by Jed Brody
That argument goes by fast, so let’s decompress it. We’ll now approach the situation as if it were a logic puzzle in which you have to figure out where to seat people at a table, given that Mr. A insists on sitting next to Ms. B but at least two seats away from Dr. C, and so on. In our case, each photon has one of the four sets of hidden properties, and the other photon also has one of the four sets. Thus, there are sixteen different combinations. I’ll represent this in the table below, where the rows represent the hidden properties of one photon (which I’ll call the row photon), and the columns represent the hidden properties of the other photon (which I’ll call the column photon). G and R stand for green and red.
1→G. 2→G. 1→G. 2→R. 1→R. 2→G. 1→R. 2→R.
1→G. 2→G.
1→G. 2→R.
1→R. 2→G.
1→R. 2→R.
We can eliminate some of these sixteen possible combinations by studying the three observations. Observation A is that if both switches are set to 1, at least one light flashes green. So, we can eliminate the cases in which both switches are set to 1 and both lights flash red. I’ll mark these forbidden cases in the table with an A, indicating that Observation A forced us to eliminate them:
1→G. 2→G. 1→G. 2→R. 1→R. 2→G. 1→R. 2→R.
1→G. 2→G.
1→G. 2→R.
1→R. 2→G.
A
A
1→R. 2→R.
A
A
Just to be clear why we did this, let’s look at the A in the bottom right corner (last column, last row), where both photons have the hidden property (1→R. 2→R.) If both switches are set to 1, both lights flash red. This violates Observation A. The other three cases where A appears in the table occur where both photons have the property 1→R.
Now let’s look at Observation B: If the switches have different settings, at least one light flashes red. Now we can eliminate all the cases where both lights flash green when the switch settings are different. I’ll mark these cases with a B:
1→G. 2→G. 1→G. 2→R. 1→R. 2→G. 1→R. 2→R.
1→G. 2→G.
B
B
B
1→G. 2→R.
B
B
1→R. 2→G.
B
B
A
A
1→R. 2→R.
A
A
Let’s clarify what we just did. Start with the B in the upper left. In this case, both photons have the hidden property (1→G. 2→G.). No matter what, both lights flash green. They flash green when the switch settings are the same; more significantly, they also flash green when the switch settings are different, in violation of Observation B. Next let’s move one space to the right in the table, where the row photon still has the property (1→G. 2→G.), and the column photon has the property (1→G. 2→R.). The row photon always causes its analyzer to flash green. If the column photon’s analyzer is set to 2, then there’s a red flash on the column photon’s side. This isn’t a violation of Observation B. But if the row photon’s analyzer is set to 2, and the column photon’s analyzer is set to 1, both lights flash green. This does violate Observation B, so we have to forbid this combination of hidden properties. Similar reasoning leads to the other appearances of B in the table: any time there is a possibility of two green flashes occurring when the switches are different, we have to forbid it.
Using Observations A and B, we’ve eliminated eleven combinations of hidden properties, but five still remain. Can any of these five combinations satisfy Observation C? Observation C requires some photon pairs to cause two green flashes when both switches are set to 2. In other words, in some pairs, both photons must have the 2→G property. There are four combinations in the table such that both photons have the 2→G property. I’ll mark them with an asterisk (*):
1→G. 2→G. 1→G. 2→R. 1→R. 2→G. 1→R. 2→R.
1→G. 2→G.
B*
B
B*
1→G. 2→R.
B
B
1→R. 2→G.
B*
B
A*
A
1→R. 2→R.
A
A
All four combinations that could satisfy Observation C are strictly forbidden, either by Observation A or Observation B. Thus, local realism once again fails to explain our observations. And once again, we can ask, what alternative explanation can we give? We can attribute the results to spooky action at a distance: the two photons are in some kind of communication with each other, such that the measurement of one photon influences the measurement of the other. Or, we can assert that observation is the only scientific reality; we recognize that quantum predictions are invariably accurate, but we refuse to speculate about underlying causality. In other words, we acknowledge that local realism doesn’t work, but we don’t suggest any alternatives.
I survey additional interpretations of quantum physics in chapter 6.
Three Entangled Photons
Now we’ll look at a case of three entangled photons: a trio discovered, appropriately, by a trio of physicists: Daniel Greenberger, Michael Horne, and Anton Zeilinger.10
Each of the three photons enters an analyzer similar to the analyzers of the previous example: each analyzer has a switch that can be set to Setting 1 or Setting 2, and every time a photon arrives, two outcomes are possible. Instead of flashing lights, we now have a digital display that shows either +1 or −1 (figure 13).
After the three photons enter the three analyzers, we record whether +1 or −1 is shown on each of the three displays. We repeat the experiment many times for various combinations of switch settings on the three analyzers. For simplicity, we’ll restrict our attention to cases in which an odd number of switches is set to Setting 1. In other words, either all switches are set to Setting 1, or one switch is set to Setting 1, and the other two are set to Setting 2.
Figure 13 Three particles arise from a common source and separate, traveling to three analyzers. Each analyzer has a switch with two settings and a digital display that shows +1 or −1. This figure illustrates a possible combination of switch settings and digital results.
We observe the following two facts:
Fact 1. When one switch is set to Setting 1 and the other two are set to Setting 2, −1 is shown either on one display or all three displays. (So +1 is shown on either two displays or no displays.) Since an odd number of displays shows −1, the product of the three displayed numbers must be −1: if −1 is on one display, the product of the three numbers is −1 × 1 × 1 = −1; if −1 is on all three displays, the product is −1 × (−1) × (−1) = −1.
Fact 2. When all three switches are set to Setting 1, −1 is shown either on two displays or no displays. Since an even number of displays shows −1, the product of the three displayed numbers must be +1: if −1 is on two displays, the product of the three numbers is −1 × (−1) × 1 = 1; if −1 is on no displays, the product is 1 × 1 × 1 = 1.
These observations are consistent with the quantum prediction, which we don’t need to get into. Our purpose is to show that the observations are incompatible with the assumption of local realism.
As we’ve seen in previous examples, local realism implies that each photon carries within it hidden properties. The outcome of a measurement is determined by the photon’s hidden properties and the switch setting at the analyzer that it enters; the photon isn’t influenced by the other photons or the switch settings at their analyzers. Thus, each photon must have one of these four hidden properties:
Property I:
(Setting 1→+1. Setting 2→+1.)
Property II:
(Setting 1→+1. Setting 2→−1.)
Property III:
(Setting 1→−1. Setting 2→+1.)
Property IV:
(Setting 1→−1. Setting 2→−1.)
This is a complete list of t
he hidden properties a photon might have; there are two possible outcomes for each of the two possible switch settings.
It will be convenient to use symbols to represent the numbers displayed on the analyzers. Let the letters A, B, and C represent the three photons. Let’s use a subscript to represent the setting of the switch. So, if Photon A’s analyzer is set to Setting 1, A1 represents the displayed number. If Photon A’s analyzer is set to Setting 2, A2 represents the displayed number. We can think of A1 and A2 as the hidden properties themselves, which now really are hidden variables. Thus, the four possible hidden properties of Photon A are the four possible combinations of values of A1 and A2:
Property I:
(A1 = +1. A2 = +1.)
Property II:
(A1 = +1. A2 = −1.)
Property III:
(A1 = −1. A2 = +1.)
Property IV:
(A1 = −1. A2 = −1.)
B1 and B2 are defined similarly for Photon B, and C1 and C2 are defined similarly for Photon C. We have two hidden variables for each photon, and there are three photons, so we have a total of six hidden variables. Each hidden variable has two possible values, so there’s a total of 2 × 2 × 2 × 2 × 2 × 2 = 64 combinations of values. Do we have to explicitly consider all sixty-four combinations?
Luckily, we can quickly show that none of the sixty-four combinations is consistent with the two facts. According to Fact 1, whenever one switch is set to Setting 1 and the other two are set to Setting 2, the product of the three displayed numbers is −1. We can look at the three possible cases, remembering that the subscripts indicate the switch settings:
Fact 1 (three cases):
•Photon A’s analyzer is set to Setting 1, and the other two are set to Setting 2. This gives us A1B2C2 = −1.
•Photon B’s analyzer is set to Setting 1, and the other two are set to Setting 2. This gives us A2B1C2 = −1.
•Photon C’s analyzer is set to Setting 1, and the other two are set to Setting 2. This gives us A2B2C1 = −1.
Let’s see if the three equations above can tell us what A1B1C1 is. We’ll use the fact that 12 equals 1, and (−1)2 also equals 1. Since each hidden variable (A1, A2, B1, B2, C1, and C2) is either 1 or −1, the square of any hidden variable is always 1. Thus we can write, for example, A2 × A2 = A22 = 1, B2 × B2 = B22 = 1, and C2 × C2 = C22 = 1.
We’ll just multiply A1B1C1 by 1 three times:
A1B1C1 = A1B1C1 × 1 × 1 × 1.
We’ll replace one 1 using 1 = A2 × A2, we’ll replace another 1 using 1 = B2 × B2, and we’ll replace the third 1 using 1 = C2 × C2:
A1B1C1 = A1B1C1 × A2 × A2 × B2 × B2 × C2 × C2.
Next, we’ll just reorder the variables on the right:
A1B1C1 = A1B2C2 × A2B1C2 × A2B2C1.
The three terms on the right side are precisely the three expressions that came out of Fact 1. As we saw above, each of those three terms on the right is −1:
A1B1C1 = (−1) × (−1) × (−1) = −1.
We thus conclude that A1B1C1 = −1: when the three switches are all set to Setting 1, the product of the three displayed numbers is −1. But this contradicts Fact 2, which states that the product of the displayed numbers must be +1 when the three switches are set to Setting 1!
Facts 1 and 2 arise from quantum theory and are confirmed by measurement. But local realism, once again, fails to accommodate experimental facts. In fact, local realism predicts the exact opposite of Fact 2 (when Fact 1 is a given). Yet again, we are forced to discard the assumption that each photon, independently of the other photons and their analyzers, is predestined to behave in a particular way when encountering a measuring device. Each photon behaves as if it’s monitoring the switch settings at all three analyzers, and colluding with the other two photons to satisfy the observed facts.
Each photon behaves as if it’s monitoring the switch settings at all three analyzers, and colluding with the other two photons to satisfy the observed facts.
5
Reconciling with Relativity
The two pillars of modern physics are relativity and quantum physics. These two fields of discovery unsettle not only our common sense but also seem even to unsettle each other. Let’s examine Einstein’s relativity to identify, and resolve, an apparent conflict with quantum entanglement.
The Shocking Truths of Einstein’s Relativity
Physicists before Einstein recognized that light is an electromagnetic wave. In fact, the speed of light is hidden like a buried treasure within the laws of electricity and magnetism. These tried and true laws predict a very specific speed for light in empty space: 670 million miles per hour.
The two pillars of modern physics are relativity and quantum physics. These two fields of discovery unsettle not only our common sense but also seem even to unsettle each other.
Einstein thought very carefully about this fact. He recognized that the laws of physics must be the same for everyone, no matter where they are or how fast they’re going. Specifically, the electromagnetic laws, which predict the speed of light, must be the same for everyone, no matter where they are or how fast they’re going. This means that everyone, no matter how fast they’re going, must measure the same speed for light in empty space. This single insight, apparently innocent, revolutionizes our understanding of space and time.
Imagine a baseball pitcher who can throw a ball at 80 miles per hour. Now the pitcher throws a ball at 80 miles per hour from a cart moving 40 miles per hour (figure 14). The ball is moving 80 miles per hour relative to the pitcher and the cart, so the ball is moving 120 miles per hour relative to the ground. This is common sense.
Figure 14 If a baseball pitcher rides a cart going 40 miles per hour and throws a baseball at 80 miles per hour relative to the cart, the baseball’s speed is 120 miles per hour relative to the ground.
Now replace the cart with a spaceship going 335 million miles per hour, relative to Earth. Replace the baseball with light from the spaceship’s headlight (figure 15). The light is traveling 670 million miles per hour relative to the spaceship. Is the light going 1,035 million miles per hour relative to Earth? Einstein says no! The light is moving at exactly the same speed relative to both the spaceship and Earth, even though the spaceship is moving at half the speed of light, and in the same direction as the light, relative to Earth.
To understand how light travels at the same speed relative to all observers, we have to rethink time and space. In fact, although all observers must agree on the speed of light in empty space, they disagree about other basic measurements of time and space. If one observer is moving at high speeds relative to another observer, the observers will disagree over the results from three types of basic observations: the time interval between two events, the lengths of objects, and even the order in which certain events occur.1 Both observers are correct! Neither one is under some sort of delusion. The speed of light is rigid, so time and space must be fluid.
Figure 15 The spaceship travels at 335 million miles relative to Earth. The light from the spaceship’s headlights travels at 670 million miles relative to the spaceship, and at exactly the same speed relative to Earth. Source: the image of Earth comes from publicdomainvectors.org
One of Einstein’s astonishing truths is that the length of a car, for example, depends on who’s measuring it. Specifically, the length of the car depends on how fast it’s moving relative to the person who’s measuring it. So if we ask, “What’s the length of that car?” the question is ambiguous and incomplete, much like the question, “What’s the distance to Atlanta?” We might want to know the distance to Atlanta from wherever we are when we ask the question ... or we might want to know the distance to Atlanta from Alpha Centauri. Analogously, we might want to know the length of the car when we’re sitting in the car ... or we might want to know the length of the car when it zooms by at half the speed of light. The car’s length is not a property of the car alone; the car’s length depends also on the relative speed of the observer.
These relativistic effects are significant only at relativistic speeds, meaning speeds close to the speed of light. In daily life, the speed of one person relative to any other person is much less than the speed of light. Therefore, in daily life, we all agree on time intervals, lengths, and the order in which events occur. So relativity, like quantum physics, is part of the mysterious structure of the universe revealed only when technology enhances our awareness beyond everyday perception. Let’s see how Einstein’s mysterious truths arise from the single innocent fact that everyone agrees on: the speed of light in empty space.
The car’s length is not a property of the car alone; the car’s length depends also on the relative speed of the observer.
Time Dilation
There’s a standard way to prove that observers disagree about the time interval between two events. Imagine a room with a laser on the floor, aimed at a mirror on the ceiling. Consider the time interval between these two events: the emission of a brief pulse of light from the laser, and the arrival of the light back at its source after reflecting off the mirrored ceiling.