Similarly, the BB configuration does not appear in the output mist when the two balls Owl drops in are [W,B][W,B] (bottom right corner figure). Therefore if he sees both balls are black he announces “it is not the case that both balls were initially prepared in the misty state [W,B].”
Ball 2 is white: If Owl sees Pooh’s ball is white after the CNOT-PETE-CNOT series, he collapses the two balls according to these rules:
Let’s say the balls were W[W,B] at the very start, when Eeyore and Christopher Robin separately packed them. After passing the three boxes this has evolved to the situation in the top right corner figure of this diagram. Therefore, Owl seeing Pooh’s ball is white means Einstein’s ball is now in the misty state [W,B]. If Owl now passes Einstein’s ball through a PETE box, he will definitely observe it to be white.
If, on the other hand, the balls were [W,B]W at the very start (bottom left corner figure), then, after Owl observes Pooh’s ball is white, Einstein’s ball is in the misty state [W,–B]. If Owl now passes Einstein’s ball through a PETE box, he will definitely observe it to be black.
Thus, if Owl sees Einstein’s ball is black he safely announces “it is not the case that Einstein’s ball was initially W and Pooh’s ball initially [W,B].” If he sees Einstein’s ball is white he announces “it is not the case that Einstein’s ball was initially [W,B] and Pooh’s ball initially W.”
This is quite a complicated procedure to verify works (and after all that, Pooh and Einstein have to go back to Eeyore and Christopher Robin to check if Owl was right). It has the feature that Owl does not even have to look at the mysterious, apparently invisible real state of the balls, he just passes them through some simple boxes and observes their color. We don’t know what goes on inside the boxes—perhaps they can look at real physical states, but Owl does not need to (although he probably would claim that he can).
Recapping, rexplaining
The basic question we are addressing is: although we have no clue what the underlying real physical properties of a ball in a misty state might be, can we deduce something about the relationship between misty states and the presumed real states? Einstein gave an argument that the relationship must be “many-to-one,” that is, there are many different misty states corresponding to a single real state. His method was to show that one person’s arbitrary choice in handling a ball in their possession can cause a remote ball (arbitrarily far away from the first one) to end up in one of many different misty states. Assuming distant subjective choices should not change what is “really going on” with a ball, his conclusion follows. Einstein’s argument requires the two balls in question initially be entangled, it makes a strong presumption of locality and (in light of the nonlocal correlations generated by measurements on entangled states discussed in Part II) it can be considered suspect.
As with Einstein, Pooh’s goal is deduce something about the relationship between misty states and real states of a single ball. Pooh-Bear’s argument also used two balls. For both Einstein’s and Pooh’s arguments two balls are necessary because it is known that observations on just a single ball cannot be used to distinguish between the possibilities of whether the misty states overlap on the real states (whatever they may be) or not.
Pooh’s argument explicitly relied on the balls not being initially entangled; in fact, it relies on them having been prepared independently (by Christopher Robin and Eeyore), and this assumption of preparation independence replaces Einstein’s “obvious” locality assumption that an arbitrarily distant choice in handling one ball cannot change the real state of another ball.
Before his argument dealing with two balls Pooh first considered the example of a single lunchbox that can be prepared in two different ways, “sweet” or “healthy,” such that there is a real state (honey) common to both possible preparations. If you happen to open a single lunchbox like this and see honey, there is no way to tell which of the two preparations was used. A similar situation arises with balls in misty states—in general two different misty states of a single ball cannot be distinguished with certainty. But with the balls, unlike the lunchboxes, we cannot see the proposed real states. As such it is unclear if the inability to distinguish them is because there are overlapping real states, that is, real states in common to both, just as Einstein argued there must be. Alternatively, it could just have to do with some fundamental restriction on the kinds of boxes we can build and observations we can make.
What Pooh argued was that the presence of honey—a real state in common to both possible preparations of a single lunchbox—also made impossible a different type of game you can play with two lunchboxes. This is the slightly strange game where Owl has to say which of the four possible combinations of two lunchboxes was not prepared.
Putting balls inside the lunchboxes using different preparations of misty states is a game changer (and less palatable). Unlike the case of preparation of sweet versus healthy lunches, if a ball is placed in each lunchbox prepared in one of two different misty states, then there does exist a way for Owl to win the game. This was shown explicitly for the case where the ball was prepared either W or [W,B], but something similar can be done for any pair of possible misty state preparations.
Pooh’s conclusion is that there cannot be real states compatible with two different misty states, so that there are completely distinct regions of real states for any misty state, much like this diagram from earlier:
That is, Pooh’s argument (should you accept its premises) proves the misty states are actually real in the sense described near the beginning of this part of the book.
Final thoughts
There are a many, many arguments that I have not covered, both for and against thinking of the mist as real. I have tried to explain, and get you interested in, the challenges on both sides—not to indoctrinate you with my own ever-fungible opinions.
I personally live in cognitive dissonance: on a day-to-day basis I talk about the physical properties of the photons (from which I am currently trying to build the mystical computer discussed in Part I) as if they are as tangible as any of the physical properties of the human-scale objects in the room around me. They are not. I suspect I should treat the misty states as states of knowledge, but to be understood within a more general framework of theories of inference than our present theories find comfortable.
The extreme view along these lines is to say we can only use the misty states to infer about potential happenings within the world that we may, by our personal choices, bring to being with our observations. That we should not attempt to connect those happenings to... well to stuff actually going on. While this extreme view may seem strange to us now, there was a time that the gods were attributed responsibility for the events of the world. Thousands of years ago the radical view (best captured by Lucretius in De Rerum Natura) was to propose that actually the things important to humans were just flotsam of the motion of microscopic constituents. This perspective eventually won out, and has served science incredibly well—it is difficult (for me) to let go of.
But because all attempts to provide an underpinning narrative for what is “really going on” when we do experiments in the mist—nonlocality being my personal bugbear—I have no confidence in the “correctness” of any of the physical properties our theories are premised upon. It seems crazy that I should either abandon the idea there is stuff doing things out there that doesn’t care about me, or believe my experiences of the world can depend on things happening in a galaxy far, far away.
In the end I guess I cannot escape my naive realistic belief that there is stuff there, and it has physical properties of some form independent of my concerns. I am willing to contemplate the possibility that perhaps all the physical properties I implicitly consider fundamental are merely artifacts of the evolution of human perceptions, their presumed importance an anthropocentric mistake. Confronted by nonlocal correlations, I sometimes even wonder if space and time—important properties for monkeys who need to find food and mates—might b
e no more relevant to the true underlying reality of the universe than the pleasant smell of a banana.
Summary of Part III
Although we have only discussed the color of a ball as a property, any physical property can be “put into” a misty state.
The relationship of the mathematical object of a misty state to “real physical properties” is contentious.
One option is to take the mist itself as a real physical property. This results either in some weird mechanical properties of the mist (if we accept collapse) or problems to do with explaining our own experiences within a single universe.
Another option is to take the mist as a state of knowledge. Within this option there are two possibilities. The misty states might be states of knowledge (like rocky states) where the macroscopic-scale events they represent arise from underlying real states. Or the misty states (not at all like rocky states) might be a new type of inferential object, which does not need a connection to underlying “real states.”
If there are real underlying states (i.e. properties) of physical objects, then, Einstein argued, there are multiple misty states corresponding to a single real state. His argument assumed locality. This assumption is particularly questionable as he also needed entangled states, which we now know can produce nonlocal correlations.
Without using entangled misty states, but also with a different locality/independence assumption, Pooh-Bear gave an argument for the opposite conclusion to Einstein— namely that real states correspond to unique misty states.
Epilogue
Quantum. It is a word I read hundreds of times a week in scientific papers. But here I was in an airport pharmacy, buying underarm deodorant, reading that magical word through the mental fog of travel fatigue. My armpits definitely need to be “quantum dry,” so I purchased it. I long ago gave up being irritated by stupid use of the q-word in everything from “quantum auto-body repair” to “quantum financial services.” As a word that actually originated in describing the very smallest quantities of stuff, calling your product a “quantum leap” makes about as much sense as advertising it as “miniscule progress you definitely won’t notice but which you should pay more for.”
Although this has been a book about quantum theory—the most important theory of modern physics—I have avoided using the word “quantum” per se. Primarily this is because I want to avoid what is now extremely loaded jargon about which a ridiculous number of misconceptions abound. Many of these misconceptions are due to poor use of language describing much more precise mathematics and/or experimental facts. My goal here has been to teach those facts and avoid the jargon. There is also the danger of thinking that quantum theory is difficult to sort out because we cannot “see” the microscopic things to which we normally apply it. But in fact you can see with your naked eye the fluorescence from a single atom. We could therefore build boxes very much as I have presented, where instead of black and white balls we would see something like a small dot of red or green light bouncing off (or emitted by) an atom. We have no indications whatsoever (and in fact many to the contrary) that quantum theory will fail to be the correct description of the world up to human scales. The difficulty of building PETE boxes for large-scale objects appears to be only one of engineering. If somehow quantum theory fails to be the correct theory at larger scales that would be a remarkable discovery.
The “misty states” are just “quantum states,” and the phenomena of nonlocality and entanglement and superposition would often be called “quantum nonlocality” and “quantum entanglement,” and “quantum superposition.” I have taken “the laws of quantum physics” to be the axioms of the theory as typically taught to an undergraduate physicist (often associated with what is called a “Copenhagen Interpretation” although they are basically just the rules taken at face value, and not really an interpretation of those rules per se).
As Part I hopefully convinced you, we are entering an age of amazing new technologies based on these counterintuitive phenomena. Actually there are many things we already use every day that rely on quantum phenomena. The internet is powered by lasers, which require quantum physics. The GPS system relies on atomic clocks, which require quantum physics. Although remarkable and important, these, and many other, “old quantum technologies” are basically only making use of “quantum superposition”— the phenomenon which can be exhibited by a single ball in a misty state of black and white. The challenge is to build larger, entangled, misty states with which we can do even more valuable things. It is a challenge I am having fun taking on, and perhaps you can join in one day too.
It is also fun to try and come up with “natural” explanations of quantum weirdness, but hopefully you now realize that this is an extremely challenging task, and should be approached with both care and precision.
History, Context and Further Reading
I intend to put some material on www.qisforquantum.org, both to help understand misty states better, as well as to try and connect the formalism of the mist to regular quantum theory for students who go on to university level study.
Although the misty balls do not easily describe every possible part of quantum theory, they are an extremely powerful subset, universal for quantum computing, and so in principle can simulate and calculate everything in the full theory. This means there are many other parts of quantum theory I did not try to cover that can be readily learned in this diagrammatic form—the uncertainty principle, teleportation, secret key distribution, all the main quantum algorithms, superdense coding and so on.
Part I
Let me relate some other jargon, which you will hear as you go about your physics reading, to things you now know from this book. “Wave-particle duality” is just one manifestation of the phenomenon that a ball can sometimes be “definitely white” and sometimes “definitely in a misty state of black and white.” A ball that is definitely black or white is like a particle. A ball in a misty state is “wavy” because, just as a peak in a water wave can interfere with a trough to yield no wave at all, a ball configuration with a negative-sign label can cancel with one with a positive sign label. The Wikipedia articles Wave–particle_duality and Interference_(wave_propagation) are a good start if you want to understand a bit more. However, once we are considering two or more balls, there are very many other ways in which misty states are not at all like water waves (or any other regular physical waves), so the usefulness of this wave/particle way of thinking is debatable.
“Quantum coherent” means (in the terms we’ve been using) “misty”; “relative phase” or “quantum phase” refers to the negative-sign labels. The weirdness of quantum superposition is often prosaically illustrated in terms of Schroedinger’s cat being in a (misty state) superposition of dead and alive. Schroedinger actually used the cat to illustrate the even weirder phenomenon of quantum entanglement, discussed in Part II.
The remarkable (to me) fact that I can go so far in building up proper quantum theory for you using only basic arithmetic is due to the proof by Shih (arxiv.org/abs/quant-ph/0205115) that the Toffoli plus Hadamard gates are universal for quantum computing, which means in fact they could be used to accurately calculate any quantum phenomena. The Toffoli gate is just our CCNOT box, the Hadamard gate is the infamous PETE box. (The infamous Pete person is at peteshadbolt.co.uk). The CSWAP box can be used in place of the CCNOT, and its universality for regular computation was discovered by Fredkin. The Wikipedia articles Toffoli_gate, Quantum_gate#Hadamard_gate and Fredkin_gate have more information.
If you are interested in the interplay between our processes of logical reasoning and computation, then it is worth reading the introduction to Turing’s famous papers which formalized them, along with one of Andrew Hodges’ many books or essays about Turing at www.turing.org.uk/publications/. If you want to go the next step to understanding more quantum information theory try Scott Aaronson’s Quantum Computing Since Democritus (www.scottaaronson.com/democritus/).
The robbing-a-bank example I used to i
llustrate the power of multi-ball misty state (quantum) computation is called the Deutsch-Jozsa algorithm; I stole the idea of using a bank robbery to explain this algorithm from Naomi Nickerson.
Part II
The remarkable discovery that quantum theory allows for the generation of nonlocal correlations is due to Bell, and is typically called Bell’s Theorem. Coincidentally, his version makes use of the “Bella” misty state. The particular game with the psychics that I set up exhibits a special case of a version of quantum nonlocality due to Hardy (doi.org/10.1103/PhysRevLett.71.1665). Hardy’s paper is difficult to understand; I provide the link for completeness only. The Wikipedia article on Bell’s theorem isn’t great at the time of writing, and the internet is full of incomprehensible, wrong, or overly technical stuff about quantum nonlocality. Good luck. However Bell’s original papers, some of which are not overly technical, are still a beautiful read. They are collected in the book Speakable and Unspeakable in Quantum Mechanics.
It is more common when discussing nonlocal quantum correlations to consider a game due to Clauser, Horne, Shimony, and Holt that the psychics win if they give the same color answers (BB or WW) if either one, or both, of them are told tails, and opposite color answers (BW or WB) when both are told tails. There are no other rules per se. Any local strategy can only win this game 75% of the time (known as the CHSH inequality), but with the psychics sharing balls prepared in the Bella mist it can be won over 85% of the time. I may add a fuller discussion to the webpage for the book if enough people are interested in it.
Q is for Quantum Page 11