with locality. In particular, it seems to both require and prohibit fasterthanlight transfer of information.
To make things worse, the requirements and prohibitions on quantum FTL signalling are proven mathematical facts known as Bell's Theorem and the NoCommunication Theorem respectively. They derive from the postulates we use to define quantum mechanics as a mathematical model. If we found experiments that contradicted either theorem, they would still be correct. Instead, our belief that reality is quantum mechanical would fall (or at least need some adjusting).
We have a paradox here, but it is a paradox in the wellfoundedbutseeminglyabsurd sense instead of in the leadstoacontradiction sense. The need for FTL signalling and the ban on FTL signalling apply to subtly different cases.
Signalling Required: Bell Tests and the CHSH Game
In the 1960s, John Bell discovered that there are coordination games where quantum entanglement is a useful resource. Games where you can win more by taking advantage of reality being quantum mechanical. Classical players, without FTL signalling, simply can't win as much as quantum players at these games.
The smallest coordination game where quantum entanglement is useful is known as the "CHSH" game. In this game, the goal is for two isolated players (Alice and Bob) to make moves that satisfy $(m_a \text{ xor } m_b) = (r_a \text{ and } r_b)$. The $r_a$ and $r_b$ values are the outcomes of coin flips by referees. $r_a$ is only told to Alice, while $r_b$ is only told to Bob. The $m_a$ and $m_b$ values are Alice and Bob's respective moves. In other words, Alice and Bob want to make the same move unless the referees both flip heads. This is difficult because each player sees only one of the coin flips.
It is claimed that no classical strategy can expect to win the CHSH game more than 75% of the time (or less than 25% of the time), but quantum strategies can win up to 85.3% of the time (more specifically, $\cos^2 \frac{\pi}{8} = \frac{1 + \sqrt{2}}{2\sqrt{2}} \approx 85.35534...\%$ of the time).
(Another game where quantum entanglement is useful is the Merminperes magic square game. It's the simplest game where the quantum strategy is better and always wins.)
Checking that no deterministic strategy beats 75% is easy. There's only four deterministic strategies each player can follow: "always false", "always true", "match referee", and "contradict referee". That gives sixteen possible strategy combinations for two players; few enough that you can just check each one. (They all win either 25% or 75% of the time.)
But what about probabilistic classical strategies? Maybe something interesting happens if each player randomly switches between the deterministic strategies, possibly based on the referee's coin flip? And what about that analogy people always give for entanglement, where you put the result of a coin flip into two boxes and opening one box instantly tells you what's in the other box? Maybe the pervasiveness of that analogy means that shared random bits are useful here?
Write Your Own Classical CHSH Strategy
Instead of trying to convince you with words that probabilistic classical strategies (including ones with preshared random bits) can't beat 75%, let's try something different.
Below this paragraph, there is an interactive widget with two text areas.
The text areas accept arbitrary javascript code, and the entered code is used in simulations of the CHSH game to determine the players' strategies.
All you have to do is assign true or false to the move
variable.
You can base the value you assign on the refChoice
variable (i.e. the referee's coin flip), on values in the sharedBits
array (i.e. the preshared coin flips), or on whatever else you want (I dunno, the time?).
The goal is to win statisticallysignicantly more than 75% of the time. Your results are summarized in the table at the bottom of the widget, and will update automatically as you edit the code. Go ahead, give it a try:
var sharedBits = [Math.random() < 0.5, Math.random() < 0.5, …] 
var move; //ALICE's move
var move; //BOB's move

So... assuming you actually tried, how did you do? Did you beat 75%?
(Did you consider cheating? Maybe you can win by escaping the sandbox.)
The basic intuition I get, from playing with the widget, is that each quadrant gets 25% of the "result fluid" filling up the individual case boxes. When you try to shift fluid within a quadrant, you always end up causing an equivalent shift in an adjacent quadrant. This is fine in the topleft, because scoreincreasing shifts there correspond to scoreincreasing shifts in the adjacent quadrants. But in the bottomright, scoreincreasing shifts correspond to scoredecreasing shifts in the adjacant quadrants. There's a wall at 75% because at that point every improvement in the bottomright is countered by a loss in the topright or bottomleft (or viceversa).
An actual proof that 75% is classically optimal is left as an exercise for the reader.
Quantum CHSH Strategies
The best quantum strategies for the CHSH game are actually quite simple, operationally speaking. It's understanding the underlying model that's a bit complicated.
When given a qubit, you can do two things: turn it, and measure it. When two qubits are entangled, their measurements will be correlated. The amount of correlation depends on the relative angle between the measurements that were performed on each qubit.
Quantum mechanics says, and experiments confirm, that if one of the entangled qubits is measured along an axis $v$, the other is measured along an axis $w$, and the angle between $v$ and $w$ is $\theta$, then the proportion of agreeing outcomes is $\cos^2 \frac{\theta}{2}$. (This only applies to the first measurement on each. After that they're not entangled.)
 Measurements along axes that differ by 0°, i.e. along the same axis, agree 100% of the time.
 Parallelbutoppositeindirection axes, i.e. ones that differ by 180°, agree 0% of the time.
 Perpencidular axes, which differ by 90°, agree 50% of the time and disagree 50% of the time (i.e. are uncorrelated).
 Axes that differ by 45° or +45° will agree ~85.3% of the time while axes that differ by 135° will disagree ~85.3% of the time.
(Actually, I'm simplifying a bit. The correlations depend on which entangled state you start with. Physicists' favorite entangled state is the "singlet state" $\frac{1}{\sqrt{2}} \ket{01}  \frac{1}{\sqrt{2}} \ket{10}$ where the qubits have opposite phase and value, but we'll be using the state $\frac{1}{\sqrt{2}} \ket{00} + \frac{1}{\sqrt{2}} \ket{11}$ where the qubits simply agree. The difference does matter, but you don't have to care about it here.)
The trick quantum strategies can use is that, if you start with the measurement axes separated 45°, rotating by 90° once gets you to +45° but rotating by 90° twice gets you to 135°. This is useful in the CHSH game, because both the game and this 45to45to135 trick correspond to approximating an AND gate while using only local operations. One player choosing to rotate their qubit has no effect on the rate of agreement (both 45° and 45° agree 85.3% of the time), but both players choosing to rotate swaps us into the disagreeing85.3%ofthetime case.
Write Your Own Quantum CHSH Strategy
You can try out quantum strategies for the CHSH game using the widget in this section.
You enter javascript into each of the text areas, and it will be used into simulations of the CHSH game.
The difference compared to the last widget is that, here, you have access to two new functions: turn
and measure
.
The measure
function measures the player's qubit's value in the computational basis (along the Z axis).
If you just use move = measure()
for both strategies, the qubits act like a classical preshared random bit; to get other correlations, you need to turn
them first.
The turn
function takes an axis (X
, Y
, or Z
), as well as an angle in degrees, and applies a rotation operation to the qubit.
For example, if you wanted to measure along the Y axis instead of the Z axis then you would turn(X, 90)
before measuring.
(Unfortunately, explaining exactly what each rotation operation does to the state is outside of the scope of this post. See Converting Rotations into "Nice" Qubit Operations and Entangled States are Like Unitary Matrices if you're interested in that.)
I prepopulated an optimal strategy into the widget, but feel free to erase it without looking and try to find one yourself. Or maybe see if you can find strategies that do better than 85.3%?
sharedQubits = √½·00〉+ √½·11〉 
var move; //ALICE's move
var move; //BOB's move

So quantum strategies can demonstrably outperform local classical strategies at the CHSH game. But before we move on to interpreting what this means, and jumping to conclusions about secret FTL signals, let's cover the NoCommunication Theorem.
Signalling Prohibited: No Communication Allowed
What does it mean for Alice to communicate information to Bob? As with the Bell inequalities, we can think of this in terms of winning a game.
The game is much simpler this time: there's only one referee, and only one of the players makes a move. The goal is to satisfy $m_b = r_a$. Once again, $r_a$ is the outcome of a coin flip performed by a referee and known to Alice. Also, $m_b$ is the yesorno move performed by Bob. You win when the moves correlate with the flips.
The NoCommunication Theorem states that Alice and Bob can't expect to win this game more than 50% of the time, even if they have access to preshared entangled qubits.
Oh hey, another bound on the rate a game can be won that should apply to any strategy. That means it's time to
Write Your Own Quantum Communication Strategy
The third (and final) interactive widget is found below.
It works exactly like the quantum widget for the CHSH game explained earlier.
Only the game has changed.
Both players can still turn
and measure
, but this time only Alice cares about refChoice
and only Bob needs to assign a value to move
.
The goal is for Bob's move to correspond to the referee choice that was given to Alice.
People in the comment sections of pop science articles about entanglement are always suggesting strategies for this game.
Maybe whether or not Alice measured the system can be detected by Bob, so entropy can squeak through by having Alice only measure when refChoice
is true?
Or maybe, since the axis that Alice measures will affect the states that the system can instaneously collapse into from across the universe, measuring along a different axis can be detected by Bob?
Can you come up with a way to win more than 50% of the time?
sharedQubits = √½·00〉+ √½·11〉 

Don't try too long.
My intuition for why this can't be done rests on the fact that quantum operations commute when applied to separate qubits. If "Alice does X then Bob does Y" causes result Z, then result Z must also occur for "Bob does Y then Alice does X". I'm not going to prove this, even though it's not hard, because it would require explaining in detail how quantum operations are represented mathematically and how to expand singlequbit operations into wholesystemoperations. Suffice it to say that $(A \otimes I) \cdot (I \otimes B) = (I \otimes B) \cdot (A \otimes I)$, or that Alice's operations accumulate as leftproducts while Bob's accumulate as rightproducts.
Because remote quantum operations commute, any strategy for sending quantum information instantaneously could also send that information into the past. If Alice could send information to Bob by twiddlyfwaddling her qubit before he blorgamorped his qubit, the math would also allow for Bob to receive the message while Alice waited as long as she wanted to before doing the twiddlyfwaddling. This is especially bad because it's not solved by giving the universe a preferred rest frame (i.e. we're not bootstrapping the FTL signalling into time travel by using special relativity; the time travel comes directly from the quantum mechanism we assumed).
You might not have a distaste for time paradoxes being allowed but I find it to be a compelling reductio, so I'll bluntly leave my explanation for nocommunication at that. If you want more notindepth discussion of the NoCommunication Theorem, you can read this or this or this or this.
(In the specific case of Bell tests, the reason the nonlocal correlations can't be bootstrapped into communication is that you need to compare results. You can encode information into whether the CHSH game was won or lost, but the other party can't tell if they won or lost until you get back together and compare notes. Comparing notes requires getting those notes to the same place and the same time, and doing that is bounded by the speed of light.)
Intepretation
Alright, enough explanation and infodumping. Let's address the paradox. We have two results:
1) Bell tests show that a classical system would need FTL communication to simulate a quantum system. There are useful nonlocal correlations.
2) FTL communication is impossible in quantum mechanics.
How can these possibly both be true at the same time? I mean, they're not exact logical negations... but it still seems kind of weird. Why doesn't beating Bell tests prove that we can communicate?
The answer is very simple, if unsatisfying: reality ain't classical. If reality was classical, if it matched our preconceptions, Bell tests would in fact prove that reality needs FTL communication to do what it does. But it's not, so it doesn't.
Different interpretations of quantum mechanics concretely resolve this issue in different ways. Some interpretations have a notion of locality and use a different effect to explain the correlations (e.g. ManyWorlds, Retrocausal). Other interpretations just use blatantly nonlocal effects (e.g. Collapse, PilotWave). Still other interpretations just don't give a damn (i.e. SUAC). But, ultimately, QM's postulates and the theorems we've discussed force every interpretation to have some amount of weird. At best, you get to pick which brand of weird you find the least offensive.
Physicists are telling the truth when they say quantum mechanics is counterintuitive. Quantum mechanical systems have nonlocal correlations, but you can't use those correlations for communication. You can prove both of those statements, for yourself, from QM's postulates. You'll still be left with the blunt fact that that's pretty weird. You can learn the math, and how it applies to reality, but you're never going to find out that the math secretly predicts what evolution operating in the manymanyparticles regime has shaped us to expect apriori about locality. Those apriori expectations are simply mistaken. How unfortunate.
(Of course, calling reality weird is a bit funny. If anything, reality should set the standard for normal. We're the weird ones with our weird expectations, left pondering and bemused by reality's taste in regularity.)
Summary
Quantum strategies using entanglement can win some coordination games more often than would be possible classically. If the universe were classical, this would imply the players were secretly communicating. But communication is not one of the coordination games quantum strategies are better at. This is counterintuitive.