by Jed Brody
This is not exceptionally light reading, but it is recreational, for the analogies to quantum systems take the form of logic puzzles. Unlike most logic puzzles, these quantum analogies are not arbitrary or contrived but symbolize the predictions of quantum physics and thus the ultimate nature of reality. Besides analogies, we will study actual laboratory observations, including the first experimental contradiction of local realism.
All physicists agree on the mathematical predictions of quantum mechanics (which is what physicists call quantum physics). And all physicists agree that experiment resoundingly confirms these quantum predictions. But there is no consensus about how to interpret quantum mechanics. By the end of this book, you will understand the reasoning that forces us to discard everyday assumptions, and you will be able to draw your own conclusions.
Before journeying into the microscopic world, let’s think about common objects like computers and coins. In the context of these familiar objects, we’ll develop the concepts that we’ll need later. In subsequent chapters, we’ll study particles that are directly governed by the unnerving and enchanting quantum laws.
Hidden Variables
Let’s begin by imagining a computer that displays a random number every time we tap the space bar. The displayed number is only one digit, so there are ten possibilities: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each of these numbers is just as likely as the rest, appearing about 10 percent of the time. Suppose that we’re really intrigued by these random numbers, and we want to develop a theory to explain what’s going on inside the computer.
Our first attempt at a theory can simply be a summary of our observations: each number from 0 through 9 occurs about 10 percent of the time. This theory is a probabilistic theory: it gives us the probabilities of future outcomes, but it doesn’t predict exactly what each outcome will be. This is a disappointment. We suspect that if we knew the algorithm used by the computer program, we could predict the exact outcome every time we tapped the space bar.
So, we begin to speculate about possible algorithms employed by the computer program. Suppose the seemingly random numbers are based on a hidden clock inside the computer. The clock, we imagine, has a precision of milliseconds, so the time starts at 0.000 seconds and proceeds to 0.001 seconds, and so on. What if the seemingly random number is simply the last digit of the hidden clock at the instant we tap the space bar? So if the hidden clock has the value 143.852 when we tap the space bar, the computer displays the last digit, 2. If the hidden clock has the value 5762.267 when we tap the space bar, the computer displays 7. The displayed numbers appear random because the last digit of the hidden clock is completely uncorrelated with our casual decision to tap the space bar.
Let’s further develop this theory, even though it’s a just a guess. Let’s say that the last digit of the hidden clock is a hidden variable. The theory that we’re developing is a hidden variables theory. Let’s call the hidden variable λ (lambda), so when the hidden clock has the value 143.852, λ = 2. When the hidden clock has the value 5762.267, λ = 7. Let’s say that the number displayed on the computer monitor is N. So our hidden variables theory is simply
N = λ:
the number displayed on the monitor equals the hidden variable.
In the hidden variables theory we just came up with, λ isn’t so hidden because it happens to equal the displayed number N. We can imagine an alternative theory in which
N = 9 − λ
and λ is still the last digit in the hidden clock. But now, when the hidden variable λ = 0, the displayed number N = 9. Our new hidden variables theory generates seemingly random numbers just as well as the original theory. In fact, there are lots of other equations we could think of to explain the random number N in terms of the hidden variable λ. Here’s just one more example: N = λ+1 unless λ = 9, in which case N = 0. There are many equally plausible hidden variable theories (equations for N in terms of λ).
We can even consider other sources of the hidden variable λ. Maybe λ, instead of being based on a hidden clock, is somehow based on the number of people watching internet videos of puppies at the moment we tap the space bar; the computer program could search online to find this information. Or λ could be based on the price of gold, the temperature in Nairobi, or countless other quantities found online or inside the computer.
We can imagine many quantities that λ could be based on, and we can imagine many ways to calculate N in terms of λ. In all cases, we remain confident that the computer program uses some algorithm to determine the displayed number N based on some hidden number λ; the computer doesn’t pull N out of thin air.
We originally defined N as the number displayed on the monitor, but we’ve now redefined it as a quantity calculated from λ; λ changes over time, though at any moment λ has a precise value. Since N can always be calculated from λ, N too has a precise value at every moment, even though we’re not tapping the space bar all the time. So N is a quantity that exists, at least mathematically, at every moment, even when we’re not observing what it is.
Let’s compare the hidden variables theory with our initial probabilistic theory, which simply stated that each number from 0 through 9 occurs about 10 percent of the time. The probabilistic theory is true but incomplete; it doesn’t predict the displayed number with certainty, so the theory’s missing some facts. The hidden variables theory is complete because it does predict the displayed number with certainty, at any moment. For example, we could propose the hidden variables theory—that the displayed number N is the last digit of the number of people currently watching internet videos of puppies—and then we could test this theory by finding out how many people were watching puppy videos at the moment we tapped the space bar. The hidden variables theory may be correct or incorrect, but it always predicts exactly what N is at every moment.
Let’s suppose we come up with an accurate hidden variables theory. Then, the probabilistic theory is consistent with the hidden variables theory. Specifically, the hidden variables theory predicts N with certainty, and the probabilistic theory correctly gives the probabilities of the different possible values of N.
Flipping a Quarter
Imagine flipping a quarter. (But you can do more than just imagine: the equipment required for this experiment costs only 25 cents! If you can’t afford a quarter, you can try selling this book for a quarter. If no one thinks it’s worth a quarter, try a dime. A nickel? A penny? Anyone? Please?)
You flip the quarter into the air using your thumbnail. While the quarter is spinning in midair, you use one hand to slap it onto the back of your other hand. Don’t remove the top hand yet; keep the quarter concealed. Without looking at the quarter, you already know that the outcome of the coin toss is either heads or tails. We’ll say that these are the two possible states of the quarter.
Do you believe that the state of the quarter is already determined, even though you can’t see it through your hand? Or is the state of the quarter not only unknown but also unknowable prior to observation? Is direct observation the only reality? Does the definite state of the quarter come into existence only when it’s observed? Most people probably believe that the state of the quarter is fixed the moment you slap it between your hands. We don’t know the state of the quarter until we look at it, but the observation merely reveals the state that the quarter is already in. If we believe that the concealed quarter has a definite state even though no one’s observing it, then we believe in realism. According to realism, physical states exist totally independently of whether anyone is observing them (or whether any laboratory instrument is measuring them). If we don’t know the state of the quarter hidden by our hand, it’s only because we’re ignorant; the quarter has a definite, physical state that we happen not to know.
Let’s try to outline a theory to predict the quarter’s state. Our first attempt might be to simply specify what we observe after repeated trials: about half the time we get heads, and the other half of the time, we get tails. This is a probabilistic theory, a
nd we want to try to do better. We want our theory to conform to physical reality. If the quarter has an exact physical state even before we observe it, then our theory should predict the exact physical state before we observe it. The probabilistic theory is accurate, but it’s incomplete because it doesn’t determine individual outcomes with certainty.
Is the state of the quarter already determined, even though you can’t see it through your hand? Or is its state not only unknown but also unknowable prior to observation?
If we want to predict the quarter’s state before observing it, we have to do a lot of physics. We have to consider all the forces on the quarter: the flick of our thumbnail, the weight of the quarter, possibly the effect of air resistance, and the force of the top hand as it comes down onto the quarter. We also have to know the position on the quarter at which each force acts: Do we flick the quarter right near the edge, or toward the center? We need to know the initial orientation of the quarter. There may be other parameters that we haven’t even thought of. We don’t actually want to go through all the work of developing a complete theory, so we don’t need to list every parameter that influences the final state of the quarter. Instead, we refer collectively to all the influential parameters as hidden variables.
Without even constructing a hidden variables theory, we can say some things about it. It is complete: it predicts with certainty the final state of the quarter every time. It conforms to realism: it assigns a final state to the quarter regardless of whether anyone is observing it. It is consistent with the probabilistic theory: it predicts heads half the time, and tails half the time. In the hidden variables theory, though, nothing is truly random.3 Instead, common variations of the hidden variables (the speed and angle at which the thumbnail flicks the quarter, etc.) lead to a prediction of heads just as often as tails.
Shaking Two Quarters
Now imagine shaking two quarters between your hands. (The cost of the experiment just doubled. At this rate, you’ll need a particle collider by the end of the book.) You know that one of the quarters is dated 1999, and the other is 2000. You shake the quarters so well that you don’t know which is which. Without looking at the quarters, you separate them, one in each hand, concealed by your fingers.
Before you open either hand to observe a quarter, is the date of that quarter already a physical reality? Or do the properties of the quarter come into existence only when you observe them? Once again, we’re asking about realism: Is the date of the quarter in your left hand an objective, physical reality even if no one can see it or know with certainty what it is?
If we reject realism in this case, something very strange happens. If a specific property (the date) of the quarter in your left hand comes into existence at the moment it’s observed, then the quarter in your right hand must simultaneously acquire the other date. The observation of one quarter, absurdly, affects the other quarter.
Common sense rejects absurdity and demands that the observation of one quarter has no effect on the distant quarter (even if the distance is only an arm’s length). This is the everyday assumption of locality: the observation of one object has no effect on a distant object. In fact, locality implies realism. If observing one quarter has no effect on the other quarter, and the two quarters are always observed to have different dates, then both quarters must have had their dates all along. This combination of locality and realism is local realism: objects have properties that exist regardless of whether anyone’s observing them, and they’re unaffected by observations of distant objects.
Let’s contrast a few different points of view. Suppose you shake the two quarters and conceal one in each hand. After a moment of quiet contemplation, you open your left hand to reveal the quarter dated 1999. What does this imply about the quarter in your right hand, which you still haven’t opened?
•Local realism: The quarter in your left hand was the 1999 quarter all along, and the 2000 quarter was in your right hand all along. The observation of the 1999 quarter merely informs us where each quarter was all along. The observation has no effect whatsoever on either the quarter you observe, or on the other quarter.
•Observation transforms physical reality: The quarter in your left hand became the 1999 quarter at the moment you observed it. The quarter in your right hand simultaneously transformed into the 2000 quarter. We dare not ask what the dates on the quarters were prior to observation.
This combination of locality and realism is local realism: objects have properties that exist regardless of whether anyone’s observing them, and they’re unaffected by observations of distant objects.
•Direct observation is the only reality: Unobserved properties don’t exist; anything outside our senses is senseless, and is imagination, not reality. The quarter in your left hand acquired its property (the date, 1999) the moment you observed it. The unobserved quarter in your right hand still has no properties (beyond what you can feel), even though you know that whenever you observe it you’ll see the date 2000.
It seems that all of these viewpoints, however implausible, cannot be refuted by evidence: How can we acquire any evidence of how something is before it’s observed, given that we gather evidence only through observation? Astonishingly, astoundingly, physicists have discovered:
•Local realism imposes constraints on measurable quantities. (Many versions of this surprising fact are proven in detail later in the book.)
•Measurements of entangled particles violate the constraints imposed by local realism.
•Therefore, local realism is not a valid assumption for entangled particles.
This book will empower you to draw your own conclusions from the (almost) incontrovertible fact that quantum mechanics is incompatible with local realism.
Let’s apply local realism to the two quarters. (Since quantum mechanics doesn’t apply to quarters in an obvious way, local realism is a legitimate viewpoint.) If the 1999 quarter was in your left hand even before you peeked at either quarter, then some combination of forces drove that quarter into your left hand. In principle, we should be able to develop a theory of hidden variables to predict which quarter ends up in which hand. The hidden variables include the starting positions of the quarters before you shake them together, the exact shape of your cupped hands, and the rate and intensity of the shaking. We want to emphasize that the observation of the quarter in one hand does not affect the quarter in the other hand, so we call our theory a local hidden variables theory. The local hidden variables theory is consistent with the simple probabilistic theory that merely gives the 50 percent probability of each possible outcome. The local hidden variables theory goes beyond the probabilistic theory to predict the exact outcome of each trial.
Before proceeding to real particles and their entanglement, let’s review the main points from this chapter.
•A hidden variables theory assumes that a seemingly random process is actually governed by influences that we’re inadequately aware of. If we knew all the influences exactly, we’d be able to predict all outcomes exactly.
•An accurate hidden variables theory is consistent with the observed probabilities of outcomes. For example, a hidden variables theory describing a coin toss should predict heads half the time—as well as predicting exactly which tosses will be heads.
•Realism is the belief that objects have properties that exist totally independently of whether anyone is observing them (or whether any laboratory instrument is measuring them; I use “observe” and “measure” interchangeably).
•Locality is the belief that the observation of one object can’t affect a distant object.
•If two quarters are shuffled and then separated, locality implies realism: if the observation of one quarter has no effect on the other, and the two quarters are always observed to have different dates, then they must have had their dates all along, prior to observation.
•As subsequent chapters will demonstrate in detail, the assumption of local realism imposes constraints on measurable quantiti
es. These constraints are violated by measurements of entangled particles.
2
An Experiment to Challenge a Philosophy
We know that electrons have an electrical property: negative charge. Electrons also have a magnetic property called spin. In the first experimental observation of spin, a beam of silver atoms was aimed through a magnetic field that was stronger in some places than others.1 (We can create this kind of magnetic field by taking a horseshoe magnet and sharpening one of the poles.) If we send silver atoms between the poles of the magnet, we find that some atoms are deflected toward the north pole, and some are deflected toward the south pole (figure 1). No atoms pass straight through. Just like a coin toss, exactly two outcomes are possible.
This experiment is generally performed with neutral atoms—not isolated, negatively charged electrons. The deflection resulting from an electron’s charge overwhelms the deflection resulting from its spin. But the beauty of thought experiments is that we can ignore inconvenient details such as these. Let’s do a series of thought experiments in which the electron’s deflection results from its spin, and we ignore the deflection caused by its charge.
Figure 1 The arrows represent silver atoms passing through a magnetic field, with some deflecting toward the north pole (a), and some deflecting toward the south pole (b).
We imagine an experiment with special pairs of electrons. Suppose that the two electrons are emitted from a common source, and they travel in different directions. Each electron encounters a magnetic field. If the two magnets have the same alignment, the two electrons are always deflected in opposite directions: one to the north, and one to the south (figure 2). Because the behavior of one electron is linked so powerfully to the behavior of the other, we say the electrons are entangled.
