Math error in Brian Greene book?

I’m reading The Fabric of the Cosmos right now, and there’s something in it that puzzles me. I think Greene may have made a mistake in one of his analogies. Either that or–more likely–I’m not getting something.

Anyway the analogy is an attempt to explain Aspect’s experiment proving quantum non-locality. I’m really interested in the analogy itself, though. I’m aware calculating quantum mechanical probabilities is different than calculating “conventional” probability. For those of you with access to the book, this is from pp. 107-110 of the Vintage paperback.

Anyway, the story is Mulder and Scully (Greene loves his pop-culture references.) are on vacation in widely seperate locales. Both receive a mysterious package containing hundereds of small titanium cubes. The cubes can be opened three ways we’ll call top, right and front. Inside the cube is a sphere which glows red or blue at random. It will glow a different color depending on what door is opened first.

Mulder calls Scully and claims that the packages have been sent by aliens. The aliens have a mysterious tech that works like this: If Mulder opens, say, the front, and it glows red, then the device will communicate a signal to Scully’s Cube making it glow red if she opens up the front. If she opens up the right or top, the Sphere will flash at random. Scully thinks this is dumb. The boxes were simply programmed from the outset to flash the same colors if the same doors are opened. Since there is no way to test for either hypothesis–the flashing is disabled if the boxes are tampered with–it’s undecidable.

Mulder puzzles on this for a while, then comes up with an idea. They will open up the boxes in order (they are labled) but open the doors at random. If Scully is correct then they should see the same color more than 50% of the time.

For example, say the box is programmed to flash RED(top) RED(right) BLUE(front). There are nine possible ways M and S can open the doors. TT, TR, TF, RT, etc. etc. Five of those ways give the same color: TT, TR, RT, RR, FF. The same reasoning would hold for any cube which might flash both colors, since they must have two doors flashing one color and one door flashing the other. Thus M and S must see the same color at least 5/9ths of the time. If and cube is either RRR or BBB they will see the same color always, so this will only increase the percentage of same color sightings above 50%.

So far so good. Mulder has shown that if Scully is right and the boxes are pre-programmed they will see the same color more than 50% of the time.

But my problem is that if Mulder is right and the boxes are not preprogrammed, they will still see the same color more than 50% of the time.

Say Mulder opens the front door and it glows red. Now Scully has a 1/3 chance of opening the front door, in which case she will also see red. Scully has a 2/3 chance of opening a different door, in which case the sphere flashes at random, giving her a 50-50 chance of seeing red or blue. So Scully has a total chance of 2/3 of seeing red. Thus she will see the same color as Mulder 2/3 of the time.

Granted, 2/3 is a different number than 5/9. But under Scully’s hypthesis, enough cubes could have been programmed RRR or BBB to raise the percentage of same color sightings to 2/3. In short the experiment cannot decide whether Mulder or Scully are correct.

Why am I wrong?

He’s saying that you will still get the same properties of both of them having a 50% chance to see either coolor if you make the assumption that RRR or GGG are forbidden states for the sphere. In that case, 10 out of 18 cases will result in them picking the same color. If you instead say that RRR or GGG are allowed states, they would then get the same color 16 out of 24 cases.

Greene’s math isn’t wrong, but I think it’s too skimpy to satisfy someone (like yourself) given to trying to work out the math and check him yourself. A much better treatment is given in (and, some might say, is half the point of) Jeffrey Bub’s Interpreting the Quantum World.

punoqllads I don’t think that helps since all he’s saying is that if they see the same color more than half the time, Scully’s hypothesis is correct. My question is that it seems that under both hypotheses they should see the same color more than half the time. Greene goes into the calculations of the actual experiment in an endnote, and I can see how it worked out in that case. But the math in the analogy should work on it’s own terms, otherwise why put it in there at all?

Mathochist Thanks for the book tip. I’ll check that out. Again though, I think the analogy should stand on its own, without reference to the experiment it purports to explain.

On the whole, I like the book, it was just a little perturbing to find this in it. The book gives a good overview of modern cosmology, though it does leave one with a hunger for more of the details.

It’s a trade-off. If he explains the analogy in far more detail he can satisfy the people who want to check it but haven’t already seen the answer. On the other hand, he’d lose the people who don’t have the ability/interest to check it, and that’s ultimately the larger audience.

Let me see if I can give a better explanation…

Either the boxes are identically programmed, or they’re random-but-mysteriously-connected. If they’re programmed then for six of the programs they’ll see different colors in four of the nine ways of opening the pair of boxes. In the other two cases, they’re guaranteed not to see different colors (only one color to see). So, if we make no assumption on the distribution of the programs, P(different colors) <= 4/9, and it only manages to be 4/9 if the uniform programs don’t show up at all. If we assume all eight programs show up equally likely, then P(different) = 1/3.

Now, in the random-but-connected part, I think I see your error. You make the (perfectly reasonable) assumption that the randomness takes the form of guaranteeing identical answers if the same doors are open, but giving completely uncorrellated results if the doors aren’t. The real case, which Greene wisely didn’t go into, is that the boxes each contain one of a pair of entangled quantum states. To explain exactly what this means and how probabilities come from it is difficult and that’s why he doesn’t talk much about it. The important part isn’t exactly what the random hypothesis predicts for the probability, but that the programming hypothesis says the probability of getting different colors is definitely less than 1/2.

Why is that all that matters? Because when you actually run the experiment in the real world you get the same color with probability 1/2. Greene isn’t trying to prove Mulder right, which is why he doesn’t go into exactly what Mulder’s hypothesis is. Greene is only trying to prove Scully wrong.

Mathochist Thanks, that makes sense now, and I understand your earlier point.

That doesn’t sound right. Even though picking the same door makes them come up with the same color each time, the probability of them getting the same color is 1/2? Consider a set of 10 trials, where by coincidence Scully and Mulder happen to pick the same door all 10 times. You can’t have Scully and Mulder finding the same color all 10 times yet having 50% of the trials yield different colors. Or am I misunderstanding what you’re saying?

Of course it sounds screwey. It’s positively bizarre from a classical standpoint, but we’re talking quantum mechanics here, remember? Sit down, write out the EPS state and the correllation operator and work it out yourself.

No, my example is more than just screwy, it’s a logical contradiction. You can’t have 100% of the samples matching up and 50% of the samples not matching up at the same time. But that’s exactly what you seem to be saying.

Consider Scully and Mulder each rolling a die 4 times, for 4 trials. If they roll a 1 or 2, they open the top. If they roll a 3 or 4, they open the front. And, if they roll a 5 or 6, they open the side.

Now, it’s theoretically possible that they each roll the same door on the same trial, and open up the same door. They would then each see the same result as each other, all 4 times. But what you seem to be saying is that they will see different results 2 out of those 4 times. So it can’t happen. Therefore, it is impossible for Scully and Mulder to each roll a die 4 times and get the same number as each other all 4 times. Or I’m misunderstanding what you’re saying.

Yes, you’re still approaching it from essentially a classical viewpoint. If you work it out using the rules of quantum mechanics – maybe moving it back to the original formulation of measuring spin-states of electrons would help – you get 1/2.

OK, I’m a little confused by this. (I don’t have the Greene book handy, so I might just be misunderstanding the whole thing.) It sounds like the book is trying to contrast quantum mechanics with local hidden-variables theories, none of which make all of the same predictions as QM; but I don’t see how the example makes this point.

From reading the comments so far, it sounds like the system being modeled is supposed to be the maximally-entangled Bell singlet state, measured along one of the three Cartesian coordinate axes (Top, Right, Front). This state has the property that measurements of the two particles along the same axis give opposite results (each individual measurement being unbiased), and measurements of the two particles in orthogonal axes are uncorrelated.* (So to make the analogy nicer, one of the boxes will invert its responses so that the “opposite” results give the same color. Then measuring along the same axes gives the same colors; measuring along different axes gives uncorrelated results.) If this is going to be a useful example then we shouldn’t be able to replicate this classically, as with Scully’s suggestion.

But consider the following classical example (I can’t tell from reading the thread whether this is the same as Scully’s proposal): Each pair of boxes is set identically with three independent fair coins. That is, there are exactly eight different kinds of box pairs: RRR|RRR, RRB|RRB, RBR|RBR, RBB|RBB, BRR|BRR, BRB|BRB, BBR|BBR, BBB|BBB (listing the top, right, and front colors for Mulder’s and Scully’s cubes). Each box will flash the given color for the first door opened (and after that, randomize itself; only one measurement is allowed on each box). Doesn’t this give exactly the same statistics as with the Bell-state case? When Mulder and Scully open corresponding doors, the colors they see are always identical; when they open different doors, then (because the two random variables representing the two doors are independent fair coins) the colors are uncorrelated (they see the same color 1/2 of the time). In each case they should see the same color 2/3 of the time. I don’t see any measurements Mulder and Scully can do on this system to distinguish the two cases.

(*)Note that this is different from the choice of axes used in the Aspect experiment. In Aspect, some of the axes are nonorthogonal (e.g., Mulder might measure along “north” and Scully along “northeast”), so that the measurements along different axes give correlated but not deterministic results. These probabilistic results are what are hard (OK, impossible) to replicate with a classical local hidden variable theory.

Yes, it’s a direct (if incompletely delineated) translation of the classical Bell no-go theorem for deterministic hidden-variables theories. Scully’s hypothesis is deterministic hidden-variable, and the inequality described is exactly Bell’s inequality. What’s left out is the exact description of Mulder’s hypothesis (Copenhagen) and that it predicts exactly the 1/2 result that agrees with experiment.

Here’s the logic from the book, with the logical connectives hammered on hard:

If
the boxes are preprogrammed (there are deterministic “hidden variables” behind QM)
then
Mulder and Scully will see the same color at least 5/9 of the time – more than half, at any rate –
but
when we run the experiment we find they agree half the time
therefore
Scully is wrong and the boxes cannot be preprogrammed.

To go further and say that the experiment endorses Mulder’s position (rather than merely refuting Scully’s) is beyond that section of the book.

OK, I think I’ve been misunderstanding the description of the “actual” (quantum) behavior of the boxes. Let me try to describe my understanding from reading this thread (I haven’t read the book):

  1. The “experiment” being done: Mulder opens one of the doors on his box #n, and Scully opens (uniformly at random, and independent of Mulder’s choice) one of the doors on her corresponding box #n. They then compare colors, and count (over many trials) the number of times they saw the same color.

  2. The mysterious behavior of the boxes: If Mulder and Scully open corresponding doors, the colors they see are always the same (50% RR, 50% BB). But if they open different doors, the colors they see are uniformly, randomly distributed (25% RR, 25% RB, 25% BR, 25% BB). [This is the way I interpreted the OP; I then interpreted this in the language of Bell states, as measurements made along one of three orthogonal axes for the singlet state.]

But given this behavior, we can calculate how often the actual boxes will show the same color (since I’ve given the classical probability distributions for the results). 1/3 of the time they pick the same door and see the same color; 2/3 of the time they pick different doors, and half of the time they see the same color. So 2/3 of the time they see the same color. This is the same result as in the Scully-like proposal I described above, and is different from the result you quote (that they agree exactly half the time).

So now (trying to reconcile these statements) I think I was misunderstanding the description of the quantum boxes. It is certainly possible to choose three (nonorthogonal) axes for a possible measurement on halves of a Bell state in such a way that if corresponding doors are opened the colors are always the same, but if different doors are opened the results are randomly (but not uniformly) distributed, with the overall effect that when performing the experiment described above, they agree exactly half of the time. (In particular, the axes are chosen so that when different doors are opened, the colors are the same 25% of the time and different 75% of the time: 12.5% RR, 37.5% RB, 37.5% BR, 12.5% BB.) This is more like the Aspect experiment, in which (as I mentioned earlier) it was important that the measurement axes not all be orthogonal. Orthogonal axes would not have provided a Bell-inequality violation.

Well, that’s just it. Greene doesn’t describe the quantum state explicitly. He’s just showing that Scully’s assumption leads to a lower bound on the probability of matching colors, and asserting that experiment doesn’t bear this out.

Sure. I was just confused because from my initial understanding of the boxes they behaved entirely classically (colors agreeing half the time when different doors are picked, instead of overall). This made it seem like rather a bad example for differentiating quantum and hidden-variable theories. :slight_smile: It’s a much better example now that I understand what it actually is.

You don’t need quantum mechanics to understand why what you just said is wrong. You appear to be falsely assuming that if the probability of something is P, and we conduct a finite number of trials T, then it is guaranteed to happen P*T times. But this is false.

In other words, just because you can do a bunch of trials and get the same result every time, it doesn’t prove that the probability of agreement is greater than 50%.

Consider flipping two coins. There’s obviously a 50% chance of agreement. But we could flip 10 pairs of coins, and have them both come up the same all 10 times. However, the fact that we can achieve 100% agreement doesn’t disprove our claim that the probability of agreement is 50%.

However, the more trials you do, the less likely it is that the percent agreement you measure will differ significantly from the actual probability of agreement. So if you do enough trials, you can eventually be fairly certain the number you’re getting is close to the true probability.

You seem to miss the point of the Bell experiment. The inequality says the probability is above 5/9, the experiment says it’s below (with a negligible chance that it’s really over 5/9).

Greetings all,

I’m new here – found this forum by googling for information about “Brian Greene Mulder Scully illustration”.

I have not read this entire thread, but I thought I would post something I’ve done related to this subject, and see if I get any interest.

I was fascinated by Greene’s illustration when I first read it years ago, and being a software engineer by profession, I immediately set to writing a program to demonstrate it. To my dismay, however, I could not get my program to show any appreciable difference in the outcome based on whether the cubes were pre-hardwired, or worked as described in Greene’s illustration.

However, I was thinking about it recently, and decided to write another program. This time I think I was able to prove that there is a measurable difference, but I’m not sure what my data means and would appreciate any comments.

Here’s what my program does: It performs the experiment outlined in the book with 1000 cubes, first using hard-wired cubes (pre-set to show pre-determined colors behind the doors), then using cubes that work like Greene describes (the first time a door is opened on a cube, it’s colors are chosen at random, and the resulting set of colors is passed to the its corresponding cube in the other set, so that it will show the same colors).

The program then goes on to perform this same test (1000 cubes) 10,000 times. For each test, it stores in a file the percentage color agreement. So, for example, a value of “50” for “Percentage Agreement” on a line means for that set of 1000 cubes, Scully and Mulder saw the same color cube-for-cube 50% of the time.

Then, I take that file into Excel and graph some stuff. Initially, I graphed the data straight up, and it was as I suspected. This image shows the “Percent Agreement” graphed across the 10,000 tests. Blue is the hard-wired cubes, and red is the “entangled” cubes. The black line in the center is the trendline Excel shows for both lines. The graph is predominantly red only because Excel graphs the red line on top of the blue line.

Then, I took a different tack. For each line, I calculate the average percentage agreement up to that point, and then graphed this “cumulative average” for both the hard-wired, and “entangled” tests. The result is shown in this image. There you can clearly see that there IS a difference between the outcomes based on whether the cubes are hard-wired or behave in the “entangled” fashion. The difference it tiny – amounting to no more than about 0.1% over the 10,000 tests. However, regardless of scale, can it really be denied, looking at that graph, that there is a real difference in outcomes between the two tests?

If anyone would like to see my source code (in C#) I’d be happy to provide it.

It’s a zombie, but oh well.

Something is wrong with your program–it is showing a 2/3 probability for both cases. I couldn’t say for sure about the 0.1% difference. There’s a good chance it’s just noise–just based on the number of trials, you’d expect around a 0.03% noise level, and 0.1% isn’t all that much higher. But it might also be some systematic problem like rounding error.

Anyway, the important part is that you’re getting the wrong number for the pre-coded cubes. How are you distinguishing between the different possibilities? There are 8 total, corresponding to red/blue for each of the 3 axes. There’s no need to run 10k trials; just the 8 are sufficient, though you may want to bump the 1000 up to something larger.

Without having worked it out, it seems possible that averaging the 8 possibilities together gets you a 2/3 probability. So maybe that’s where it’s coming from. But that’s not the same as the minimum possible number, which is 5/9. You’ll need to graph the 8 possibilities separately to make that distinction.

Now that I think about it, I think there are only two distinct pre-coding types: 6 (out of 8) of a 2-1 split (red-red-blue, blue-red-blue, etc.) and 2 of all the same color (red-red-red). The 2-1 splits all (I think) come out as 5/9 probability of matching colors, while the same-color ones obviously have probability 1. (6/8)(5/9) + (2/8)(1) = 2/3, so I guess that’s where your number is coming from.

Your other 2/3 number is correct and based on how 1/3 of the time the axes are in alignment and so the colors match, while 2/3 of the time the axes aren’t aligned and so the colors match with 1/2 prob. (1/3)(1) + (2/3)(1/2) = 2/3.