by Jed Brody
In subsequent chapters, we’ll discuss experimental methods for creating pairs of entangled particles. In the laboratory, it’s easier to do experiments with entangled photons than with entangled electrons. For now, however, we’ll again set practical concerns aside.
Figure 2 A pair of entangled electrons with opposite magnetic properties: if the magnets are aligned, one electron must be deflected to the north pole, and one must be deflected to the south pole.
Imagine observing many pairs of entangled electrons passing through the magnetic fields. Half of the time, the electron on the left is deflected north and the electron on the right is deflected south. The other half of the time, the opposite occurs: the electron on the left is deflected south and the electron on the right is deflected north.
Here’s the key question: Does each electron have the magnetic property that is eventually observed (predisposition to northward or southward deflection) all along, or do the electrons “make up their minds” when they reach the magnets? In other words, does the measurement show us what the electrons were like all along, or does the measurement fundamentally change something? This may seem like a silly question, or at least one that cannot be empirically tested. What difference does it make whether the electrons were in the measured state all along, and what’s the big deal if they’re in an undecided state until the last minute? Indeed, I’ve been in an undecided state about what to order in a restaurant, and it’s only the “measurement” taken by the server that forces me to make up my mind.
This is the big deal: If the electrons make up their minds at the last minute, they must make opposite decisions. If one chooses northward deflection, the other must choose southward. How can they coordinate this, in defiance of locality, when they’re in different places? If the electrons make up their minds at the last minute and they always make opposite decisions, they’re like twins with a telepathic link, if you’ll forgive the analogy. This is “spooky action at a distance”—and Einstein argued strenuously against its existence.
To preserve locality (and avoid spooky action at a distance), we’d better hope for realism, which asserts that the electrons all along have the properties we end up measuring. In 1935, Einstein argued that quantum mechanics is an incomplete theory,2 like the probabilistic theories about quarters we considered in chapter 1. His argument is that the electrons are deflected in opposite directions because they were created with opposite magnetic properties. Quantum mechanics can’t tell us in advance which way each electron will be deflected; it can only tell us that each electron has a 50 percent chance of being deflected northward, and the other electron must be deflected southward (if the magnets are aligned).
To preserve locality (and avoid spooky action at a distance), we’d better hope for realism, which asserts that the electrons all along have the properties we end up measuring.
A complete theory should predict the magnetic property of each electron at the outset, if we believe that each electron has a definite magnetic property at the outset. We don’t know the details of this hypothetical, complete theory, and we don’t even know what factors predetermine the properties of the electrons. Since the unknown factors are hidden variables, the hypothetical, complete theory is a hidden variables theory. More specifically, we are interested in a local hidden variables theory: a theory that predicts the outcome of the measurement of a single electron, without any dependence on the other electron in the pair. A local hidden variables theory is thus an expression of the assumption of local realism.
For decades, physicists assumed that a local hidden variables theory could, in principle, complement quantum physics, filling in missing information and replacing probabilities with certainties. But the issue seemed academic or philosophical, and not subject to experiment: a local hidden variables theory determines the state of an electron before you measure it. Is it possible to measure the state an electron is in, before it’s measured? Seemingly it is not.
In 1964, John Bell made a stunning theoretical discovery, called Bell’s theorem.3 His original paper languished in obscurity for years, but enthusiasm for his discovery swelled over the course of decades. Bell showed that any local hidden variables theory imposes a constraint on measurable quantities. The constraint on measurable quantities is now called a Bell inequality. If the constraint is violated by measurement, then a local hidden variables theory cannot be valid. Moreover, because quantum physics predicts violations of the Bell inequality, quantum physics is fundamentally incompatible with any local hidden variables theory. So Einstein’s hope was in vain: a local hidden variables theory cannot complement quantum mechanics; it can only contradict it. And since measurable quantities determine whether a Bell inequality is violated or not, an experiment can be performed to determine whether the real world is consistent with quantum mechanics, or with a local hidden variables theory; we can’t have both. To reiterate these key points:
•A Bell inequality is a constraint on a measurable quantity. An experiment can be done to test a Bell inequality. The experiment will either satisfy the Bell inequality or violate the Bell inequality.
•If the experiment satisfies the Bell inequality, the experiment conforms to local realism but contradicts quantum physics.
•If the experiment violates the Bell inequality, the experiment may conform to quantum physics but must contradict local realism.
Experimental results indeed violate Bell inequalities, thereby confirming quantum mechanics and overruling any possible local hidden variables theory. What does this mean? Either locality fails, or realism fails, or both fail: either one electron is influenced by the distant electron or the distant magnet, or the act of measurement creates a definite magnetic property that the electrons did not previously have, or both of these strange phenomena occur. (This is not a comprehensive list of interpretations of quantum mechanics, but it’s representative of possible consequences of rejecting local realism.)
Bell’s original theorem is too mathematical to prove in this book. Luckily, physicists have developed simplified versions of Bell’s theorem. We will see some of these in chapter 4, where we will prove that measured data contradict the philosophical assumption of local realism.
For now, we will investigate the fact that Bell established, without proving why it’s true. Bell asks us to imagine rotating the magnets encountered by the electron pairs. So, for example, we could rotate one magnet 180° relative to the other. If we do this, we find that the two electrons are always deflected toward the same pole: both to the north, or both to the south (figure 3).
Now we can imagine rotating the magnets to any angle, not just 180°. Let’s say a magnet is set to angle 0° if the south pole is directly above the north pole (like the magnets in figure 2). So the magnet is set to angle 180° if the north pole is directly above the south pole. We can do experiments in which either magnet is set to any angle.
Either locality fails, or realism fails, or both fail.
Figure 3 If one magnet is rotated 180° relative the other, the two electrons are always deflected toward the same pole: both north (as shown), or both south.
Bell asks us to think about just two numbers: +1 and −1. If the two electrons are deflected toward the same pole (both north or both south), we write down +1. If they’re deflected toward opposite poles (one north and one south), we write down −1. So, if the two magnets are aligned, the electrons are always deflected toward opposite poles (figure 2), and we always write −1. If one magnet is flipped 180° relative to the other, the two electrons are always deflected toward the same pole (figure 3), and we always write +1.
Bell asks us to think about three different magnet angles. Let’s choose 0°, 45°, and 90°. Then he asks us to do the following:
•Set one magnet to 0° and one to 90°. Watch a bunch of electron pairs as they pass through the magnets. For each pair, write down +1 or −1 to indicate electron pairs deflected toward the same poles or opposite poles, respectively. Then, average all these numbers. Call thi
s average number A.4 (Its value will be in the range of −1 to +1.)
•Set one magnet to 45° and one to 90°. Again watch electron pairs as they pass through the magnets, and write down +1 or −1 for each pair. Average all these numbers. Call this average number B.
•Set one magnet to 0° and one to 45°. Once again watch electron pairs as they pass through the magnets, and write down +1 or −1 for each pair. Average all these numbers. Call this average number C.
Now we have three numbers, A, B, and C, based on experimental measurements. Bell proved that the assumption of local realism requires
−1 − C ≤ A − B ≤ 1 + C.
That’s it! That’s the result of Bell’s theorem, the original Bell inequality. So, we now have a method to test a philosophical assumption. We simply perform measurements and calculate the three average numbers A, B, and C, as instructed above. Then we put these numbers in the Bell inequality. If the result is true, then we’ve satisfied the Bell inequality, and the data are consistent with the assumption of local realism. But if our data violate the Bell inequality, then we’ve contradicted local realism.
Let’s recall that local realism is an everyday assumption: observation merely reveals properties that an object already had, and the properties of an object are unaffected by the measurement of a distant object. This is the assumption that leads inexorably to the Bell inequality. The Bell inequality is, in fact, violated by measurement, so the assumption of local realism cannot be valid. We’ll explore the mysterious implications later in the book.
Next, we will look at the entanglement of light. Entangled light was used in the first experimental tests of Bell inequalities.
3
Entangled Light
A crowd of unruly students seeks admission to a prestigious yet absurdly strict boarding school. This school rigorously upholds a tradition whose purpose is long forgotten: all students must carry a baton at all times, and they must hold it precisely upright. Each eager candidate in the crowd dutifully holds a baton, but these students, hopeful for admission, as yet lack discipline: only a few hold the batons upright. The scowling headmaster admits only half the students. The closer each student’s baton is to vertical, the more likely the student is to get in. The batons of admitted students are nudged to the proper, vertical orientation, which the students assiduously maintain for the duration of their studies at the boarding school.
The time arrives for the students to apply to college. They seek admission to prestigious yet absurdly strict colleges that also require students to carry batons at all times, the tradition being as inviolable as it is arbitrary and meaningless. Unhappily, not all colleges require the batons to be carried vertically. Some require the batons to be held at an angle of 30° from the vertical, others require 45°, still others require 60°, and some even require horizontal batons. The students have trouble convincing the colleges that after long years of meticulously holding vertical batons at their boarding school they will be able to adjust to new angles. In fact, the likelihood of college admission declines dramatically as the required angular difference increases:
•A student from this boarding school is 100 percent likely to get into a college that requires batons to be held vertically.
•A student from this boarding school is 75 percent likely to get into a college that requires batons to be held at an angle of 30° from the vertical.
•A student from this boarding school is 50 percent likely to get into a college that requires batons to be held at an angle of 45° from the vertical.
•A student from this boarding school is only 25 percent likely to get into a college that requires batons to be held at an angle of 60° from the vertical.
•A student from this boarding school has no chance of getting into a college that requires batons to be held horizontally.
The batons are analogous to the electric field in a light wave. If all light in a light beam has the same electric field direction, the light is said to be polarized. If the electric field in the light wave always points in a vertical direction, that indicates vertical polarization. Most light sources emit unpolarized light, however, in which the electric field can point in any direction perpendicular to the light ray.
Now let’s think about the particles of light called photons. In unpolarized light, the photons, like the unruly applicants to the boarding school, are polarized in all different directions. A polarizer, like the scowling headmaster, transmits only the photons able to comply with the required angle (figure 4); the remaining photons are absorbed or reflected. Polarizers can be made of a variety of materials, including sheets of plastic. An ideal polarizer transmits exactly half the photons in unpolarized light. All the transmitted photons are polarized identically, in the direction enforced by the polarizer.
Consider only the photons transmitted through a polarizer. If these polarized photons encounter a second polarizer, the transmission through the second polarizer is exactly analogous to the college acceptance of the baton-toting students:
Figure 4 (a): The electric field in unpolarized light can point in any direction perpendicular to the direction that the light is traveling. So here, imagine light is traveling into the diagram. (b): Any light passing through a vertical polarizer is (c) vertically polarized. (d): The fraction of vertically polarized light that passes through a second polarizer depends on the angle between the transmission directions of the two polarizers.
•A photon has a 100 percent probability of passing through a polarizer whose transmission direction is the same as the photon’s polarization.
•A photon has a 75 percent probability of passing through a polarizer whose transmission direction is 30° different from the photon’s polarization.
•A photon has a 50 percent probability of passing through a polarizer whose transmission direction is 45° different from the photon’s polarization.
•A photon has a 25 percent probability of passing through a polarizer whose transmission direction is 60° different from the photon’s polarization.
•A photon has a 0 percent probability of passing through a polarizer whose transmission direction is 90° different from (perpendicular to) the photon’s polarization.
Quantum theory is consistent with these empirical facts. So if we like, we can think of these facts as quantum predictions.
Now that we understand polarization, we can understand measurements of entangled photons. To create pairs of entangled photons, we make use of a process in which a single photon splits into a pair of photons.1 This splitting occurs when light travels through certain materials, such as crystals of beta barium borate. For example, a violet photon might split into two identical infrared photons, which travel away in different directions.2 Only a small fraction of incoming violet photons split; most of them pass straight through.
We’re going to be interested in the polarization of the pairs of infrared photons. Suppose we’re shining a beam of polarized violet photons at a beta barium borate crystal. Let’s polarize the violet photons 45° from the vertical. We observe that for a certain angle of the beta barium borate crystal, the infrared photons produced are vertically polarized (figure 5). If we rotate the crystal 90°, the infrared photons are horizontally polarized (figure 6).
Figure 5 Some violet photons split in a beta barium borate crystal, producing a pair of infrared photons. For a particular orientation of the crystal, the infrared photons are both vertically polarized.
Figure 6 At a different angle of the crystal, the infrared photons produced are horizontally polarized.
Now we get two beta barium borate crystals and orient the first one to produce vertically polarized infrared pairs, and the second to produce horizontally polarized infrared pairs (figure 7). Each incoming violet photon has a possibility of splitting in the first crystal (producing a vertically polarized infrared pair), and an equal possibility of splitting in the second crystal (and producing a horizontally polarized infrared pair). Thus, the infrared pair is either v
ertically polarized or horizontally polarized.
How can we confirm this? We can put single-photon detectors in the path of the infrared photons. When a photon reaches a detector, the detector produces a brief electronic pulse. If both detectors pulse at the same time, we call it a coincidence (which is the technical term for simultaneous detection at both detectors). It’s very likely that a coincidence indicates a pair of infrared photons that split from the same violet photon.
Figure 7 The violet photon may split in one of the two crystals, so the infrared photons produced may both be vertically polarized, or they may both be horizontally polarized.
Next, we put a polarizer in front of each detector (figure 8). If we make one polarizer horizontal and one vertical, we never see any coincidences. (This is an idealization, but let’s keep things simple.) This confirms that no infrared pair consists of one vertically polarized photon and one horizontally polarized photon.
If we make both polarizers vertical, we see lots of coincidences, indicating vertically polarized pairs. If we make both polarizers horizontal, we see lots of coincidences, indicating horizontally polarized pairs. Let’s say we see about 100 coincidences per second when both polarizers are vertical, and about 100 coincidences per second when both polarizers are horizontal. When we remove both polarizers, we see about 200 coincidences per second: both the vertically polarized pairs and the horizontally polarized pairs.