If you are seriously looking for a nugget I would suggest trying to understand the nonlocal game example (which was hidden in the write up)
"But first, to see how the games work, let’s imagine two players, Alice and Bob, and a 3-by-3 grid. A referee assigns Alice a row and tells her to enter a 0 or a 1 in each box so that the digits sum to an odd number. Bob gets a column and has to fill it out so that it sums to an even number. They win if they put the same number in the one place her row and his column overlap. They’re not allowed to communicate.
Under normal circumstances, the best they can do is win 89% of the time. But under quantum circumstances, they can do better.
Imagine Alice and Bob split a pair of entangled particles. They perform measurements on their respective particles and use the results to dictate whether to write 1 or 0 in each box. Because the particles are entangled, the results of their measurements are going to be correlated, which means their answers will correlate as well — meaning they can win the game 100% of the time."
These notes might help to understand the basic idea: https://www.scottaaronson.com/qclec/14.pdf In particular, after understanding why the best they can is win 89% on normal circumstances, the easier examples at the beginning of the notes provide intuition of how a quantum strategy might help for the 3-by-3 grid game. Edit: also see the write up by ahelwer below.
It's a bit more complicated to follow but absolutely worth the work.
Why not just imagine Alice tells Bob how she's going to choose what to write in the boxes? That works 100% of the time too.
I don't see a distinction between "Alice and Bob communicate" and "a pair of entangled particles gives Alice and Bob a piece of shared information that, by prior arrangement, they both interpret in the same way."
Alice and Bob can pre-share arbitrary amounts of classical information and still not win more than 89% of the time.
What would your strategy be for sharing information before the game starts?
The crux is that Alice and Bob don't know which row/column the other is filling, and they are given symbol-selection rules which would contradict each other if used to fill the whole square, preventing a pre-ordained strategy (e.g. a pre-seeded RNG) with 100% success rate. So a 100% strategy would intuitively need some way of communicating _after_ they are given their row and column, and the math shows that the quasi-communication granted by entanglement is sufficient.
I'm sure my explanation is also insufficient, but there should be enough information at the link to convince you if you're curious.
[1] https://en.wikipedia.org/wiki/Quantum_pseudo-telepathy#The_M...
> They're apparently the same phenomenon
You're the one deciding that they're the same phenomenon, by choosing to only imagine using the entangled particle pair to seed a PRNG on each side using the same method.
The key thing to realize is that Alice and Bob have quantum mutual information, not classical, and that "quantum" here is not just technobabble but actually has meaningful consequences.
If Alice and Bob choose to destroy the quantum information and extract the same bit(s) of classical information from their respective halves of the entangled pair(s) before the game starts, then the quantum case has been reduced to the classical case and you can think of them as the same phenomenon. But if instead they keep the quantum information around past the start of the game and decide what to do with it based on the hand they're dealt, they can get a win rate that's impossible with only pre-shared classical information.
So entangled particles give you fundamentally different properties to a seed to an RNG (or any other arbitrary local hidden-variable setup you can come up with).
The problem can be rephrased as "does there exist an ideal PRNG that is not discernible from a true quantum RNG". Discerning between real RNG and PRNG is akin to solving the halting problem and one of the lynchpins if the proof.
If communication really needs to be forbidden, just let them agree on an RNG beforehand and share a seed. This seems to eliminate any quantum magic involved, which suggests I don't really understand the problem, which suggests that they haven't really explained it.
Can you explain how this will allow them to always win the game?
It may help to point out that (someone correct me if I'm wrong here...) this only works if both Alice and Bob are able to use their hidden information to choose the bases they're using to measure the qubit; if they were stuck with measuring at pre-determined angles, this trick doesn't work.
Disclaimer: zero quantum research background, only cs and a hint of math
Alice doesn't know which column is Bob's column, and Bob doesn't know which row is Alice's row.
The description of the game in the article is incredibly simplified. The original Bell game is that you take two particles and separate them spacially so that any local (slower-than-light) communicaton cannot interfere with the experiment. You then measure their spins in two orientations (the absolute orientation isn't important, only the angle between the two measurements is important) and repeat a large number of times. Then, sum how many times the two measurements had the same spin (+1) or opposite spins (-1) and divide it by the number of measurements to get the "correlation" (scare quotes because this is not the same as what statisticians mean when they say "correlation"). Repeat for a large number of different relative angles and plot "correlation" vs angle.
Quantum mechanics predicts (and experimental data produces) an inverse cosine "correlation" curve. However, it is simply not possible to produce that "correlation" curve using a local, hidden variable theory of quantum mechanics. Why? With some slight hand-waving, it's because the two particles don't know along which (relative) angle the other particle will be measured ahead of time -- if you permit non-locality (faster-than-light communication or "spooky" action at a distance) then this problem goes away. Now, the proof that this is the case is far more involved than this (and to be honest I'm not sure I understand it well enough to explain it). But hopefully that gives you some idea why the "just use an RNG" method is not sufficient.
Since the two particles are created from the same source, why couldn't it be that the two particles are affected by the initial state of their source so as that to appear correlated?
But that's effectively just a description of the problem. It doesn't really help you come up with a local hidden-variable theory of QM could produce the same "correlation" vs angle curve.
The problem (in computer science terms) is that you need to come up with an algorithm (which for a given particle (A or B) takes some state (called lA or lB), and an angle parameter (oA and oB)) that gives you an output which has the statistical distribution which agrees with QM if you randomly sample lA and lB values. In other words, write an algorithm such that the "correlation" of Measure(oA, lA) and Measure(oB, lB) is always equal to precisely -cos(|oA - oB|) without using global variables. It turns out this is (provably) impossible.
NOTE: In the real experiment you're actually measuring in 3D, so the angles should actually be unit vectors and the angle difference is actually (a function of) the dot product.
If the source is responsible for the correlation, what other explanation do we need? why we need to explain it with a local hidden-variable theory of QM?
Why isn't it enough that the source is responsible for the correlation?
And since it's impossible to write that algorithm, why not simply suppose the values of iA and iB are such that the correlation is possible?
Wouldn't global variables actually be non-local communication, instead of local? they are global variables, just like entanglement is 'global' for the universe.
Ah, okay -- that's a very good question (I skipped over a bunch of context on what problem we are discussing in my first responsd).
The problem is that all quantum mechanical measurements appear to be random. We might know (through our understanding of quantum mechanics) that measuring a particle in a particular state will give us spin-up 75% of the time, and spin-down 25% of the time -- but we cannot predict what any single measurement will give us. As far as we can tell, each measurement is as close to a real-life RNG as we can find.
This apparent randomness (and thus inability to predict individual experiments -- only being able to predict the statistical distribution of many experimental trials) has unnerved scientists for several decades, and thus many "deeper theories" have surfaced to try to explain where this randomness is coming from. Hidden-variable theories are one such class of theory, and are based around the idea that there is just some additional fundamental quantum numbers associated with quantum states which we don't know about -- these are the most natural-feeling theories because these kinds of extensions are a common theme in the history of physics. A local hidden-variable theory would be the most ideal because it would allow for the unification of relativity and quantum mechanics -- but alas, it is not to be.
> why not simply suppose the values of iA and iB are such that the correlation is possible
Punting the problem to the state doesn't help -- the state variables are decided ahead-of-time when the particle states are first created. Having your algorithm just use a value in the state still counts as an algorithm.
I really recommend trying to come up with a toy algorithm to solve this problem, to get an idea of where the hidden complexity arises.
The key problem (as you may discover yourself) is that you can't tell what angle the other particle will be measured with, and thus you don't know what statistical weight to apply to the state you have internally.
> Wouldn't global variables actually be non-local communication, instead of local?
Yes, "global variables" would be non-local. Hence why you can't use them (Bell's inequalities only prove the non-existence of local hidden-variable theories of QM).
In Bell's thought experiment, the angle between the axes of two particles is well known before any particle is fired. Why do you say the following?
> you can't tell what angle the other particle will be measured with
If the angle is 90 degrees, the correlation would be 50%, if the angle is 180 degrees, the correlation would be 100%, and when the angle is 0 degrees, the correlation would be 50%.
What are you saying (or rather, the theory says) is that since we have the above numbers, if there were local variables, then the correlation, for example, at 90 degrees should have been 25%, and the correllation at 270 degrees should have been 75%, but since in both cases the correlation isn't that, we can't have local variables?
Edit:
What about this algorithm?
struct ParticleState
{
double otherAngle;
};
double algorithm(ParticleState state, double angle)
{
return -cos(abs(angle - state.otherAngle));
}
This algorithm returns precisely -cos(|oA - oB|) as you suggested.
The relative angles I'm referring to are of the measurements (or rather, the relative angle between the two measurement vectors). Those angles can be thought of as being unknown when the particles are created (because you can imagine the experimenters for A and B randomly picking an angle after the particles have been separated).
We do "know" that the particles have a particular spin along a particular axis when they're created (more accurately, we know that the spin axes have a particular property -- that they have opposite spins along some axis), but we aren't necessarily measuring along that axis (and if you measure a quantum state along a different axis, you get random-looking results that follow a particular statistical distribution which is predicted by quantum mechanics).
To give an example of this property. Imagine a particle which we measure to have a +1/2 spin in the Z-axis. If you then measure it along the X-axis you'll get a result of +1/2 50% of the time (note though, that subsequent measurements of the same particle in the same state will give you the same result because measuring the particle collapsed its state to be along the axis you measured). Now, we aren't measuring along the Z-axis in this experiment (instead we're depending on a different property of the particles to calculate their correlation) but the idea is very similar.
> If the angle is 90 degrees, the correlation would be 50%, if the angle is 180 degrees, the correlation would be 100%, and when the angle is 0 degrees, the correlation would be 50%.
Sure, but if particle A is measured along -47 degrees from the Z-axis (angle chosen at random after the particles are separated), and particle B is measured along +133 degrees from the Z-axis (also a random angle chosen after the particles are separated), how would A (or B) individually know what statistical distribution to produce when you measure it? All the A particle knows is that you measured it at -47 degrees from the Z-axis and it doesn't know what angle B was measured against (heck, the scientist doing the measurement might not know what angle B is being measured against).
> 'other angle' is the angle of the other entangled particle.
The angle is not a property of the particle, it's a property of the measurement we take.
Like imagine the 3x3 grid suddenly changes after Alice and Bob write down their first message.
No, it does not. Let’s suppose Alice’s pre-negotiated (or predetermined by a seeded PRNG) choices for rows 1..3 are:
(A11 A12 A13)
(A21 A22 A23)
(A31 A32 A33)
And Bob’s choices for columns 1..3 are:
(B11) (B21) (B13)
(B21) (B22) (B23)
(B31) (B32) (B33)
If they are given row 1 and column 1, the intersect should match, i.e. A11 == B11. The same goes for any row and column selection: Aij == Bij. That is not, however, consistent with the requirement that the sum of all numbers for Alice must be an odd number (each row must add up to an odd number and there are 3 rows), and the sum of all numbers for Bob must be an even number (each column must add up to an even number and the sum of three even numbers is even). So, that’s the real catch for the classical setup. And yes, quantum correlation can beat it.
The point of the analogy is that in the quantum version there is a way to work around this unavoidable constraint in the classical problem.
The fact that they can't know which row/column they're going to be assigned makes it impossible to know with more than 8/9 certainty what the correct move would be. Even if they agree ahead of time what the values should be, the fact that they only know half of the information makes it impossible (classically) to win more often than 8/9 of the time. They have no control over and no way of knowing which row/column the other was given.
If I'm understanding correctly, the "quantum version" lets you increase that precision arbitrarily depending on how many measurements you make, up to 99.9999999...%
Really neat stuff
1. Every row has an even sum
2. Every column has an odd sum