Comment by cyphar | Hacker News Reader

Comment by cyphar | Hacker News Reader

Parent Full thread

Most of Bell's thought-experiments result in fairly unintuitive answers, so it's quite understandable that the distinction would be unclear to someone who hadn't seen this problem before.

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.

> it's because the two particles don't know along which (relative) angle the other particle will be measured ahead of time

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?

The nature of the source is the reason they are correlated in that particular way (that is effectively what entanglement is achieving, at least in this simple example). In particular, the source in the classic Bell inequalities produced particles that had opposite spins (guaranteed by conservation of angular momentum).

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.

I don't get it.

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.

> If the source is responsible for the correlation, what other explanation do we need? Why isn't it enough that the source is responsible for the correlation?

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).

I still don't get it.

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));
    }

'other angle' is the angle of the other entangled particle.

This algorithm returns precisely -cos(|oA - oB|) as you suggested.

> 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?

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.

Reply on news.ycombinator.com