After checking that each PETE box on its own is behaving properly, what is the next natural thing to do? Well, we get suspicious that perhaps the ball that enters the second PETE box has somehow been messed around with, even though we have exhaustively tested the boxes on their own and they seem to be operating fine. For example, perhaps the PETE boxes detect the presence of each other and change their behavior somehow? To check we do the following experiment. We slightly pull apart the two PETE boxes so we can shine a light through the gap. The light lets us determine the color of the ball emerging from the first PETE box, just before it drops into the second. What do you think we see?
We find that the ball that comes out the first box is half the time white and half the time black, as we know PETE boxes do. So it really does appear to be behaving normally. And now, if we let that ball we have observed keep going into the second box, it emerges from the second PETE box half the time white and half the time black. That is, if we observe after the first box whether the ball is actually black or white, then the second PETE box starts to behave unpredictably (randomly) again. It no longer outputs the ball always the same color as it was dropped into the first box.
We then turn off the light so we can’t see the ball in-between the boxes, and immediately the two PETE boxes act perfectly predictably (non-randomly) again—a white ball always emerges white, a black ball always black.
Fine, perhaps the light is screwing things up? Well, we try many, many other less-invasive methods of observing the color of the ball after it emerges the first box, just before it enters the second. We find that no matter how smart a technology we employ, if the method we use is capable of determining the color of the ball emerging from the first box, then it causes the second box to have the random, sometimes-black-and-sometimes-white, completely unpredictable output. If the method we employ cannot tell us the color of the balls (e.g. we use too dim a light), then they behave in the fully predictable way where the color that emerges from the second box is the same as the one going into the first.
We conclude from all this that somehow, just by our peeping, we have affected the process. It may remind you of baking a cake: if you open the oven door and peep in while the cake is rising, the cake goes flat; but if you wait patiently until the cake is cooked, it rises as it should. In that case we know the reason our observation changed things—we let cold air in. However, we don’t know the reason our observations affect the ball between the PETE boxes. What we are sure of is we cannot be passive observers of the balls exiting a PETE box—and since the balls and boxes are ultimately made up of physical stuff, this becomes a realization that we cannot always be passive observers of the physical world.
This portends a major shift in how we view our interventions in the world. How strange is that? Well it’s certainly a step away from the classic scientific view of the universe in which we believe we are not ultimately that important—and so can make sense of things either much bigger or much smaller than us by presuming they conduct their business in (understandable) contempt of our actions.
Later I will explain why, while such “observer dependence” is interesting, the fact that observations have consequences is not necessarily a complete breakdown of the whole intellectual edifice upon which science has stood successfully for centuries. (After all, perhaps everything we do is like baking a cake.) By contrast, such a dramatic conclusion is ultimately where the PETE box will try to lead us by the end of this book.
Another conclusion, almost as dramatic as the first, is that the PETE box’s behavior portends a failure of the very logic that underpins how we think. This has exciting consequences. For example, it lets us envision radically new types of computers and other technologies—although in return it is hard to understand the full potential of these technologies, precisely because they don’t sit well at all with our “sensible” logic.
Now it takes a few steps to justify this second conclusion. The first step is to try and express what we think is “happening” to the ball—what do we think is “really going on”; what do we think the “facts of the matter are”; what is “the status”? Or, to use language that is meant to capture all of these: what is the “real state” of the ball when it exits the first PETE box?
The word “state” is itself pretty loaded jargon to physicists, and later we will discuss some more precise notions of the state of a physical system. So far all we can be sure of is that the balls that we observe come in at least two distinct states, black and white. Other colored balls are possible to build, so to be cautious we try yellow, red, and grey balls (which are arguably something in between black and white). We find that PETE boxes simply don’t work at all if we input any color other than either black or white: nothing drops out the bottom at all. So it is natural to expect that, even if we don’t observe the ball when it exits the first PETE box, it actually is either black or white. However, if the state of the ball exiting the first PETE box really is black “or” white, then we have a black “or” white ball entering the second PETE box, and in that case the ball exiting the second box would randomly sometimes be black and sometimes be white. But it is not—this is the whole conundrum!
We are forced to conclude that somehow the logical notion of “or” has failed us. The other natural logical notion when faced with only two possibilities is to say perhaps the ball is black “and” white. Now intuitively this is nonsense—a single ball which is black “and” white is as ridiculous as a cat which is both fat “and” skinny. It is trying to combine two things that are mutually exclusive, while using a logical notion which requires the possibility that they are not.
Confronted with this conundrum, physicists simply invented a new word to describe the ball after it exits the PETE box. We say the ball is “black superposed with white,” or more colloquially it is “in a superposition of black and white.”
Superposition is a completely new possible state of physical being, and a completely new state of logical being, for two distinct alternatives. Sometimes you will loosely hear a superposition referred to as “black and white,” but this is either ignorance or laziness. You know better now.
I will now step away from talking only about experimental observations, to explain precisely how our current physical laws describe such experiments. The first step is to find a way to represent superpositions—these new possible states of physical/logical being. The way we do it seems very arbitrary when you first encounter it, so first a small aside to motivate you to try and learn the details, rather than just skipping over them:
What follows in this book is the only option we know
What I will explain from now on is the only way that we have found to quantitatively describe what is going on with the balls and the PETE boxes. I cannot stress this point too carefully. Many people (myself included) have tried to devise alternative explanations, and often succeed in finding something that looks very different. Once we examine it more deeply we find, however, that it is either exactly (though non-obviously) equivalent to the description I will teach you, or it is in conflict with experimental observations and therefore worthless.
More precisely, what follows in the rest of this book is the only method known to work once we consider experiments that involve multiple balls passing through arbitrary combinations of the PETE boxes together with the CSWAP, NOT, CNOT and other boxes described above.
For the case of a single ball falling through stacked PETE boxes you can actually find other potential explanations which are quite simple. Here is one. Perhaps both black and white balls can have a small sticker on them that we cannot see, but which a PETE box can see. This means there are four possible balls: white, with or without a sticker; and black, with or without a sticker. Perhaps when we create a white or black ball it also, randomly and with equal likelihood, either does or does not have a sticker stuck onto it as well. A PETE box changes the color of a ball if it has a sticker, and not otherwise. This means two PETE boxes either both change the color or both d
o not change the color—either way the output is the same as the input. To explain what happens when we observe a ball, perhaps our act of observation causes the sticker (if it is there) to be removed, and then randomly, with equal likelihood, a new sticker either is or is not added onto the ball.
I wouldn’t waste too much time trying to understand how this simple single-ball model works, I am just telling it to you because it is possible, and so when you find something equivalent on your own please don’t send me lots of emails telling me you’ve solved everything. Yet.
Representing superpositions, the new state of physical/logical being
To represent a superposition of black and white—and capture this new type of ambiguity between the ball being black and the ball being white—we draw a cloud into which we list both possibilities, separated by a comma:
Because the terminology “cloud computing” is already used for something well-known (and irrelevant to our discussion), let me call this new state a “misty” state. (My father would say it’s very mist-erious.) We could say, colloquially, that the white ball “splits” into a superposition, i.e. misty state, of both white and black.
Although the misty state seems to contain two alternatives simultaneously, we already know if we observe (look at) the ball, it reveals itself as only one of the alternative colors in the mist—and importantly, it does so completely at random. Thus, if we do observe the ball’s color, the mist disappears and we get left with just a regular black or white ball. The ordering of the two color configurations in the mist is irrelevant, just like the ordering is irrelevant when you list all the possible things you might get given for lunch.
It may seem that we should use exactly the same ambiguous representation for the state which emerges from a PETE box when we have dropped a black ball through it, because it also equally likely appears black or white when observed. But it must somehow be represented differently, because it must capture the fact that after a second PETE the ball always emerges white or black. This means there must be some difference between a mist originating from a white ball and a mist originating from a black ball.
You could probably come up with many alternatives to distinguish the two possible mists diagrammatically. But, as mentioned in the preceding section, any method that works in general is ultimately equivalent to the following:
Here the black ball in the mist has a “-” (a minus, or negative) sign in front of it. I think of it as a “-1,” a negative 1, somehow associated with, or labeling, this configuration within the mist. But it isn’t a “physically different” type of black ball; if we looked at the ball at this stage we would just see it as randomly either white or black. No matter what we do, we won’t be able to see anything to tell us that when it we see it black it is actually “negative-black.”
After the PETE box outputs a ball in this misty state, what happens when we drop that ball (without looking at it) into another type of box? The basic rule is that you apply the box to each configuration within the mist independently. For instance, here is what happens when we pass the two different misty states we have just encountered through a subsequent NOT box:
The ordering of the ball configurations within a mist does not matter, and so the mist that emerges for the case on the left-hand side is identical to the one which entered. On the right-hand side we see the negative-sign labels can apply to white balls as well. The NOT box does what it always does—it acts on the color of the ball, and it ignores the negative sign, which just comes along for the ride.
To see how the negative-sign label affects things, we need to look at what happens when we drop a ball that is already in a misty state (having gone through the first PETE box) through a second PETE box. Following the dictum that you simply act the box on each configuration within the mist, each ball within the mist splits, depending on its color, according to the precise rules for the PETE box given above. This gives a larger bunch of alternatives within the misty state; a mist within the mist. Somewhat intuitively, we find that a mist within a mist is mist—the boundaries fade into each other. Here is the whole evolution:
In these two figures we see the white ball split into a misty white and black ball, while the black ball splits into a misty white and negative-black ball. In the figure on the right, the ingoing black ball already had a negative-sign label, and when it splits that label is inherited by the whole misty state it splits into, hence the minus outside the cloud. This means it actually splits into a negative-sign labeled white ball, and a negative-negative-sign labeled black ball. Just like when your mother used the logic that if she makes two negative comments about the state of your bedroom she is actually being a positive influence, the two negative-sign labels combine to a positive label, which we depict as no label at all. What we see happen to the balls in the figure on the right is a bit like when we do math of the form:
–(2 + –3) = (–2 + 3).
Now, if it is ever the case within a mist that the configurations of balls (in this example we have one ball; later we will consider multi-ball mists) are such that two of the alternative color configurations are identical, except one has a negative-sign label and the other doesn’t, then both of them vanish. We say they “interfere” or they “cancel each other out,” like when we do math of the form:
+42 + –42 = 0
—except that the mist somehow describes material objects, not just ethereal numbers. We can see this happening in both figures above. In the left figure, the two black balls vanish in a puff of interference; in the right figure it is the two white balls that disappear. The only alternatives left in each mist are two configurations of the ball that are the same color, which means if we looked at the ball now we would certainly see it as that color.
Putting together all these rules we can calculate the effect on a white ball falling through two stacked PETE boxes, which yields the “illogical” behavior that it always emerges white:
A very similar figure could be drawn for an initially black ball, to show it always will emerge black. I have drawn a tube to connect the two PETE boxes. This is because our act of observing the color of the ball burns away (destroys) the mist, so in practice when we connect many boxes we must use something like such tubes to prevent us looking at the balls before we want to.
Do not be distressed if the negative-sign is mysterious. My own mother, commenting on an early draft of this book, wrote: “I cannot get into the Why of that minus applied to a black-misty, seems so unfair but I am bashing on with the reading in rebellious acceptance, believing it can somehow be justified and maybe even explained!” Being confused by all this runs in the family.
Is the mist really a “state of physical being”?
Before moving on to examine the exciting power of misty states, a word of caution. The mist itself is never directly observed. I have called it a “new state of physical/logical being”. However, amongst physicists these days the extent to which the misty state is “physically real” is very contentious. Everyone agrees that writing the mist on a piece of paper and using it to work out what we will observe in our experiments (performed with actual physical objects) is valid. So, in that sense, the mist is definitely a new state of “logical being” that somehow relates to a state of “physical being.” But (a few? some? many?) physicists believe that the mist should not be thought of as “physically real” in and of itself. They would say it is only a tool for calculation—it is something we humans use to make predictions about experiments, and the PETE box should not be thought of as spitting out or responding to a physical misty state itself.
To perhaps labor the point, the mist in the diagrams above can be thought of as either (i) representing an actual physical process of some real stuff (drawn as a mist, but obviously not made of tiny water droplets) passing through the boxes; or (ii) “just diagrams describing an experiment” where the only physical thing is the color of the balls entering and exiting the boxes (once observed).
It is fascinating that we have this incredib
ly precise theory, which, as we shall see, is going to let us build marvelous new useful devices, and yet we are still arguing about what it all really means. My goal in this book is not to bias you on which side of the argument to sit. In Part III, by which time you will have understood all the key points of the theory, I will introduce you to some of the arguments for and against both viewpoints.
In Part I we will stick to investigating its awesome potential within devices that do not care about our philosophical consternations regarding how they work.
Computers without mist
Obviously our lives have already been revolutionized by computers and the many related technologies all based upon the same basic principles of micro-electronics. Pause and count the number of objects within ten meters of you which contain such electronics. In fact, in the interests of science, I recommend you grab a hammer, smash your TV or computer or phone, and dig out some of those little black “computer chips.” Inside those tiny devices electric currents run around and combine together to produce the movies you watch, the game worlds you immerse yourself in, all the stuff you view on the internet, and so much more.
Yet what goes on inside those chips is actually just an electrical version of dropping black and white balls through the boxes we have encountered already—all the boxes, that is, except the PETE box. All the diversity of computational experience arises from electrical currents in one of two possible distinct states (a high and low voltage, but they may as well be called black and white electricity blobs) running through tiny boxes (etched into silicon), and coming out in one or other of the two possible states according to rules like those for the NOT, CNOT, CSWAP, and CCNOT described above. So I am going to call these non-PETE operations the “computer rules.” The computer rules are simple to state, but they are what we call “universal”; from just this simple set of rules you can create the inexhaustible complexity of our computer-based technologies.
Q is for Quantum Page 2