Every day, a man comes to my door with a United States nickel in his hand. He asks me whether I’d prefer to examine the heads side (which is always painted either black or white) or the tails side (which is always painted either red or green). I choose each day according to my whims.
And the same thing happens to my sister. Different man, different coin, but each day he’s there with a painted nickel, offering to let her examine either the heads side or the tails side.
Sometimes we call each other to compare notes on the colors we’ve seen. Here’s what we’ve concluded:
- Our heads sides are never both white.
- Whenever one of our tail sides is green, the other one’s heads side is white.
We have thousands of observations to support these conclusions: On days when we both examine our heads sides, we never both see white. On days when we examine opposite sides and one sees a green tail, the other always sees a white head.
The Brain Teaser: Today we both chose Tails and both saw green. What colors were on our Heads sides?
Solution: By point 2) above, they were both white. But by point 1) above, that can’t happen. So….?
So what now?
Possible Resolution I. There are no such coins. You’re right. There are no such coins. But there are subatomic particles that behave exactly like these coins. It’s easy to set up an experiment where every day, my sister and I each receive an electron. We can examine the spin of our electrons in either the up/down direction or the left/right direction. And here’s what we find:
- Our electrons are never both spin down.
- Whenever one of our electrons is spin right, the other is spin down.
- There are days when both of our electrons are spin right.
By points 1) and 2) and the same simple logic we used for the coins, there can never be a day when both electrons are spin right. Nevertheless there are such days. In fact, on those days when we both choose to make left/right measurements, our electrons are both spin right about 8.3% of the time. That’s not a huge percentage, but it’s sure not zero either.
So Possible Resolution I doesn’t work, at least if we replace the coins with electrons.
Possible Resolution II: The coins are very very sneaky and they like to screw around with our minds, so they change their own colors depending on the choices we make, just to fool us. Sometimes they are both white on the heads side, but not on the days when we’ve both decided to check the heads side — so that Rule 1 is false but we get fooled into believing it.
Aside from its inherent implausibility (intelligent coins? intelligent coins with nothing better to do than to mess with our heads?), this resolution falters on the fact that my coin has no way of knowing what choice my sister is making.
Possible Resolution III: Neither side of either coin has a color until we decide to examine it, so that on a day when I examine my tails side, it makes no sense to ask about the color of the heads side in the first place.
Therefore I am not allowed to pose this brain teaser in the first place.
If that strikes you as implausible, I invite you to devise another Possible Resolution. Plenty of people have done so. None, though, has ever devised a Possible Resolution more plausible than Possible Resolution III.
Welcome to quantum entanglement.
If you liked this example, you’ll love Chapters 14 and 15 of The Big Questions.
Technical Appendix: For the cognoscenti, I wrote R=B+W, G=B-W, and assumed the coins are prepared in the initial state BB+BW+WB. Rule 1 follows. You can check that this state is equal to both 2BR+WR+WG and to 2RB+RW+GW. Rule 2 follows. You can check that this state is also equal to 3RR+RG+GR-GG, so that when both parties make R/G observations, GG comes up 1/12 (about 8.3%) of the time.
If the 2 coins could communicate and change color based on this communication then it would be easy to imagine them running an algorithm that delivered the described results. I think this would also be consistent with the statement ‘Neither side of either coin has a color until we decide to examine it’ since the algorithm used would have no need to assign a color to the unseen side.
I perhaps have not fully grasped the implications of III but I am not seeing how just the ‘Neither side of either coin has a color until we decide to examine it’ condition could allow the described results if there was not also some collusion between the 2 coins.
Rob Rawlings: Excellent point. But, precisely because of this concern, experiments have been conducted in which, in order for this to work, the communication would have to take place faster than light — which, to those well versed in physics at least — is the most implausible thing of all.
I don’t see how the electron version follows from your Technical Appendix. Your electron version is only about electrons being spin down or right. Your Technical Appendix is about R,G,B,W, which don’t seem to stand for anything in your post.
Roger: B is up, W is down, R is left, G is right.
I don’t quite follow how you are preparing the electron states, but regardless, electrons do not “behave exactly like these coins.”
Coins have two sides. Electrons don’t. It doesn’t even make much sense to say things like “Whenever one of our electrons is spin right”. You can do a measurement to try to detect a right component of spin, but that measure destroys whether an attempt to detect a down component of spin would have succeeded. So it doesn’t make any sense to talk about some electrons having spin right, and some having spin down.
You can prepare an electron so that an attempt to measure spin right will succeed, but that is not what you are doing. I think that if you restate the scenario to be more precise about what you mean by spin, then the paradox disappears.
Roger:
The analogy is quite carefully constructed, and I stand by it unreservedly.
With electrons: As you say, you can measure a left/right component of spin, but that measurement destroys your ability to determine what would have happened if you’d measured an up/down component.
With the coins: You can measure the red/green property of the tails side, but that measurement destroys your ability to determine what would have happened if you’d measured the black/white property of the heads side. (This is exactly why I set things up so that the postman is willing to show you only one side.)
Note the exact parallelism.
“So if you measure the position, you don’t know what you would have measured for momentum.”
I recently read an interesting analogy with radar range finder/speed gun.
Imagine you want to measure the speed and distance of an aircraft coming towards you with a Doppler effect radar. If you use a long pulse with lots of waves you will be able to measure the change accurately to get the speed, but the distance measurement will be imprecise. Alternatively, if you use a short pulse with a few waves you will be able to measure the distance accurately, but you don’t have many waves to determine the shift in wavelength to measure the distance. The more you know about the velocity, the less you know about the position. Slightly more technically it is because a wave with a concentrated Furrier transform has to be spread out in time and vice versa.
Roger: Your most recent comment seems to contain all of the content of your long and repetitive string of comments that precede it, so I’m deleting most of those.
You seem to persist in restating my point while pretending that I said something different.
So I will summarize my response to all your previous posts here: OF course there are no such coins. But there are electrons that behave *exactly* like these coins (and yes, the analogy is perfect — in both cases we have a two-dimensional state space for one object; a tensor product of two such spaces for the state space of a pair, and an initial state of BB+BW+WB where (B,W) is a basis for the one-particle state space).
The point is that a) if you saw coins that behaved like this, you’d be shocked. b) electrons do behave like this. c) that’s why newcomers to quantum mechanics are often shocked. d) newcomers to quantum mechanics who aren’t shocked have almost always failed to understand what’s so surprising here.
You’ve repeated most of these points several times, while continuing to suggest that I somehow said something that contradicts these points. If you want to post again, please tell me specifically whether you are disagreeing with a), b), c) or d) or quote the exact line where you think I said either not-a), not-b), not-c) or not-d).
There’s also a Possible Resolution IV, called “superdeterminism”: at some point in the distant past, the same causal influences that determined the coins’ colors also set in motion a chain of events that eventually lead to the neurons in both your and your sister’s brains being in such a configuration that you never both choose to measure up/down when the spins are both down. It’s true that most physicists consider this to be far less plausible than Possible Resolution III, although Nobel laureate Gerard ‘t Hooft is one exception. But it does get around the faster-than-light communication problem with Possible Resolution II.
Steve: Yes, you are saying a), b), c), and d). I am not trying to put words in your mouth.
I do disagree with b), c), and d). I was responding to your invitation to submit a resolution of the paradox.
I don’t know much about quantum entanglement, so just checking: When you say “We can examine the spin of our electrons in either the up/down direction or the left/right direction”, are you saying you can’t do *both*? Once you examine in the up/down direction, you lose the ability to examine in the left-right direction, and vice versa? (If so, then it might be helpful to edit to clarify this.)
Roger (10): But if you disagree with b), you are objectively wrong. The coin story and the electron story (with the electrons prepared in the specific initial state specified in the technical appendix) are isomorphic under a mapping that takes:
“examine the heads side” to “make an up/down spin measurement”
“examine the tails side” to “make a right/left spin measurement”
“black” to “up”
“white” to “down”
“red” to “left”
“green” to “right”
The statistics assumed in the coin problem exactly match the statistics you’d get from the electrons in the given state. To verify this is a straightforward exercise in (very elementary) linear algebra.
Bennett (#11). Correct. You cannot simultaneously measure both the up/down and the left/right components of spin. In fact, one way to prove that you can’t is that if you could, you’d be up against the paradox in the initial brain teaser—there would be no pattern of observation that fit either the predictions of quantum mechanics or the results of experiments (which are the same as the predictions of quantum mechanics).
QM is weird. Or it seems so to us who are used to dealing with the big world.
Bennett: Yes, that is the crux of the paradox. I posted a comment about that, but it has been deleted.
So where does this leave us? On its face this seems pretty slim pickings: II with its intelligent conniving electrons, III including FTL collusion, or IV in which (quantum uncertainty notwithstanding) certain decisions made by the Landsberg siblings’ brains are unfailingly correlhated with the properties of specific, otherwise-unrelated electrons?
Arch1:
So where does this leave us?
It leaves us with quantum mechanics, which correctly predicts the result of this experiment, among kajillions of others, and which is incompatible with classical intuitions (as the puzzle in the post is intended to illustrate), but is perfectly consistent with different intuitions which take a while to cultivate, but are not impossible to get used to.
Thanks. However I am still wondering about the apparent open-endedness of the set of Possible Resolutions. While it is certainly confusing to think about, the scenario doesn’t seem to have that many ‘moving parts’, which naively suggests that the set of alternative axiom sets which could produce the observed behavior might be small enough to exhaustively enumerate. Or is this a pipe dream?
“QM is weird. Or it seems so to us…”
Yes. This is why it’s a good idea to frequently remind students that QM is not an obscure genre of hard sci-fi, but the best known description of the world we live in, and as such it’s more normal than any random intuition of randomly selected naked ape species form a random pale blue speckle of dust.
Are we to assume that each player sees a 50-50 distribution,
black-white and red-green?