Quantum Entanglement
Page 6
N(30°,−30°) is the same as N(0°,60°) because the angular difference between the two polarizers is 60° in both cases. The probability that a photon pair passes through both polarizers depends only on the angular difference between the two polarizers. Thus, the coincidence count also depends only on the difference between the two polarizers, and N(30°,−30°) = N(0°,60°) = .
We now have explicit expressions for all three terms in our Bell inequality:
N(30°,0°) ≤ N(0°,60°) + N(30°, −30°),
which becomes
This simplifies to
which is clearly untrue. Experiment shows that the coincidence count on the left side is indeed 50 percent larger than the sum on the other side, confirming quantum mechanics and overruling local realism.
Everyday observations, in our familiar macroscopic world, do conform to local realism. If we actually performed the observations of pairs of twins, the Bell inequality for twins would be satisfied, not violated. We can imagine how the twins would have to behave to conform to the quantum predictions. We have to imagine that prior to observation, the twins’ height, hair color, and eye color are undetermined and somehow undecided, not merely unknown. Before either twin is observed, there’s a 50 percent probability that a twin will be tall once height is observed; likewise, there is a 50 percent probability that a twin will be short. Similarly, brown hair and blond hair are equally likely, and brown eyes and blue eyes are equally likely.
Suppose the twins’ names are Jordan and Pat. If Jordan is observed to be tall, then Pat, for all practical purposes, immediately becomes tall. Even more strangely, once Pat’s hair or eye color is observed, Pat’s height reverts to being unknown, and there’s a chance that a subsequent observation will reveal Pat’s height to be short (no longer tall like Jordan)!
This last effect is indeed true for entangled photons: as soon as the polarization of one photon is observed, the polarization of the other is known to be the same.6 But no further measurement of either photon affects the other. The entanglement is severed the instant it takes effect, exactly when a measurement of one photon creates a definite polarization for both. This provides a possible explanation for why we don’t observe entanglement on a macroscopic scale: the very many particles making up macroscopic objects are constantly interacting and in some sense “measuring” one another, thereby disentangling as fast they entangle.7
The everyday assumption of local realism is contradicted by observations of entangled photons. Moreover, if macroscopic objects started behaving like entangled photons, we would be assailed by a host of bizarre effects. We’ll look at other constraints imposed by local realism—constraints that are violated by measured results.
Three Observations by Maudlin
This next example is from Quantum Non-Locality and Relativity by the philosopher Tim Maudlin.8 We’ll be thinking about photons all along, and making the assumption of realism more explicitly than in the previous example. The assumption of locality will be implicit, at first.
We’ll continue to work with the pairs of entangled photons described previously (each pair is 50 percent likely to pass through horizontal polarizers, and 50 percent likely to pass through vertical polarizers). We now specify that each photon will encounter a polarizer set to one of three angles: vertical, 30° from the vertical (in the clockwise direction), and 60° from the vertical (in the clockwise direction). As we might predict from our previous discussion, the following three observations can be made:
Observation 1. If the two polarizers have the same orientation, the two photons always do the same thing: either both photons pass through the polarizers, or both are blocked by the polarizers.
Observation 2. If the angle between the two polarizers is 30°, the two photons do the same thing 75 percent of the time; 25 percent of the time, one passes through its polarizer, while the other is blocked.
Observation 3. If one of the polarizers is vertical and the other is at 60°, the two photons do the same thing 25 percent of the time.
This is based on real observations; this isn’t made up.
Let’s think about Observation 1. If the two polarizers are in the same direction, the two photons always do the same thing. How do they accomplish this? It must be, according to common sense, that the two photons share a common property. It seems intuitively obvious that the photons have this common property all along—from the moment they’re created. Our common sense and intuition are based on realism: the photons have “hidden properties” prior to observation; observation merely lets us view the properties that the photons had all along. Let’s see where this assumption leads.
The hidden properties of a photon might be this:
(Would pass through a vertical polarizer. Would be blocked by a 30° polarizer. Would pass through a 60° polarizer.)
It took a lot of words to write that. Let me represent exactly the same thing in abbreviated form:
(Vertical→Pass. 30°→Block. 60°→Pass.)
If one photon in a pair has these hidden properties, the other one must have the same properties. To confirm this, suppose one photon in a pair has the hidden properties listed above, and the other one has these properties:
(Vertical→Block. 30°→Block. 60°→Pass.)
This says that if both polarizers are vertical, the two photons do different things: one passes through the polarizer, and one is blocked. This contradicts Observation 1, which requires the two photons to do the same thing if the polarizers are set to the same angle (in this case, vertical). So the two photons in a pair must share the same set of hidden properties.
These hidden properties might be thought of as instruction sets, telling the photons what to do when they reach polarizers. Or, the hidden properties may be thought of as features that are detected by polarizers at the appropriate angles. I like to think of the hidden properties as tickets, and the polarizers are bouncers, who admit only the bearers of appropriately marked tickets. In all cases, the hidden properties are qualities inherent in the photons all along, prior to measurement. This is realism.
We actually already made the assumption of locality as well: we assumed that what a photon does at a polarizer depends only on the angle of that polarizer. We assumed that a photon’s ability to pass through a polarizer does not depend on the angle of the polarizer that the other photon encounters. This assumption seems so natural that we have to go out of our way to recognize that we’re making it. Now, we’ll see where the assumptions of realism and locality lead us.
Let’s consider again the first example of hidden properties:
(Vertical→Pass. 30°→Block. 60°→Pass.)
If all photons had exactly these properties, the two photons in a pair would always do the same thing (pass) whenever one polarizer was vertical, and the other was 60°. But this should happen only 25 percent of the time, according to Observation 3. So a certain fraction of photons might have the hidden properties listed above, but other photons must have different hidden properties.
Now, let’s list all possible sets of hidden properties, in four groups. Two sets of hidden properties are in each group.
Group 1: The two photons do the same thing, regardless of the orientation of the polarizers. (Vertical→Pass. 30°→Pass. 60°→Pass.): each photon always passes.
(Vertical→Block. 30°→Block. 60°→Block.): each photon is always blocked.
Group 2: The two photons do the same thing, unless exactly one of the polarizers is vertical. (Vertical→Block. 30°→Pass. 60°→Pass.): each photon passes unless it reaches a vertical polarizer.
(Vertical→Pass. 30°→Block. 60°→Block.): each photon is blocked unless it reaches a vertical polarizer.
Group 3: The two photons do the same thing, unless exactly one of the polarizers is 30°. (Vertical→Pass. 30°→Block. 60°→Pass.): each photon passes unless it reaches a polarizer at 30°.
(Vertical→Block. 30°→Pass. 60°→Block.): each photon is blocked unless it reaches a polarizer at 30°.
&nbs
p; Group 4: The two photons do the same thing, unless exactly one of the polarizers is 60°. (Vertical→Pass. 30°→Pass. 60°→Block.): each photon passes unless it reaches a polarizer at 60°.
(Vertical→Block. 30°→Block. 60°→Pass.): each photon is blocked unless it reaches a polarizer at 60°.
We’ve listed all eight possible sets of hidden properties. This is a complete list: for each of the three polarizer angles, two outcomes are possible (pass or block), and we listed all possible combinations of outcomes.
Our final task is to determine the fraction of photon pairs that have hidden properties from each of the four groups. Let F1 be the fraction (between 0 and 1) of photon pairs whose hidden properties are from Group 1. Then F2, F3, and F4 are similarly defined.
Now, let’s consider Observation 3, based on one vertical polarizer and one 60° polarizer. Let’s imagine leaving the polarizers at these angles for a long time. Many photon pairs encounter the polarizers, and the two photons do the same thing 25 percent of the time, as also stated in Observation 3. So exactly 25 percent of the photon pairs must be from groups that do the same thing when the polarizer angles differ by 60°. When we look over the four groups, we see that photon pairs in Group 1 and also Group 3 do the same thing when the polarizer angles differ by 60°: photon pairs from Group 1 always do the same thing, and photon pairs from Group 3 do the same thing as long as one angle isn’t 30°. (In contrast, the photons in Groups 2 and 4 do different things when the polarizer angles differ by 60°: one photon passes, and one is blocked.) So 25 percent of the photons pairs must be from either Group 1 or Group 3. We don’t yet know how the 25 percent is divided; it’s not necessarily 12.5 percent from Group 1 and 12.5 percent from Group 3. We just know that the total fraction of photon pairs from Groups 1 and 3 combined is 25 percent. We can write this fact mathematically as:
F1 + F3 = 0.25.
Next, we’ll apply the same logic to Observation 2. Let’s consider Observation 2, in the specific case of one vertical polarizer and one 30° polarizer (a 30° difference). We imagine leaving the polarizers set to these angles for a long time, so that many photon pairs encounter these angles. According to Observation 2, the two photons in a pair do the same thing 75 percent of the time. In other words, 75 percent of the photon pairs are from groups that do the same thing when one polarizer is vertical and the other is at 30°. We find that photon pairs from Group 1 always do the same thing, and photon pairs from Group 4 do the same thing as long as one polarizer isn’t 60°. (Photon pairs from the two other groups do different things when one polarizer is vertical and the other is at 30°.) This means 75 percent of the photons pairs are from Groups 1 and 4:
F1 + F4 = 0.75.
We repeat for the other case of Observation 2, with one 30° polarizer and one 60° polarizer (still a 30° difference). Again we imagine leaving the polarizers set this way for a long time. The two photons do the same thing 75 percent of the time, so 75 percent of the photons must be from groups that do the same thing when one angle is 30° and the other is 60°. We find this behavior in Groups 1 and 2. Therefore, 75 percent of photon pairs must be from Groups 1 and 2:
F1 + F2 = 0.75.
Finally, we use the fact that the sum of all fractions is 1 (100 percent of the photon pairs come from one of the four groups):
F1 + F2 + F3 + F4 = 1.
If you subtract the first three equations from the last one, you find
F1 + F2 + F3 + F4 − (F1 + F3) − (F1 + F4) − (F1 + F2) = 1–0.25–0.75–0.75.
This simplifies to
−2F1 = −0.75
or
F1 = 0.375.
Plugging this into the first equation, F1 + F3 = 0.25, yields F3 = −0.125. But we can’t have a negative fraction of photons! Therefore at least one of our assumptions is disproven. More specifically, either realism is false, or locality is false, or both are false.
If realism is false, the photons do not have their observed properties prior to observation; prior to observation, the photons are in a mysterious indeterminate state. If realism is false, the measurement of the photons definitely changes something. But what changes? An objective physical property of the photons, or just our knowledge of the photons? Does measurement create objectively real states, or is direct observation the only reality accessible to science? These questions are answered differently in different interpretations of quantum mechanics. There is no consensus among physicists.
There is a way to save realism: we can discard locality. In this case, each photon’s behavior depends on the angle of both polarizers: the polarizer it encounters, and the polarizer encountered by the distant photon. But polarizers can be made out of sheets of plastic. Why should a photon be affected by a distant piece of plastic? The only thing linking the photon to that piece of plastic is the fact that the other photon encounters it. I find it especially spooky to imagine one photon being affected by the other photon’s polarizer.
Or, we can discard both realism and locality. This permits an interpretation of quantum mechanics that I find simple and expedient: measurement creates objectively real states. Because entangled photons share a single state, the measurement of one photon immediately creates an objectively real state for both photons. This interpretation is nonlocal because the measurement of one photon physically alters the distant photon (for all practical purposes). Although this interpretation allows for objective reality, the objectively real state is created by the measurement and does not exist prior to the measurement. Thus, this interpretation is not consistent with realism, which requires measurable properties to exist prior to measurement.
A further sampling of interpretations of quantum mechanics appears at the end of the book.
Hardy Makes It Easy
We’ll take a look now at an example that doesn’t require us to think of proportions of photon pairs doing one thing or another. We only need to know that some outcomes never occur, and some outcomes occasionally occur. Lucien Hardy came up with the basic idea for this example, but I’m following a version by N. David Mermin.9
Consider entangled photon pairs from a certain source. These photons are entangled differently from the cases we’ve seen previously; they’re not 50 percent horizontally polarized and 50 percent vertically polarized. We aren’t concerned with the specifics of the new entangled state. We need to know only that this new state can be created experimentally.
The two photons in a pair travel in different directions, each to its own analyzer. There’s a switch on the analyzer that allows the experimenter to select one of two settings: Setting 1 or Setting 2. All we need to know is that the analyzers are wired to lights: a green light and a red light. Each time a photon arrives at an analyzer, either the green light flashes, or the red light flashes (figure 12).
Figure 12 The two photons in a pair go their separate ways, each to an analyzer. The analyzer has two settings, Setting 1 and Setting 2, and two lights, red and green. Each time a photon arrives at an analyzer, one of the lights flashes.
After recording a large number of results for all possible combinations of switch settings, we make the following observations:
Observation A. When both switches are set to Setting 1, at least one light flashes green.
Observation B. When one switch is set to Setting 1 and the other is set to Setting 2, at least one light flashes red.
Observation C. When both switches are set to Setting 2, occasionally both lights flash green.
How do the photon pairs manage to comply with these rules? One possibility is that the two photons are in communication with each other: spooky action at a distance, which is a familiar concept by now. But we’ll stubbornly cling to our cherished common-sense notion of local realism. We’ll attempt to explain the photons’ behavior under the assumption that each photon all along has a hidden property that is detected by the analyzer, independent of the other photon or the other photon’s analyzer.
Since there are only two switch settings, there are only four possible sets of
hidden properties for each photon:
(Setting 1→Green. Setting 2→Green.)
(Setting 1→Green. Setting 2→Red.)
(Setting 1→Red. Setting 2→Green.)
(Setting 1→Red. Setting 2→Red.)
Equivalently, we can say that each photon has one property from each of the two columns below:
Setting 1 Column Setting 2 Column
Setting 1→Green.
Setting 2→Green.
Setting 1→Red.
Setting 2→Red.
Unlike previous cases, it is not necessary that the two photons in a pair share the same hidden properties. In fact, it’s clear from Observation A that if one photon causes a red flash when its analyzer is set to Setting 1, then the other photon must have a different property. So we need to figure out which properties one photon may or may not have, based on the other photon’s properties.
Let’s look at Observation B first. Since there’s at least one red flash when the switch settings differ from each other, if one light flashes green on Setting 1, the other must flash red on Setting 2. In other words, if one photon has the (Setting 1→Green) property, the other must have the (Setting 2→Red) property. Similarly, if one photon has the (Setting 2→Green) property, the other must have the (Setting 1→Red) property. Thus whenever both photons have the (Setting 2→Green) property (which occasionally happens, according to Observation C), both photons must also have the (Setting 1→Red) property. But this contradicts Observation A because then both lights would flash red when both Settings are 1!