The second time you flip it the coin comes up heads. Alice takes the box labelled STORAGE 2, but this time she does not hold it above the PETE box, she just pulls the lever, and a white ball drops out the bottom of the PETE box. Alice says, “My second answer is white.” For someone supposedly being telepathic, Alice is acting quite brusque and business-like. “Next!” she says impatiently. “There are a lot of games to play.”
After a full day, with the test drawing to a close, you have seen the same procedure repeated thousands of times. Each time you play, Alice takes the next unused storage box from the stack. If the coin you flipped shows heads, she just pulls the lever, while if it is tails, she holds it above the PETE box and pulls the lever. Either way, she announces “black” or “white” according to the color of the ball that drops out.
After the end of the four thousand repetitions of the game you and Randi meet up and exchange stories. He had the exact same experience with Bob as you had with Alice—Bob also based his black/white announcements on the color of a ball which fell out of a storage box, one that was first held above a PETE box if the coin flip showed tails.
You and Randi now sit down and start the laborious process of counting up how many times the psychics won. That is, how many times did they both say black when you two both flipped tails? You also need to check that their other answers obeyed the two rules. The first pages of your notebooks might look like this:
When you add it all up, you find that the psychics have won eighty times—which far exceeds the minimum twenty times you agreed on. They will win the cash and the gold bars! A bit panicked, you (not Randi, he’s chilled about everything) do the comparison again—maybe having that overly-strong American Pale Ale while you were doing it wasn’t a great idea, and you messed up counting? (For that matter, are you old enough to drink a beer yet?) You find that you were correct the first time, the psychics have exceeded the requisite twenty wins by a long way without ever breaking either rule.
The probability of eighty wins following a strategy where you risk a one-in-four chance of being caught every time you cheat, according to your friend in the bank, is absolutely and utterly, mind-numbingly ridiculously small. It is so small that you have a much better chance of winning a game where I take one grain of sand, mark it somehow, and hide it anywhere on any beach in the whole world, or anywhere in the Sahara desert as well. You then walk around the whole world blindfolded, sifting through all those sandy beaches, dragging yourself through that lovely desert, and at some point you grab a single grain of sand. The chance that you grab the same grain that I marked is still much greater than the chance the psychics can win the game eighty times and not be caught cheating, if they really are using one of the strategies discussed above and just gambling on not being caught each time they play.
I don’t know about you, but if it was me I would be very, very suspicious at this point. Randi you can likely trust, although maybe given what is at stake even he should be suspect. Because remember, what is at stake is not just money and gold, it is something far more important: it’s the possibility of some kind of psychic connection that defies common sense. Something at least as strange as telepathy.
But, in fact, stranger.
What went wrong?
Assuming for the moment you trust Randi, your suspicions will first fall on the isolation rooms. All it takes to win the game every single time is for information about the coin flip outcome in the other room to be available. For instance, perhaps hidden inside the STORAGE or PETE boxes is a cellphone of some form, which sends a message to the other room? While you have completely shielded the rooms to all known types of signals, there could be ones you don’t know. In fact this type of cheating is not how they do it, and a bit later I will explain how we try to take extreme measures to ensure that it is not what they do, but first let us see how they really are doing it.
The answer, as you may have guessed already, involves misty states of balls somehow.
A tangled question: how did they do it?
Alice and Bob each have a pile of four thousand storage boxes (numbered 1, 2,…, 4000). Inside each storage box is a single ball. Each ball in each correspondingly numbered box—the one in Alice’s room and the one in Bob’s room—has been carefully prepared to already be in this misty state, where the first ball is in Alice’s storage box, and the second is in Bob’s storage box:
The storage boxes are very carefully designed so that they don’t (even inadvertently) observe the color of the ball they contain. As we know, looking at the color of the ball will destroy the mist. Toward the end of this part of the book I will explain how this particular misty state can be prepared. One of the remarkable features of the mist is that, although the balls need to be brought together in order to create a mist like this, once it has been created they can be separated as much as we like without the mist being affected (as long as we keep all balls safe inside storage boxes).
Alice and Bob both get heads: When both coin-flips in both rooms result in heads, Alice and Bob each simply release a ball from its numbered storage box, observe its color (destroying the mist), and use that as their answer. As a result, they will answer WW, WB or BW with equal likelihood. They will never answer BB, since that is not one of the configurations within the mist. This ensures they will always obey Rule 1. They need a new pair of storage boxes/balls for each time the game is played, because the mist gets destroyed by the observation.
Alice gets tails, Bob gets heads: If Alice gets told tails, she passes her ball through a PETE box before she observes its color. Calculating what happens is simpler than some of the calculations in Part I:
From this we can see that Rule 2 will be obeyed when Alice gets tails and Bob gets heads: the forbidden configuration BW does not appear in the misty state—it disappeared when it was cancelled out by interference. This means that when they now observe their respective balls, regardless of which answer they give (WW,WB or BB), it will be valid under the rules.
I will leave you to do the calculation for the opposite case—where Bob gets tails and Alice gets heads. It is very similar, and for this case you should find that the forbidden configuration WB does not appear in the misty state.
Both Alice and Bob get tails: The most interesting case—the “winning” case, so to speak—is when both of them get tails. Writing out all the steps is a bit long and messy, but what happens here is the heart of Alice and Bob’s “telepathy.” Here it is written in our more compact notation from Part I:
The mist inside the two storage boxes initially:
[WW,WB,BW]
Passing each ball through a PETE box:
[[W,B][W,B],[W,B][W,–B],[W,–B][W,B]]
which is the same as
[[WW,WB,BW,BB],[WW,–WB,BW,–BB],[WW,WB,–BW,–BB]]
which is the same as
[WW,WW,WW,WB,BW,–BB]
That is, after both Alice and Bob have passed their ball through a PETE box, the final misty state is:
We see that the BB configuration is there in the misty state. This means sometimes they will observe both balls are black and win the game.
I have shown only the calculation for one pair of storage boxes, but the pairs of boxes are all the same and so, when the game is played many times, eventually some BB outcomes will be observed. Even just seeing the BB outcome once is amazing, given the fact the two rules are always obeyed.
In the above two-ball misty state, it would seem that the likelihood of seeing the BB outcome is one in six, because there are six configurations in the mist. It is not—there is a subtlety to do with computing probabilities in cases where some configurations appear more often than others (as here, where WW appears three times while WB, BW and BB each appear only once). At the end of this part of the book I will show you how to do that type of calculation and we will find it is a one-in-twelve (about 8%) probability of seeing BB, given two balls in the misty state above. For the moment, let us just be amazed it happens at all, because we couldn’t come u
p with a strategy to win the game, and so just saying, “Oh, well, the psychics use magical misty states of balls,” only defers the question: How do the balls actually manage to do it? Unless, of course, the balls themselves are telepathic—that is, they somehow know what is happening to the other ball?
Now you know how the psychics “cheated” you out of Randi’s money and your gold. But didn’t they do it in a remarkable way? It’s worth at least a million dollars to understand this feature of the universe (and if it’s the first time you have genuinely understood it, please feel free to post me a cheque!). In fact, the more one thinks about what has happened, the more disconcerted one gets. Let us delve a bit further into the conceptual problems that all this raises about how the world works.
Nonlocality of correlations
We have seen above a demonstration of what physicists call “nonlocality”: When we observe a misty state of a ball in one location, the outcome of that observation can depend on what is happening to another ball in a different location.
The words “can depend” in the preceding sentence invoke a notion of causality—what Alice is doing to the one ball is “directly affecting” Bob’s ball, or vice versa. For a number of reasons, this description is already controversial and not accepted by all physicists. One of those reasons is that there is no need for Alice and Bob to observe their balls at exactly the same time, and so which direction the cause happens seems to somewhat arbitrarily depend on the timing of who measured first.
A less arguable way of describing the situation would be: the outcome obtained when we observe a misty state of a ball in one location is inextricably linked with what is happening to another ball in a different location. Even the words “what is happening to” in the preceding sentence would make some physicists uncomfortable.
I don’t think anyone would argue with this version: the outcome obtained when we observe a misty state of a ball in one location is inextricably linked with the outcome obtained when we observe another ball in a different location. This is purely a statement about the experiments we do. Just re-imagine the combination of Alice+tester and Bob+tester as merely experimental physicists who are choosing randomly between two different experiments to perform on their balls—either to observe them directly, or to put them through a PETE box and then observe them. They see are what are often called “nonlocal correlations” between the colors of the two balls at the separate locations.
Correlations per se are not strange. If someone gave Alice and Bob each a box containing a ball and assured them that the balls were both the same color, then when they open their boxes they will see that the colors are correlated—in this case, both the same color. But in that situation each ball would “really have” a color prior to being observed; it is just that Alice and Bob do not know what it is.
Such an explanation will not work to explain how the colors of the balls can be correlated in such a way as to respect the two rules of the psychics’ game, yet sometimes both emerge black when they have both been passed through a PETE box. Can you see why?
To perhaps over-labor the point, you can imagine that if the two balls did “really have” a color prior to observation, then they would need to choose their colors using a strategy that would mirror exactly the kind of thinking that Alice and Bob went through in the imaginary conversation above. Instead of giving black/white answers to heads/tails questions, the two balls are either being observed directly, or being passed through a PETE box and then being observed. But they still have to choose a color “to actually be” once they are observed:
AN IMAGINED CONVERSATION BETWEEN THE TWO BALLS
BALL1: Awesome, those suckers accepted our proposed test. Obviously we aren’t actually psychic, but I’m sure we can win this game.
BALL2: Let’s work out a strategy. Actually, we both know I’m not the smoothest bearing in the barrel Ball 1—I better let you work it out.
BALL1: Fine. By Rule 1 when we both get observed directly, we cannot both be black. So how about I will be white when we get observed directly, and you can be black.
BALL2: Woah, slow down there Ball 1. I think I better write this down, it sounds complicated already.
Ball 2 hunts for pencil and paper, finally finds one and draws a diagram.
Ball 2 shows it to Ball 1.
BALL2: (Proudly) See, I made a table of what we should do. I drew a picture of each possibility—being observed with or without passing through the PETE box first, and I’m drawing a black circle to mean “be black” and a white one to mean “be white” next to each to indicate the color we should be.
BALL1: Yes Ball 2, it’s very pretty. Let’s keep going. Since we only win by both being black when we both get passed through the PETE box, when we see we are going through the PETE box we should always be black.
Ball 2 duly makes a note of this as well.
BALL1: So there you go, we have a solution. Now let’s think about what to spend all that money on?
Ball 2 is looking a bit puzzled, scratching his head. Ball 1 has started to daydream.
BALL2: Uh, Ball 1, I think there is a liiiiiittle problem….
Hopefully you get the idea—if two intelligent creatures cannot work out a strategy to obey the rules but win the game (without being telepathic) then what chance do the two balls have on their own? Let me reiterate that explanations of the form “Well, we already know from experiments with just a single ball and a PETE box that we shouldn’t think of a ball as really having a color when it isn’t observed” are not particularly helpful in understanding the nonlocal nature of the correlations in the colors we do observe. Sure, the balls do not have to pre-decide which colors they will be, but we implicitly also granted that option to Alice and Bob—they didn’t have to try and work out a concrete strategy beforehand, they could have said, “Let’s just decide once we know what the coin flip in our room is.” In that case, they would have needed even more telepathy to win once they were separated.
Causal nonlocality would have to be weird
Perhaps then we should consider more seriously the possibility that putting one of the balls through the PETE box actually causes, by some mechanism, the color of the other ball to be affected.
In physics, prior to encountering misty states and nonlocal correlations, it was always the case that the causes of things were mediated by “physical stuff.” Sometimes the physical nature of the causal mechanism is obvious. When you grab your cat’s tail and drag it along the floor, there is a complex story involving atoms and the forces between them that can explain the events that occur, right up to you getting scratched and later bleeding to death. In other cases, the physical stuff that acts as a causal intermediary is not so obvious. When you use your phone to call the ambulance for help (not in time, unfortunately) the radio waves that your phone emits and absorbs are not obvious to your senses. But they can be detected and manipulated—that is what your phone is doing—and they are so physical they can pull and push the molecules that make up a cat almost as well as you can (for example if you had, equally unadvisedly, tried to warm up the cat in the microwave). Radio waves are undeniably “physical stuff.”
Events that are caused by physical stuff have certain common features, none of which turn out to be true for a mechanism that could give a causal explanation for the correlations between the colors of the balls:
Causes precede effects, so the ordering of events in time matters.
If the cause and the effect are separated, then it takes time for the physical stuff that connects them to propagate. (You do not notice the time delay between when you speak in the phone and the person you are calling hears what you have said, but it is there. If you were on Mars it could take up to twenty minutes for what you have said to be received on earth, because radio waves travel at finite speed—the speed of light—and Mars is very far away.)
You can use the connection between cause and effect to send a message, though it will always be limited to traveling no faster than the speed
of light.
It is harder to maintain the connection between cause and effect the further apart they are (the intermediary physical stuff inevitably “spreads out” in some sense and becomes weaker with distance).
How can we be sure that a causal explanation of the nonlocal correlations does not respect these four common features?
Going back to the story of you and Randi testing the psychics: we concluded that the most obvious explanation for their win would be that you and he have failed to shield the rooms properly, and they have cheated by communicating. On the face of it, there will never be a way to absolutely and completely shield a room. However, all known signals that can carry information also share common feature (ii), namely they travel at a speed no faster than the speed of light. This gives us a way to ensure that the psychics are not signaling to each other: You demand that after you tell Alice the coin flip outcome, she tells you her black/white answer (in effect the ball color) before there is time for a signal travelling at the speed of light to make it to the room where Bob and Randi are.
Because light travels so fast, even if you put the isolation rooms at the opposite ends of the earth there are only small fractions of a second during which the whole process must take place. I doubt you can even flip a coin fast enough. For this reason, when we do this experiment we use some electronic mechanism to do the coin flip and a different piece of equipment to insert (or not) the PETE box. That is, we use a mechanical version of both you and Alice. But in principle what we do is identical to the psychics’ game—a version of it where and Bob and Randi play their part on the equivalent of Mars, safely too far away to signal to Alice.
The fact that nonlocal correlations do not get weaker the further you separate the two experiments (feature (iv)) and cannot be used to send signals faster than light (feature (iii)) can also be tested experimentally. Both are also fundamental theoretical predictions of the whole misty-state description of the world, and if they proved to not be true it would be an interesting breakdown of the laws of physics as we currently understand them. Testing feature (i), namely that the ordering of the experiments in time doesn’t matter, is tricky but has been done. I do not want to go into details, but if you know about the Theory of Relativity, it shows we can set up the experiments so that two different people will not even agree on whether Ball 1 was observed before Ball 2, or vice versa, and yet the misty states still correctly predict what happens.
Q is for Quantum Page 6