The conclusion of all this is that if you want a causal explanation of the nonlocal correlations we observe, then it has to be a very strange explanation in its own right. So strange that physicists seriously consider other disturbing options. For example, one way to get around all of these conundrums is to propose that the balls (or the psychics) already know in advance all of the coin flips that you and Randi will flip. This explanation requires that no matter how you and he try to make an independent choice (you don’t need to use a coin, you could use any object, or do it purely in your mind) whatever you will choose can be pre-known to the balls/psychics. This way they can easily arrange to obey the rules, yet sometimes win, with no need for telepathy or any other causal link between them.
Such explanations are known as “super-deterministic.” They conflict with our psychological feeling of free will (which arguably need not be given much credence in a theory of physics), but more critically, they conflict with the very notion of performing independent experiments to test and verify our fundamental ideas and theories. Without such independence we have to call into question the whole process of science and, at some level, all the scientific understanding we (think we) have gained, because it is so tied up with that process. It is a significant price, but an idea that is considered seriously.
All of which is not to say that a causal explanation is impossible. One such explanation is to consider the mist itself as “real physical stuff.” So far, the mist has only played a role of encapsulating rules by which we can calculate what we will eventually observe. It has been part of our mental deduction process—a mathematical object, not a physical one. As is hopefully clear from Part I, we do not see actual mist emerging from the bottom of the boxes. But one option is to take seriously the possibility that the mist directly represents some kind of real physical object, like a radio wave or a bowl of soup, and when we separate the psychics the mist is stretched between them:
Such a mist would, for the reasons just discussed, have to have many physical properties that differ from any other kind of physical stuff we have ever encountered. It would be arbitrarily stretchable and move instantaneously when you whack it at one end, for example. The whole question of how to interpret the mist—as something physically real? as something which is just mathematics in our heads?—is one of the major schisms between physicists. Some arguments for and against both are addressed in Part III of this book.
Before we turn to these questions, you are now ready to learn a couple more important features of misty states.
Computing the likelihood of observing a particular configuration given a complicated mist
Previously I skipped over explaining why it is that when Alice and Bob observe balls in the misty state there is a one-in-twelve probability that they see both balls are black (and hence win the game). One would more naturally expect the probability to be one in six, since there are six configurations, but, unfortunately, nature is not quite that kind to us. Nobody really knows why. It’s all part of the mystery you are going to solve for us one day.
The general rule for computing the likelihood (probability) of seeing any particular configuration is the following: Square the sum of the times the particular configuration appears, and divide that square by the sum (over all the configurations that appear) of the squares of the sum of the number of times that they each appear.
Huh? Yes, I do enjoy causing you a bit of mental pain. It is much easier to understand than it sounds, and the best way to work out what is going on is with an example of a single ball in a misty state:
In this example the white ball appears two times, the black ball three times. We first square these numbers. The probability of any given configuration is then the ratio of the squared number of times it appears, to the sum total of all the squared numbers. In this example if you observe the ball you will find it white with probability four in thirteen (4/13=0.3077... so you see it white approximately 30% of the time), instead of the more natural expectation, which would be with probability two in five (or 40%). In these types of calculations, we ignore any negative-sign label—all copies of a given configuration will have the same label, since oppositely labelled copies will have cancelled out by interference already.
We can now work out the probability of the psychics winning the game, by calculating the probability of BB on the misty state that the two balls evolve to when both psychics put their ball through a PETE box:
In the event (as always occurred in Part I of the book) that all configurations in a mist appear the same number of times, doing the whole calculation is unnecessary, all configurations are equally likely. Can you prove that for yourself more carefully following the “square divided by sum of squares” rule?
Making observations on a few balls within a multi-ball mist
There’s one more rule for computing with misty states. Up until this point we have only considered the case where we always observe all the balls at once. The result of that kind of observation is that we completely destroy the mist. We are left with just one of the configurations from the mist, with a probability you have just learned how to compute. But what if we don’t observe all the balls?
Here’s a misty state being created from three initially white balls:
Have a guess at what the final state of the three balls will be if we now only observe the first ball?
As is typical for misty states, we will sometimes observe the first ball is white, and sometimes observe it is black. Let’s say we observe it is white. We still will not know anything about the color of the other two balls. It is then perhaps unsurprising that the new misty state just comprises all the pieces from the original pre-observation state in which the first ball is white, in this case WWW and WBW. Similarly, if we observe the first ball to be black then the final misty state of the three balls consists of those configurations in which the first ball is black, in this case BWW and BBB. Summarizing:
We see that by observing only one of the balls it is possible to leave the other balls in a misty state—whereas if we had observed all three balls, then there would be no mist, and no ambiguity about each ball’s color.
Remember the two-ball state that the psychics used? Of course you do, it’s how they cheated you out of your gold bars. One way the psychics can prepare that state is to observe ball 3 of the example above instead of ball 1. Summarizing:
When the third ball is found to be white, the first two balls end up in the misty state that the psychics required. You should really hate that misty state.
Entanglement
Recall the following misty state that we first encountered in Part I:
This particular misty state has many beautiful features and is considered quite special by physicists, so I will name it the “Bella” mist, after the Italian word for beautiful.
The Bella mist has the feature that it is reasonably simple to see there is no way to view it as being obtained by combining separate mists for each ball individually. That is, we saw in Part I that when you combine separate, individual, misty states of different balls you get a larger single misty state. The claim is that the Bella mist cannot be built up in this way. Here are three examples of two-ball misty states that are built up from separate misty states:
None of these examples are the Bella mist. The top two examples have only two configurations in the combined mist, just like the Bella mist, but one of the balls is always the same color in both. In the bottom example there are too many configurations. You should try several more examples to convince yourself getting to the Bella mist by combining two separate misty states is not possible.
This feature of the Bella mist—that it cannot be built from a combination of individual mists for each ball—turns out to be extremely special and useful, and so we give it a new word. We say the Bella mist is “entangled.” As with the word “superposition,” saying a misty state is “entangled” is just physicists making up a word to describe a situation which had previously never been encountered in any ph
ysical theory. Unfortunately, unlike “superposition,” “entangled” is a word that already has colloquial meaning, and that meaning is only vaguely reminiscent of the precise way in which we will use it. This happens a lot in physics and math—common words get adopted for precise usage, and it can be a major source of confusion, so be careful if you ever study these subjects more deeply.
One might have thought that the lesson from passing single balls through PETE boxes was that, if you can’t think of the color of a ball as something it “really has”—as a physical property—maybe the mist itself is a physical property. Then a single ball could “really have” a “value” of its mistiness. This view might lead one to think of superposition as the only mystery to be explained. The phenomenon of entanglement shows that this is not the full story—we have seen that an entangled misty state of just two balls like the Bella mist cannot be interpreted as arising from balls that are actually in their own individual misty states. In reality, superposition is not the only mystery.
Of course you may want to simply adjust the proposal; perhaps it is misty states of two balls that are the “real thing”? But then we can find entangled misty states of three balls that cannot be built from separate misty states of one and two balls. (Try it). The whole of Part III is about the many questions and issues surrounding the reality (or otherwise) of the mist.
A word of warning, determining whether a misty state of two balls is entangled or not is not trivial. Consider these 2 examples, which differ by only one negative-sign label:
The two-ball misty state [WW,WB,BW] that was repeatedly used by Alice and Bob to defeat you and Randi is also entangled. In fact, entanglement is provably necessary for generating nonlocal correlations. It is not possible to generate nonlocal correlations with a non-entangled misty state, basically because without entanglement the balls behave as if they are completely independent. It is often said that entangled states are special because they are “non-separable.” That is, you cannot any longer treat the entangled balls separately.
Again, some caution is required when trying to claim that this non-separability is fundamentally strange. Remember back when you went hiking, and your lucky friend had a lunch packed that consisted of either chips and a burger, or jerky and pizza but none of you knew which was the case? We could invent a way of depicting the state of your friends’ lunch box, similar to misty states. Let’s depict them in a rock (since they are more down to earth), and call them rocky states:
We see that by identifying foodstuffs with ball colors appropriately there is at least a superficial similarity with the Bella misty state [WW,BB]. The correspondence goes further: this state is “rocky-entangled.” Specifically, it is not possible to create a lunchbox in this state by taking two individual lunchboxes that have uncertainty about their contents. More precisely, imagine you are told there are four potential configurations for lunch, namely CB,CP,JB,JP. You would say that is just the combination of two separate lunchboxes, one of which contains either chips or jerky, the other of which contains either a burger or pizza:
However, there is no way to create the rocky state
Creating an “entangled rocky state” like
Once again, these similarities and analogies are useful, and perhaps (physicists argue about it) they reveal something important about how to understand misty states. But they should be treated carefully. If entangled misty states were really equivalent to entangled lunchboxes, telepathy/nonlocality would definitely not be demonstrated by winning the psychics’ game, because Alice and Bob could have just played with their lunch and there is nothing strange about that. Your lunch cannot be negative, it cannot interfere, and it is not telepathic.
Summary of Part II
We can perform far-separated, well-shielded experiments (observations) on balls in a misty state, the outcomes of which are inextricably linked, in as much as they cannot be reproduced by physical stuff (whatever it is) responding only to what is going on around it locally.
Such “nonlocal correlations” do not depend on the temporal ordering of the experiments, cannot be used to send messages, and occur even if there is no time for communication at light-speed between the experiments. This puts them in severe tension with the normal type of causal explanations in physics.
Misty states of two or more balls can be “entangled,” by which we mean they cannot be treated as if they have independent colors, or independent misty states for that matter. Entanglement underpins nonlocality.
Given misty states with unequal numbers of repeated configurations of ball colors in the mist, the rule for computing the probabilities of observing any particular configuration involves squaring numbers and dividing them. All a bit messy, but still just basic arithmetic.
Beginning with misty states comprising many balls, observing only some subset of the balls can leave the remainder of the balls in a misty state.
Part III: Q-REALITY
Realism and physics
Science works, believe (in) it or not. Fortunately for phone companies the laws and scientific principles underpinning how your cellphone works are the same whether built by a person in China or one in the USA. Although you likely take this for granted, why should the laws of physics behave the same for different people in different countries?
One way to explain this universality is to posit that there is a deeper “underlying reality” upon which physics is based. More specifically we could presume that: (i) there exists a physical world that is external to us, and (ii) that we are not particularly important to what is going on in this external world—the stuff within this universe existed before we came along and will continue to exist once any or all of us are gone.
The extent to which we can draw “ultimately true” conclusions about this underlying reality from our scientific theories is debatable, tuned as the theories are to solving pragmatic problems for human-scale creatures. Still, one might think it unarguable that “there really is something there,” and the fact that the human endeavor of science succeeds equivalently across variations in time, location and culture of the practitioner is somehow due to this.
The preceding paragraphs summarize a view known as (naive) scientific realism, of which there are multiple subtle variations. There are many, many people who would disagree with some, or all, of this view, both scientists and non-scientists. If you are interested in alternatives then start with the Wikipedia article on philosophical realism, and follow your nose until your brain hurts.
I am not qualified to enter a debate on such topics. My goal in this final part of the book is to quantitatively expose to you the extent to which the somewhat weird phenomena introduced in Parts I and II, and our physical laws that describe them in terms of misty states, yield new insights and conundrums on such questions, irrespective of your own philosophical preferences. Feel free to ignore any armchair philosophy that has crept in (I have tried hard to avoid it). I also don’t want to get tied up in knots being overly cautious about language. I hope you are both willing and able to apply your own filters without rejecting the overall message, because the quantitative and technical things that you are now able to understand about our physical theories will sharpen your understanding of your own viewpoints; as well as, I hope, rule out some things you thought were obviously true.
Physical properties
If we at least agree that there are other people than ourselves (and frankly, if solipsism is true and you are creations of my mind, then you would all be better looking and made of chocolate), and if we also agree those people are experiencing the world in a similar manner to ourselves (again, you will find people who make a big deal of the fact that this is unprovab
le), then the consistency of our conversations about what we are individually experiencing (“it’s hot,” “that book is heavy”) leads us to implicitly or explicitly form the useful notion that “things have physical properties.”
We all appreciate it can be a blurry line between properties we know are subjective (“the banana smells nice”) and those our common agreement indicates are objective (“the banana is yellow”). But it is a natural concept that there are at least some objective physical properties, of some kinds of material things, which are out there in the universe and independent of our subjective experiences. Similarly, it’s a natural concept that some of those properties are more fundamental than others. Think of the incredible diversity of textures, smells, colors, and tastes of everything that we personally experience; all this originates from less than a hundred different building blocks we call atoms. Lego can’t even make a model Millennium Falcon without double that number of different building blocks.
Q is for Quantum Page 7