Q
is for
QUANTUM
Terry Rudolph
© Terry Rudolph, 2017
All rights reserved. Without limiting the rights under the copyright reserved above, no part of this publication may be reproduced, stored in, or introduced into a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording, or otherwise) without prior written permission. While every effort has been made to ensure the accuracy and legitimacy of the references, referrals, and links (collectively “Links”) presented in this book, the author is not responsible or liable for broken Links or missing or fallacious information at the Links. Any Links in this book to a specific product, process, website, or service do not constitute or imply an endorsement by T. Rudolph of same, or its producer or provider. The views and opinions contained at any Links do not necessarily express or reflect those of T. Rudolph. Particularly the ones on interpretations of quantum theory.
For permission requests, please contact: [email protected]
ISBN: 978-0-9990635-0-7
www.qisforquantum.org
Cover design by Chris Van Diepen, createwithbodhi.com
Interior design by booknook.biz
For Xavier, Aby, Lydia, Jesse and Caleb
TABLE OF CONTENTS
Preface
Introduction
Part I: Q-COMPUTING
Black balls or white balls?
A twist on the balls
What follows in this book is the only option we know
Representing superpositions, the new state of physical/logical being
Is the mist really a “state of physical being”?
Computers without mist
Logic from the motion of matter
Mist through computer-rules boxes
Mist through both computer-rules and PETE boxes
Collisions within a fog
Good grammar, is essential
Lunchtime lesson
Misty computation can be very lucrative
The Archimedes calculation (consider this an aside)
Is there no limit to the magic?
Well, why can’t we buy one of these magical, misty computers yet?
Summary of Part I
Part II: Q-ENTANGLEMENT
A tale of testing telepathy
Playing the games
What went wrong?
A tangled question: how did they do it?
Nonlocality of correlations
Causal nonlocality would have to be weird
Computing the likelihood of observing a particular configuration given a complicated mist
Making observations on a few balls within a multi-ball mist
Entanglement
Summary of Part II
Part III: Q-REALITY
Realism and physics
Physical properties
Correspondence between mathematics and physical properties
A deeper description of the rocky state of a coin
Revisiting our first conundrum
Two variations on a misty state “being real”
If misty states are real, should we collapse in confusion?
Currency collapse, mental collapse
Einstein throws himself in completely
Questioning Einstein’s two assumptions
Why no faster-than-light communication?
Pooh-Bear creates complete confusion
How can Owl do it?
Recapping, rexplaining
Final thoughts
Summary of Part III
Epilogue
History, Context and Further Reading
Part I
Part II
Part III
Acknowledgements
Preface
This book has been written for my 15-year-old self. Well, there is plenty of advice I would give my 15-year-old self that you don’t want to hear, and there are very many things I was very interested in at 15 that are not covered here. But one thing I was interested in at that age was science, and I distinctly recall being frustrated by the lack of concrete explanations within “pop-sci” accounts of modern physics. The exciting descriptions I found in them were ultimately hollow. They were vague on details and they came loaded with jargon, questionable analogies, and somewhat mysterious pontifications about the nature of physical reality. Implicitly justifying the lack of explanation were historical anecdotes about how these discoveries confused all the famous physicists who made them as well.
As someone who wanted to tackle these mysteries (perhaps solving them before I left high school!) the situation was frustrating. Having failed to solve the mysteries by 17, I was forced to go and waste years studying physics at university. The failures kept piling up, and I have ended up a professional physicist who is still both confused and amazed by our physical laws.
A fairly recent mathematical breakthrough (not by me) suggested a very different method of presenting some of the most interesting and weird phenomena of modern physics. Through my talks to teenagers I have found it is possible for students who know only arithmetic to quantitatively—not just qualitatively—understand the most important features of many of our deepest confusions about what is going on in the natural world. In fact, my nine-year-old nephew understood why there is a genuine mystery about what could “really be” happening inside the PETE boxes introduced in Part I and a few days later asked me whether I have solved it yet. Perhaps he wanted to solve it himself before leaving primary school.
I eventually decided to write down the method I use, and here it is. Of course it is very much easier explaining this kind of thing in the back-and-forth dialogue of a classroom. It would be awesome if high-school teachers got interested in this approach and felt comfortable enough to explain these wonderful things to their students. I welcome feedback about what other resources I could provide to facilitate this.
Introduction
This is a book about physical phenomena I find deeply mysterious; about how we plan to harness them in amazing new technologies despite not really understanding them; and about where we stand in our attempts to obtain such an understanding.
We use mathematics to help us describe things going on in the physical world around us. This is not only because quantitative statements (which are precise and technical, such as: “If you fall out of a tree twice as high it will take 1.4142… times as long to hit the ground, regardless of the tree height or what size planet you are on”) are more useful than qualitative ones (which are fuzzy and vague, such as: “Well, duh, it’ll take longer becoz the tree is higher, which planet are you on?”). Rather, to a physicist, the math is an inextricable part of our understanding. Many times we have successfully predicted the existence of new physical objects and new physical phenomena based on the math alone.
The mathematical equations of physics—the “physical laws”—typically provide us with precise and beautifully intricate rules by which we understand how physical things we either observe directly, or otherwise believe exist, connect to each other. This lets us build a story, a narrative, about what is “really” going on. It does sometimes happen, however, that we are uncertain about the exact connection between the useful math and the physical world. For example, prior to the direct observation of atoms, some doubted their existence for this sort of reason.
Modern physics is in an odd situation. Some of our most important physical laws lead to a very strange (many would say completely nonsensical) narrative about what is “actually” going on when we view the mathematical objects in the theory as corresponding to something physically real.
More precisely then, this is a book about the tension between the abstract math, the observed physics, and the inferred story.
Along the way the goal is to elucidate both strengths and limitations of some of the very cool new technologies we are currently building based on these incompletely understood laws.
Unfortunately, not everyone is good at mathematics, and most have little, if any, training in physics. So to tell the remarkable story of this ongoing intellectual adventure and the controversy around it, I (and many others) typically have resorted to qualitative expositions. These are very, very limited, and appreciating them alone simply will not let you make a meaningful contribution to the discussion, despite many emails I receive from crackpots suggesting the contrary. It is like only having van Gogh’s “Starry Night” described in words to you, by someone who has only seen a black and white photograph. One that a dog chewed.
I recently came to the realization that it is possible to do much better than this. I believe I can help you properly understand most of the mysteries that swirl around our abject failure to take some mathematical equations—which unquestionably describe experiments we can do—and underpin them with a universally accepted physical narrative.
Nominally, the only math required in this book is the arithmetic of positive and negative integers. But in fact the drawings you will see in subsequent pages are mathematics. They are symbols on paper that we manipulate according to fixed rules, which have subtle relationships with each other, and which act as shortcuts for much longer and wordier descriptions. All math is really just this. What I am doing with all these drawings is what theoretical physicists do for a living—play around with numbers and equations and diagrams to try to describe certain things that we observe (or suspect) happen. When we find some that seem to explain the situation consistently, we are delighted if puzzled, and use them to do more complex calculations pertaining to related happenings. As long as the outcomes fit with the observations, we feel somewhat happy with our equations. But ultimately we would like to feel we “really understand” what our mathematics “means” in terms of stuff that actually goes on in the physical world.
I begin in Part I by presenting some simple but amazing experiments we can do, building up the math we use to describe them. From this I will be able to show you how we will soon build new types of computers, ones that think using a logic very differently to ours. We can do this even though we don’t have a deep understanding of what is going on inside them at the underlying physical level. In Part II we will tackle the strange phenomena of nonlocality and entanglement; for me these were the gateway drug to physics. In Part III we go on a trip that may make you wonder if physicists are on other kinds of drugs too, as we explore the strong incompatibility between the “physical realism” we simply take for granted, and any sensible narrative about what is actually going on.
I really should spend a few more pages spouting profound-sounding blather, both to set up the story of this book and entice you to buy it. But while you are mentally fresh I’d rather get you to concentrate on some technical things.
Part I: Q-COMPUTING
Black balls or white balls?
Imagine you have a box which has a hole in the top and a hole in the bottom. (I say “imagine,” but I want to emphasize from the start that what follows is not an analogy, but rather a description of physical devices which we could, in principle, build. They are constructible according to the laws of physics as we know them today. However, the prohibitive cost and engineering challenges of building them means we do not actually try to do these experiments this way—we use other physical setups which are less easy to describe, but which have identical functionality.)
OK, back to your box with a hole in the top and bottom. You can drop either a black or a white ball into the hole in the top, and as it falls through the color flips. If you drop in a black ball, it comes out the bottom hole white. If you drop in a white ball, it comes out black. You could label this box “flip” or “change,” but for various reasons it is traditionally labeled “NOT,” since a white ball comes out “not-white,” i.e. black, and vice versa.
Next imagine you have a different type of box with two holes in the top, and two in the bottom. You discover that if you simultaneously drop one ball in each top hole, then the balls which emerge from the bottom have their color swapped with each other:
Looking at the balls coming through the first and fourth boxes one might be unsure that a swap had occurred, but it did—white just swapped for white and black for black. Dropping balls of different colors into the second and third boxes makes this clear. We may wonder then if the box is swapping the balls themselves. To check, we can use a plastic ball on the left and a metal one on the right. We find that the ball entering a hole always drops out from the hole directly below it, only the colors have swapped.
Another two-ball box is the CNOT or “controlled-NOT.” This is a box where a NOT happens to one of the balls, the “target” ball, based on whether the other “control” ball is “switched on” by being black. If the control ball is white, nothing happens. In either case, the control ball’s color is always unaffected:
A useful three-ball box is the CSWAP or “controlled-SWAP”. Here is how it works on all possible input colors of balls:
Like the CNOT, nothing happens to any of the three balls when the control ball is white, as you see in the first row of boxes. When it is black, a SWAP happens to the color of the two target balls. In the figure, a SWAP has happened to the colors of all the target balls in the second row of boxes, but you can only see it in the colors of the target balls in the middle of the second row, since the others swapped color with a ball of the same color as themselves.
The next thing to consider is that by stacking boxes on top of each other, we can use the output of one box as the input to another. For example, we can stack two NOT boxes, and the resulting transformation is that the color of the ball stays the same:
We can repeat this stacking trick to execute more sophisticated transformations of balls. For example, consider this arrangement:
On the right, I have shown the calculation of the color of each ball progressing through the boxes, for the case when all three balls we drop in to the physical setup on the left are black. If we do a similar calculation for the other seven possible input configurations of three balls we find: (i) the first two balls always emerge the color they went in, i.e. their color is unaffected overall; (ii) when both of the first two balls are black a NOT is applied to the third ball. In all other cases the third ball is unaffected.
We could combine these three boxes into a new box, which we would call a “controlled-controlled-NOT” (CCNOT) box, since it applies a NOT to the third ball only when both the first two “control” balls are black. Because it is an important box, I recommend you write out a full schematic for the behavior of the CCNOT box much as I did for the CSWAP box a few figures ago. (Don’t worry if you cannot, I will do it for you when we encounter it again.)
A twist on the balls
So far you could be forgiven for thinking that all we have seen so far is a bit of a silly game. Yet I’m pretty sure I can convince you, in a little while, that even the simple boxes we have already encountered are actually doing something interesting and extremely important, both practically and philosophically. Before getting to that, however, I want to describe one final box, a box whose behavior is so profoundly mysterious I am really hoping it will, by the end of this book, go much further. I hope it will completely change your views on what is “real” about the physical world around you.
Traditionally this last, very strange, box is named after a person who never built or even envisioned such a device and who has plenty of other things named after him. So instead I will call it the PETE box, after my friend Pete who has spent a significant portion of his life building and testing versions of it. Like the PETE box, Pete often does strange things—for example he put together a machine that enabled a tank of goldfish to browse the internet and control a drum machine. He made the profound discovery that goldfish like seeing humans without their clothes on.
> The PETE box has only a single hole in both the top and the bottom. After playing with it for a while, we find that regardless of the color of the ball that we drop in, when it emerges from the bottom it is equally likely black or white; and from one use of the box to the next there is no pattern, no rhyme or reason, about which color the ball emerges:
Is the behavior of the PETE box really so different from the boxes above? Of course the ones above behaved perfectly predictably, while the PETE box is unpredictable—which color emerges is completely random. So far we have deliberately not asked any questions about what goes on inside the boxes we have encountered. All we have considered is what they do that we can actually observe. As described thus far, however, the PETE box’s possible inner workings are not necessarily particularly strange. We can imagine building a box with an internal mechanism which flips a coin. If the coin shows tails, it lets the ball travel through directly; but if it shows heads, a NOT box is inserted into its path:
If this was the explanation of the inner workings of the PETE box, it would not be a radical addition to our collection of boxes. However, when we stack two PETE boxes, something remarkable happens: if we drop a white ball in the top of the first PETE box it always emerges white from the bottom of the second box. Similarly, if we drop a black ball in the top box it always emerges black from the bottom of the second box:
Can you see why this behavior is puzzling? It is critical that you do. The second PETE box, regardless of whether the ball entering it from the first box is black or white, should sometimes output a black ball and sometimes a white ball, because inputting a white ball leads to a random color emerging and inputting a black ball also leads to a random color emerging. But that is inconsistent with what is happening when we stack the boxes; stacking them leads to a completely predictable, non-random output.
Q is for Quantum Page 1