It is natural (and common) to question Einstein’s first assumption—the existence of real states. As mentioned at the start of Part III, it would be difficult to question “macroscopic” reality (the stuff we observe directly, and all agree occurs). But we might question the existence of “microscopic reality.” This viewpoint calls into question the process whereby we assume (or infer) the existence of microscopic things built from ever-smaller things. In particular, if we question the existence of microscopic reality, we question the assumption that the macroscopic physical properties are built from—or are manifestations of—microscopic ones. The assumption of microscopic reality has served science well, but given the difficulties we have in finding a narrative to explain what the misty states are in terms of well-behaved, microscopic, unobserved real states, perhaps we should abandon the notion that there is any connection between them? (That is, if the real states have meaning or “exist” at all.)
Proponents of this view see the misty states as an inferential tool for predicting the outcomes of future experiments (and how our observations will affect such). In that sense, they propose, the misty states are states of knowledge, but not states of knowledge that tell us anything at all about an underlying reality that is the cause of what we see.
Einstein’s second assumption (which he called the “separability hypothesis”) was that the real state of Bob’s ball cannot depend on what Alice chooses to do with her ball when they are far away from each other. At the time Einstein made his argument, it seemed completely obvious to him (and, in fact, to the people with whom he was arguing) that there could be no question of some kind of mechanical disturbance to (the real state of) Bob’s ball based on Alice’s measurement choice.
As we now know, based on Part II, “some kind of mechanical disturbance” does have to be contemplated seriously as a possible explanation for the nonlocal correlations. Moreover, it would have to have all of the physically anathematic features that caused Einstein and his contemporaries to dismiss such a thing out of hand. (Einstein’s argument preceded the discovery that measurements on entangled misty states can generate nonlocal correlations, as discussed in Part II. Similar to the method the psychics used to get your gold, Einstein’s argument requires measurements on an entangled pair of balls. However; he did not discover the type of strong nonlocality presented in Part II, where the data appears in the outcomes of experiments, and does not rely on any interpretation of mathematical objects, mists, or philosophical prejudices.)
With one of Einstein’s core assumptions thrown into doubt in such a concrete manner, we cannot rely on his argument to tell us whether every real state of a system uniquely determines a misty state or not. Even worse, in a moment, I will explain an argument built on very, very similar (but arguably less questionable) assumptions than Einstein’s; one that reaches the complete opposite conclusion.
Why no faster-than-light communication?
Before we turn to the argument that reaches the opposite conclusion to Einstein’s, let me briefly explain why it is that we cannot send messages instantaneously by making measurements on one of an entangled pair of balls. I often hear from people confused about this.
Imagine that Alice, in the experimental setup we’ve been discussing, wants to communicate something to Bob—let’s say either “ATTACK” or “RETREAT.” What does she have control over to use as a transmitter? Well, she has one choice: she can observe her ball directly (and collapse Bob’s ball to W or [W,B] according to whether she sees her ball is black or white respectively), or she can send it through a PETE box and then observe it (collapsing Bob’s ball to B or [W,W,B] according to whether she sees her ball is black or white respectively). Let’s say she and Bob have planned that if she observes her ball directly, that means “ATTACK,” and if she runs it through the PETE box first, she’s sending a “RETREAT” signal.
What does this mean for Bob? A small calculation shows if she observes her ball directly, she will collapse Bob’s ball to [W,B] twice as often as to W. That is, the probability of collapsing to [W,B] is 2/3, while the probability of collapsing to W is 1/3. Alternatively, if she wishes to communicate “RETREAT” (and runs her ball through a PETE box first), we find that the probability of collapsing Bob’s ball to [W,W,B] is five times as likely as collapsing to B. That is, the probability of collapsing to [W,W,B] is 5/6, while the probability of collapsing to B is 1/6.
The key point is that although Alice can choose which pair to collapse Bob’s ball to, she cannot control which member of the pair it ends up in. The probabilities cannot be changed from what the misty states predict.
Bob needs to determine which pair she has collapsed his ball to. Imagine he doesn’t do anything to his ball, he just observes its color. When Alice is communicating ATTACK, the probability he sees a white ball is 1 if she collapsed it to W, and 1/2 if she collapsed it to [W,B]. His overall probability of seeing a white ball is 2/3.
When Alice is communicating RETREAT the probability he sees his ball is white is 0 if she collapsed his ball to B, and 4/5 if she collapsed his ball to [W,W,B] (remember the squaring rule). Since there is a 5/6 chance of collapsing to [W,W,B] the total probability he sees a white ball is the product of 4/5 times 5/6—which is 2/3.
This is identical to his probability if she was communicating the complete opposite message. He has gained no information when he sees his ball is white.
Even if Bob first sends his ball through some complicated array of boxes along with other balls this conclusion remains true. All he sees is that it is equally likely Alice is telling him to attack versus retreat, and so he gains no information at all.
All this is mainly strange if you consider the mist to be real—if you follow Einstein and deny that the real state of Bob’s ball is changing at all then you would not expect them to be able to communicate in this manner.
But then how would you explain how the psychics won your gold?
Pooh-Bear creates complete confusion
Any argument contradicting a conclusion of Einstein’s should be made by a figure of equivalent stature, although they will, of course, necessarily be of inequivalent intelligence. By consensus of his friends, Winnie-the-Pooh “has no brain.” It seems appropriate to attribute these characteristics to an advocate of this counterargument, and so I will call it the Pooh-Bear argument.
Pooh adopts Einstein’s first assumption, that there is such a thing as a real state of a ball. Einstein’s second assumption was that the real state of Bob’s ball cannot depend on what Alice (located arbitrarily far away) does to her ball. The weakness of Einstein’s argument, premised as it is on his second assumption, is that it requires the two balls to be entangled. This in turn requires them to, by some means or another, have already come into contact with each other prior to being separated by Alice and Bob. As such they are definitely not necessarily completely independent (entanglement is, in fact, the misty-state manifestation of this). Perhaps when they come into contact they create a magical hole in spacetime that can stretch through extra dimensions and this lets them instantaneously coordinate their actions? Any ridiculous possibility must be considered, and the fact they have had to have some kind of causal connection already to get entangled means it is hard to rule any such things out. The fact the balls have already had this connection also makes the pretty natural expectation that they have real states (physical properties) “all of their own” a lot more questionable.
Pooh replaces Einstein’s separability assumption by a separability assumption that is simple enough for even a brainless creature to understand. His assumption is: if two balls have never come into contact (for example, perhaps they were originally created at far corners of the universe), then they have physical properties/real states of their own which are independent from each other, and if you never ever bring the balls together then you can always treat them as separate physical things, completely ignoring the existence of the other one if you like.
Pooh’s separability assumption is not
a mechanical one like Einstein’s. It is an assumption that we can always isolate the things we are examining to a sufficient extent, and treat them as separate physical entities. Without the ability to do independent things we would not actually be able to make scientific progress, because the process of science is built on “independent verification.” The thought that we cannot reason and infer things based on an experiment here, because of what might be going on right now on the other side of the universe (with which we have not interacted in billions of years), reaches a new level of absurdity, and makes one wonder how we could investigate nature at all. (Although Nature has been known to be both absurd and capricious when it comes to such questions.)
Without the ability to consider a rock on Pluto as having its own properties independent of the properties of a rock here on Earth, it’s very hard to see how we could begin to organize our thoughts about the world. Never mind tie our shoelaces.
To set the scene for the Pooh-Bear argument, we turn to:
UNIMAGINABLE CONVERSATION BETWEEN POOH-BEAR AND EINSTEIN
POOH: We should think about what we will have for lunch, which is always nearby in both time and space thankfully.
EINSTEIN: Time, space? Boring. Although your clock is not working, which has started me thinking about... hang on a second, didn’t you just have elevenses?!?
POOH: (Smiling) Oh yes, I did just have a small something. Perhaps that is why my Very Little Brain is so ready to explain. Hey, that rhymes. My “brain” can “explain,” hum tiddely pom...
Pooh begins to hum
EINSTEIN: Ah, Mr. Pooh,...
POOH: Oh sorry, yes, well imagine there are two types of packed lunch. The first type either contains hunny or it contains condensed milk.
EINSTEIN: Honey?
POOH: Yes, hunny. Just one of the two, of course. I wouldn’t be allowed to have both. (Looks wistful, pats his oversize tummy) I will call that packed lunch the “sweet” option for lunch, because…
EINSTEIN: (Interrupting) I think I know why.
POOH: Sorry, I do tend to be a little slow of thought, and perhaps I over-explain. I heard you have a Very Clever Brain. Anyway, the other lunch option contains either hunny or a banana. I’m going to call that the “healthy” option, because...
EINSTEIN: (Impatiently) Yes, Mr. Pooh...
POOH: Sorry, there I go again. Now, Christopher Robin is going to pack me either a sweet lunch or a healthy lunch. And Eeyore is, completely independently, going to pack you either a sweet lunch or a healthy lunch, choosing from the same three foods. And then we are going to visit Owl.
EINSTEIN: Good, I have always wished to meet that knowledgeable creature.
POOH: There are four possible lunch combinations we could be carrying: sweet-sweet, sweet-healthy, healthy-sweet and....and....
Pooh looks a bit anxious
EINSTEIN: healthy-healthy?
POOH: (Smiling) Yes, exactly, thank you. The plan is to set Owl a challenge. He must tell us, for absolute sure, a lunch combination that was definitely not prepared.
EINSTEIN: Why can’t he tell us one that was prepared?
POOH: I don’t really know, this is the only way at the moment that I can make the whole argument work.
EINSTEIN: That is fine. So Owl must definitively rule out one of the four options. I presume he is allowed to look inside the two lunchboxes?
POOH: Oh yes, he can look inside if he wants. Now the question is, will Owl definitely be able to rule out one of the lunch combinations?
EINSTEIN: Well, let’s just consider all the possibilities. If Owl sees a banana in my lunchbox and a condensed milk in your lunchbox, he knows the combination prepared was—listing mine first and yours second—definitely “healthy-sweet.” So he can say, “The combination prepared was definitely not sweet-sweet.” Or, for that matter, he can say it was definitely not healthy-healthy or not sweet-healthy.
POOH: Exactly. In that case there are three different answers he could give, because he knows for sure what was prepared, so he can say for sure what was not prepared.
EINSTEIN: Yes, that’s a pretty easy case. The same goes for if he sees bananas in both lunchboxes, or condensed milk in both, or condensed milk in mine and a banana in yours. He knows for sure what was prepared, so he has lots of options to give an answer about what was not prepared.
POOH: What if he sees hunny in your lunchbox and a condensed milk in mine?
EINSTEIN: Hmm. That is more tricky, because honey is both healthy and sweet.
POOH: (Beaming) I know, isn’t hunny wonderful?
EINSTEIN: So, he knows from the condensed milk in yours that your lunchbox was prepared sweet. But he can’t be sure about mine because of the honey.
POOH: Remember the rule was, Owl needs to only tell us one of the combinations that was not prepared. He doesn’t actually need to know which combination was prepared.
EINSTEIN: Aha—I see. Since he knows from the condensed milk that your lunchbox was prepared sweet, he can answer that healthy-healthy was definitely not prepared. Or he can answer that sweet-healthy definitely was not prepared. Because he sees you have the condensed milk, he knows any combination involving your lunchbox being healthy was definitely not prepared.
POOH: Yes, there are only two safe options for him to answer now, but he can still meet the challenge easily.
EINSTEIN: The same will be true whenever one of us has honey and the other one does not. In all those cases Owl will be able to give a satisfactory answer because he will know for sure from the lunchbox without honey how it was prepared.
POOH: You really do think so much faster than me, Mr. Einstein. I have to be honest, that when I was first thinking about this I had to sit and write out all of the options. There were quite a few of them, and I got so absorbed I missed having a little something at eleven o’clock that day. (Looks sad at the memory)
EINSTEIN: I can also see now that there is no way that Owl can be certain of winning this challenge. If we played it many times he must eventually be stumped. Because, what can he say if he finds honey in both of our lunchboxes?
POOH: Exactly, Mr. Einstein, you have seen the problem! I think hunny in both lunchboxes would be a Very Good Thing to find, especially if you are willing to share (glances anxiously at Einstein). But when that happens Owl will be stumped. He cannot say for sure one of the combinations that was not prepared.
EINSTEIN: I’m not able to see how all this might be relevant to my argument about misty states...
POOH: Well your claim is that just as the two different “states of lunchbox knowledge”—sweet versus healthy—have a “real state” of hunny in common, the misty states have at least some real states in common.
EINSTEIN: Yes, assuming there are such things as real states, and assuming what someone does in one location can’t change a real state somewhere far away, it is simple to see that there must be real states common to more than one misty state. I don’t know why I’ve had to spell it out so many times to those...
POOH: Careful Mr. Einstein, children read these books. Well, let’s imagine we redo the challenge with Owl, but this time Christopher Robin will prepare me a STORAGE lunchbox that contains one of the infamous black or white balls. And he will either prepare it as a white ball W, or he will prepare it in the misty state [W,B]. Eeyore is independently going to do the same thing for you—prepare you a STORAGE lunchbox that contains a ball either in W or in [W,B]. You should think of W like a “sweet” lunchbox preparation and [W,B] like a “healthy” one.
EINSTEIN: I see. My claim is that W and [W,B] are not themselves the real states because by my argument there are at least some real states common to both of them. Those common real states are like the honey. Even if Owl is so wise and knowledgeable that he can see real states, whatever they may be, at least some of the time both balls will be in a honey-like real state. This makes it impossible for Owl to always say one combination that was not prepared.
POOH: Yes, yes—now we are there. Because her
e is the amazing thing. Owl always manages to win this challenge!
EINSTEIN: What? Are you sure?
POOH: Yes, we did it many, many times. In fact it doesn’t matter which pair of misty states Christopher Robin and Eeyore pick from, Owl can always meet the challenge. This is why I claim that every real state is associated with one and only one misty state.
EINSTEIN: Remarkable. Things are really not the Things that they seem to be.
POOH: Oh I am so glad you appreciate that. When you are a Bear of Very Little Brain, and you Think of Things, you find sometimes that a Thing which seemed very Thingish inside you is quite different when it gets out into the open and has other people looking at it.
How can Owl do it?
Owl needs to take the two balls from Pooh and Einstein—about which he only knows that they could be in any one of the four misty states WW, W[W,B], [W,B]W, or [W,B][W,B]—and rule out one of the four combinations.
The schematic of the procedure Owl devised (with a little help from Rabbit, Tigger and Roo) is to take Pooh’s and Einstein’s balls (without looking at them) and drop them through these boxes:
Working out what happens for the four different possible input misty states is old hat to you by now:
Holding Einstein’s ball in storage, the next stage of Owl’s procedure is to look at the color of Pooh’s ball. If he observes Pooh’s ball is black he then observes Einstein’s ball directly, but if he sees it is white he first drops Einstein’s ball through a PETE box and then observes it.
Ball 2 is black: If Owl sees Pooh’s ball is black, then he releases Einstein’s ball from storage and directly observes it. Looking at the four possible output mists in the above figure, we see that the only configurations appearing which have Pooh’s ball black are BB and WB. We further note that the WB configuration could have resulted from three of the possible combinations that Einstein and Pooh might have presented Owl with. It does not, however, appear in the output mist when both Pooh’s and Einstein’s balls started out white (top left corner figure). So, if Owl sees that Pooh’s ball is black and then sees that Einstein’s ball is white, he can safely announce “it is not the case that both balls were initially prepared white.”
Q is for Quantum Page 10