by Jed Brody
Figure 8 A polarizer is placed in the path of each infrared photon. If a photon passes through a polarizer, it reaches a detector. If the detector detects a photon, it sends an electronic signal to a circuit. If both photons are detected, they’re detected at the same time, and a coincidence is recorded.
All evidence indicates that we’re creating pairs of photons such that both are horizontally polarized or both are vertically polarized. We want to know if the photons all along are predisposed to a particular measured outcome, or if the photons are in a fundamentally undetermined state prior to measurement.
Let’s recall Einstein’s argument in favor of realism and apply it in this context: Whenever both polarizers are horizontal, for example, the two photons in a pair must always do the same thing (pass through if they’re horizontally polarized, or get blocked if they’re vertically polarized); therefore, common sense dictates that the two photons all along must have shared properties that predetermine their behavior at the polarizers. If, instead, the photons decide at the last minute what their measurement outcomes will be, they appear to be in some mysterious collusion: spooky action at a distance.
Einstein insisted on realism to preserve locality: the photons must have shared properties all along—from the moment they are created in a single location. And since these shared properties can’t be predicted with certainty in quantum mechanics, quantum mechanics must be incomplete, says Einstein: an unknown, more powerful theory should tell us exactly what each individual photon will do under all possible measurement conditions.3
Einstein tirelessly defended our common-sense assumptions, which were shared by many other prominent physicists. But we can no longer accept Einstein’s argument in favor of local realism. Bell showed that local realism imposes constraints that may be either satisfied or violated by experiment. In fact, experiment violates these constraints (Bell inequalities), so local realism is overthrown in the laboratory. John Clauser and his PhD student Stuart Freedman achieved the first experimental violation of a Bell inequality in 1972.4 They measured entangled photons, similar to those discussed in this chapter.
Freedman derived a Bell inequality that’s even simpler than the original. We’ll look only at the result here; as in chapter 2, the proof is a bit too mathematical for this book. In chapter 4, though, we’ll carefully develop additional Bell inequalities, without skipping any steps.
Einstein insisted on realism to preserve locality: the photons must have shared properties all along—from the moment they are created in a single location.
Freedman asks us to perform the following simple arithmetic:
•Record the number of photon pairs detected when the angle between the polarizers is 22.5° (figure 9a).
•From this, subtract the number of photon pairs detected when the angle between the polarizers is 67.5° (figure 9b).
•Multiply this result by 4.
•Is this greater than the number of photon pairs detected when the polarizers are removed? If so, then we’ve contradicted the common-sense assumption of local realism.
Figure 9 Two of the three measurements required for a test of Freedman’s inequality. The third measurement is made when the polarizers are removed.
To confirm that Freedman’s rules are just simple arithmetic, let’s look at a set of data. Freedman and Clauser didn’t publish their raw data, so it will be convenient to look at data taken by my student, Charlotte Selton.5 Detecting photon pairs over intervals of 50 seconds, she obtained
•1821 photon pairs when the angle between polarizers was 22.5°,
•377 photon pairs when the angle between polarizers was 67.5°, and
•4474 photon pairs when the polarizers were removed.
Let’s follow Freedman’s rules: We start with 1,821 (corresponding with 22.5°), and then we subtract 377 (corresponding with 67.5°), which gives 1,444. Then we multiply this result by 4 to obtain 5,776. This is greater than 4,474, the number of pairs detected when polarizers were removed, so we’ve contradicted the constraint imposed by local realism!
Before we rigorously develop additional Bell inequalities, it will be useful to generalize a fact about the pairs of entangled photons described in this chapter. We’ve seen that:
•Both photons are vertical, or both are horizontal.
Equivalently:
•If one polarizer is horizontal and one polarizer is vertical, there’s no possibility of a coincidence. (If one photon goes through a vertical polarizer, there’s no possibility that the other photon goes through a horizontal polarizer.)
But what if we set the polarizers to some other angle? It turns out that our rule for the entangled photons can be generalized:
•If the angle between polarizers is 90°, there’s no possibility of a coincidence. (If one photon goes through a polarizer, there’s no possibility that the other photon goes through a perpendicular polarizer.)
This fact does not obviously follow from the method of creation of photon pairs: the original violet photon splits, producing a photon pair, in one of two crystals. One crystal produces vertically polarized photons, and the other produces horizontally polarized photons. Let’s take it as an empirical fact that no coincidences occur when the polarizers are perpendicular to each other (regardless of whether the polarizers are horizontal, vertical, or diagonal). Another empirical fact is that just as many coincidences occur when both polarizers have the same angle (whatever the angle may be) as when both polarizers are horizontal, or both vertical.
Let’s measure angles clockwise from the vertical, so vertical polarization is 0°, and horizontal is 90°. (Clockwise, of course, depends on which side of the polarizer we’re looking at. We can choose either side—the side the photons are coming from, or the side they’re going to—as long as we’re consistent.) A negative angle represents counterclockwise rotation. We can now write two special cases of the rule above:
•There’s no possibility of a coincidence when one polarizer is set to 30° and the other is set to 120°.
•There’s no possibility of a coincidence when one polarizer is set to −30° and the other is set to 60°.
We’ll return to these special cases in the next chapter. There, you will see for yourself the rigorous reasoning that forces physicists to reject local realism. You can decide for yourself whether the physicists’ arguments make sense. Remember, local realism is what most of us would call common sense: objects have real properties that exist regardless of whether anyone is measuring them, and the measurement of one object can’t affect a distant object. If this viewpoint is incorrect, even on the microscopic scale, then something very weird is happening in our universe. (Or, as Philip Ball recently noted, we’re weird for thinking that quantum physics is weird.6 “Weird” means uncommon, but there’s nothing more common than the particles that literally everything is made of—and these particles obey the laws of quantum physics.)
Local realism is what most of us would call common sense: objects have real properties that exist regardless of whether anyone is measuring them; the measurement of one object can’t affect a distant object.
4
Rigorous Contradiction of Everyday Assumptions
Bell’s original inequality is easy to articulate, as we saw in chapter 2. Freedman’s version, the first Bell inequality tested in the lab, is even easier to put into words. But establishing those Bell inequalities is complicated. In this chapter, we’ll look at simplified versions of Bell’s theorem that can be proven rigorously, with very little math. The examples I chose for this chapter reveal some of my favorite quantum contradictions of common sense. I think it’s instructive to study multiple examples because the reasoning is different in each case, but the conclusion is always the same. I find these examples more persuasive in the aggregate than in isolation.
Just Two Numbers, and then Two More: +1 and −1, +2 and −2.
Let’s imagine an experiment performed with pairs of entangled photons.1 Each photon travels toward an anal
yzer that determines whether the photon is polarized in a chosen direction. Let’s imagine that the analyzer is a simple polarizer, followed by a detector to indicate whether each photon has passed through the polarizer.2
If a photon is transmitted through the polarizer, let’s represent that result with the number +1. If it is not transmitted through the polarizer, we record −1. So to perform an experiment, we simply record +1 or −1 for each of the two photons.
We imagine two physicists, Alice and Bob, stationed at the two analyzers; each physicist takes responsibility for recording data at one of the analyzers. Each physicist dutifully records +1 or −1 for each photon arriving at the analyzer. Alice is interested in determining whether photons are polarized in the 0° direction (vertical). She’s equally interested in determining whether the photons are polarized in the 45° direction. She sets her analyzer to either of these two angles. If she sets the angle to 0°, she uses the symbol A to represent the measured outcome (which is either +1 or −1). If she sets the angle to 45°, she uses the symbol A′ to represent the measured outcome (again, either +1 or −1).
There’s no need to ruminate about why she chooses these two angles and not any others. The choice of these two angles is external to this discussion. We can think of the two angles simply as two settings of a switch.
Bob is also interested in two angles: 22.5° and 67.5°. When he sets his analyzer to 22.5°, he uses the symbol B to represent the measured outcome (+1 or −1). When he sets his analyzer to 67.5°, he uses the symbol B′ to represent the measured outcome (as always, either +1 or −1).
In effect, Alice is measuring A or A′, and Bob is measuring B or B′ (figure 10). The result of any measurement is either +1 or −1.
There are four possible combinations of measurements:
•Alice measures A and Bob measures B.
•Alice measures A and Bob measures B′.
•Alice measures A′ and Bob measures B.
•Alice measures A′ and Bob measures B′.
Figure 10 A simplified diagram of the measurement of a photon pair. Alice sets her analyzer to measure either A or A′, and Bob sets his analyzer to measure either B or B′. In this illustration, Alice is measuring A′, and Bob is measuring B′.
Next, we’ll define a simple quantity, S. Let’s say that S stands for Simple. S is calculated from A, A′, B, and B′:
S=AB+A′B–AB′+A′B′.
S does not have any obvious physical meaning. It’s just a quantity that we can calculate. We want to predict the possible values of S. For example, if A, A′, B, and B′ are all +1, we find that S is +2. If we change B to −1 while A, A′, and B′ are all still +1, S is −2. If we go through all possible combinations of values for A, A′, B, and B′, we find that S is always either +2 or −2.
Now, how can Alice and Bob measure S? For each photon pair, Alice measures either A or A′, not both, and Bob measures either B or B′, not both. So it’s impossible to measure S for a single photon pair. Alice and Bob decide to take many measurements for all possible combinations of angles. So, when Alice measures A while Bob measures B, they obtain the product AB, and they find the average of this product over many measurements. Similarly they find the average of A′B over many trials in which Alice measures A′ while Bob measures B. They also find the average of AB′, and the average of A′B′. They obtain the average of every term in S, so they can calculate the average of S itself:
(average of S) = (average of AB) + (average of A′B) – (average of AB′) + (average of A′B′).
Now we make a key assumption. Even though Alice measures either A or A′, the photon is preprogrammed with both values: the photon has properties that predetermine the result of any possible measurement on it. This is Einstein’s assumption of realism. Similarly, Bob’s photon is preprogrammed with both values, B and B′, even though Bob measures only one of them for each photon. So, the four values A, A′, B, and B′ exist for each photon pair: they are hidden variables. So, the quantity S=AB+A′B–AB′+A′B′ exists for each photon pair.
Since S can only be −2 or +2, it’s obvious that the average of S must be between −2 and +2. But experiment shows that the average of S is greater than 2! Where’s the faulty reasoning that led to the false constraint on S?
Our mistake is the assumption that A, A′, B, and B′ all exist at the same time. Alice can only measure either A or A′; Bob can only measure either B or B′. The quantities that aren’t measured don’t have specific values that we can plug into S=AB+A′B–AB′+A′B′. So S doesn’t exist for a single photon pair. Our belief that S should exist for a single photon pair is really our belief that photon properties exist before we measure them. Experiment contradicts this belief: the measured value of average S exceeds the limits imposed by realism.
All of the mathematical statements are really just abbreviations for statements about physical reality. Let’s clarify this point. If we make just a single measurement of a photon, what does it mean to determine that A, for example, is +1? Does it mean that the number +1 was somehow imprinted on the photon, or riding along with the photon? Let’s consider a more familiar kind of number: my weight, which is 150 pounds. Is the number 150 somehow imprinted on my body? But, isn’t the pound an arbitrary unit? If I step on a metric scale, I see that I’m 68 kilograms. So which number am I carrying around with me? Is it 150, or 68, or something else? In fact, “150 pounds” and “68 kilograms” both represent a more fundamental fact: there are 40,000 trillions of trillions of protons and neutrons in my body. So “150 pounds” and “68 kilograms” are convenient, shorthand ways of specifying the total number of protons and neutrons in my body. (Electrons have mass too, but they’re much lighter than protons and neutrons. All the electrons in my body weigh just about an ounce.) In a sense, the numbers 150 and 68 are indeed attached to me, because my weight is a very real quantity that can be measured any time. The numbers 150 and 68 represent factual statements about how much mass is in my body, regardless of whether I’m standing on a scale.
Similarly, the equation A = +1 is really just a code, or abbreviation, for a factual statement: this photon gets transmitted through Alice’s polarizer if she sets it to 0°. Rather than saying, “This photon has a property that will cause it to pass through Alice’s polarizer if she sets it to 0°,” we just say “A = +1.”
Experiment contradicts the assumption that every photon pair has values of A, A′, B, and B′. This exact same (and false) assumption can be written in a longer form: “The photon traveling toward Alice has properties that predetermine whether it will be transmitted or blocked by Alice’s polarizer, regardless of whether she sets it to 0° or 45°. The photon traveling toward Bob has properties that predetermine whether it will be transmitted or blocked by Bob’s polarizer, regardless of whether he sets it to 22.5° or 67.5°.”
Did we assume locality, as well as realism, when we claimed that S must be +2 or −2? Yes, in a subtle way. We implicitly assumed that A, for example, is a property that doesn’t depend on Bob’s polarizer. In the definition of S—S=AB+A′B–AB′+A′B′—we assumed that A has the same value whether Bob measures B or B′.
Local realism imposes a constraint on a measurable quantity: the average value of S. Experiment violates this constraint, so at least one of our assumptions was wrong. If realism was the bad assumption, then the measurements do not reveal properties that the photons had all along; the photons were in some kind of undecided state prior to measurement. If locality was the bad assumption, then the angle of Bob’s polarizer affects Alice’s photons, and the angle of Alice’s polarizer affects Bob’s photons. Both of these possibilities are strange, and we’ll see even stranger alternatives in chapter 6.
Perhaps you’re curious about the single minus sign in S=AB+A′B–AB′+A′B′. When Alice measures A, her polarizer is set to 0°, and when Bob measures B′, his polarizer is set to 67.5°. The angle between the two polarizers is 67.5°. In all other cases, the angle between the polar
izers is 22.5°:
•AB: 0° on Alice’s side and 22.5° on Bob’s side.
•A′B: 45° on Alice’s side and 22.5° on Bob’s side.
•A′B′: 45° on Alice’s side and 67.5° on Bob’s side.
The minus sign in S is associated with the one combination of angles that produces a 67.5° difference. The quantum prediction for the average of S is about 2.8, which violates the constraint imposed by local realism (−2 ≤ average S ≤ +2). Experimental imperfections reduce the average of S below the ideal quantum prediction. The highest average S that I’ve measured is 2.66.
Do the Lights Match the Buttons?
Here’s a variation on the same experiment, described in a book by Nicolas Gisin.3 Alice and Bob install green and red lights on their analyzers. The green light flashes if the photon is polarized in the chosen direction; otherwise, the red light flashes. So, the green light corresponds with a +1 result in the previous example, and the red light corresponds with a −1 result.
Alice and Bob next install buttons to set the angles of the analyzers. Alice’s buttons are labeled A and A′, and Bob’s buttons are labeled B and B′. Each experimenter presses one button before the measurement of a photon pair. So, when Alice pushes the A button, she’s measuring A, and when she pushes the A′, she’s measuring A′. When Bob pushes the B button, he’s measuring B, and when he pushes the B′ button, he’s measuring B′. They decide to flip coins to decide which button to push so neither of them is influenced by the other person’s choice. So, the four combinations of button presses are equally likely: A and B, A′ and B, A and B′, and finally A′ and B′.
