Hidden Variables?

Because quantum entanglement violates the principle of locality, Einstein, Podolsky, and Rosen argued that the theory of quantum mechanics must be incorrect, or at least incomplete in the sense that the wave function doesn’t represent the complete state of a microphysical system. They suggested that the entanglement phenomenon might be explained instead by local hidden variables—undiscovered properties that predetermine how each of the supposedly “entangled” particles (in this case, photons) will behave when measured. Perhaps some unknown property of each photon determines what it will do when encountering any given type of polarizing filter, and the two “entangled” photons simply have opposite properties, so they do opposite things. In that case, there would be no need for the photons to communicate with each other at faster-than-light speeds.

Many physicists tried, without success, to construct a theory of local hidden variables that could explain the odd behaviors of entangled photons. In 1964, decades after Einstein et. al. had published their famous paper criticizing quantum mechanics, an Irish physicist named John Bell proved mathematically that any alternative theory like the one Einstein, Podolsky and Rosen wanted (that is, any theory involving local hidden variables that determine what state a particle will have when it is measured) must yield predictions that differ from the predictions of quantum mechanics. This is known as Bell’s theorem:

It is impossible for any non-chancy theory of local hidden variables to give all of the same predictions as quantum mechanics.

In other words, there will always be at least some experiments for which such a theory predicts different results than quantum mechanics does. No matter what sort of local hidden variables are proposed, it is mathematically impossible for the theory to agree with every prediction of quantum mechanics. Therefore, unless quantum mechanics is wrong in some of its predictions, no theory of local hidden variables can be correct.

Bell’s theorem doesn’t rule out the possibility of hidden variables (undiscovered properties) involving faster-than-light influences and/or chancy processes. But those are the very things Einstein, Podolsky, and Rosen wanted to avoid in the first place.

Although the details of Bell’s proof are too technical to explain here, a simple illustration will serve to show why local hidden variables can’t yield the same predictions as quantum mechanics:

Suppose entangled pairs of photons are emitted from a device in the center of a room. The photons are sent in opposite directions: one from each entangled pair is directed toward the east side of the room, and the other toward the west. At each end of the room, the photons hit a linear polarizing filter. The two filters are rotated at varying angles so that sometimes both filters are oriented the same way and at other times they are oriented at different angles. Some of the eastbound photons pass through the east filter and strike a detector on the other side; others are blocked by the filter and do not strike the detector. Similarly with the westbound photons and their filter.

Since each photon is emitted simultaneously with its entangled partner, the event of the eastbound photon striking the east filter and the event of the westbound photon striking the west filter are simultaneous (from the reference frame of the room), so according to special relativity neither event can causally influence the other.

What does quantum mechanics predict about the results of this experiment? Quantum mechanics gives two interesting predictions, either one of which could be explained just as well by a local hidden variable theory. But it is impossible for any such theory to yield both predictions, as I will explain.

First, suppose the two polarizing filters are oriented at the same angle, e.g. both horizontal. According to quantum mechanics, the entangled photons always have orthogonal polarizations. So if the eastbound photon passes through the filter at the east end of the room, then the westbound photon will be blocked by the filter at the west end of the room, and vice versa. This same result also happens if both filters are oriented vertically, or at 45°, or at any other angle, so long as both filters are oriented the same way. Let’s call this Prediction A:

Prediction A: Whenever the filters are oriented at the same angle, the photons will do opposite things: one will pass through its filter and the other won’t.

But what happens when the two polarizing filters are oriented at different angles? According to quantum mechanics, the probability that a photon will pass through a polarizing filter is equal to cos2(θ), where θ is the angle between the photon’s polarization and the orientation of the filter. This implies that whenever the two filters in our experiment are oriented with 120° angles relative to each other, the probability of exactly one of the two entangled photons being blocked is always ¼. To see why, notice that there are only two possible behaviors for the eastbound photon. Either it passes through the filter, or it is blocked:

  1. In order to pass through, according to quantum mechanics, the photon’s superposition must collapse to the polarization required by that filter. Since entangled photons always have orthogonal polarizations, the westbound photon must collapse to a polarization perpendicular to the east filter. And since the east and west filters are oriented at 120° relative to each other, this means that the westbound photon is now oriented at 120° - 90° = 30° relative to the filter that it encounters at the west end of the room. So its probability of passing through the west filter is cos2(30°) = ¾, and its probability of being blocked is ¼.
  2. On the other hand, suppose the eastbound photon is blocked by its filter. In order for that to happen, according to quantum mechanics, the photon must collapse to a polarization perpendicular to the filter. So the westbound photon must collapse to the orthogonal polarization, which in this case is parallel to the east filter and therefore 120° relative to the west filter. Thus, the probability of the westbound photon passing through the filter is cos2(120°) = ¼.

These are the only two possible behaviors for the eastbound photon, and in both cases the westbound photon has a ¼ chance of doing the opposite thing. Thus, quantum mechanics predicts that if the two filters are oriented 120° with respect to each other and the experiment is repeated many times, the photons will do opposite things (one passes through its filter but the other doesn’t) roughly ¼ of the time. Let’s call this Prediction B:

Prediction B: Whenever the filters are oriented at 120° with respect to each other, the photons will do opposite things roughly ¼ of the time.

Now, let’s try to explain both of the above predictions using local hidden variables. (It’s impossible, but let’s try anyway, just to see why it’s impossible.) In order for our theory to give Prediction A, we’ll need some way to guarantee that the photons always have opposite polarizations when they hit filters at the same angle. The photons need not have any definite polarization before hitting the filters, but there must be something about each photon—some unknown feature, a “hidden variable”—that predetermines what the photon will do if it hits a filter oriented at any given angle. For example, if the first photon has some feature that lets it pass through a horizontal filter, the second photon must have a property guaranteeing that it will be blocked by a horizontal filter; and similarly for all other possible filter orientations.

To make this idea more concrete, we can imagine that each photon is a tiny computer that has been pre-programmed with instructions about what to do when it hits a filter. To ensure that this (admittedly silly) hidden variable theory yields Prediction A, we could suppose that the eastbound photons are always programmed to pass through the filter and the westbound ones are always programmed to be blocked. Or we could suppose that it’s the other way around: the westbound photons always pass through and the eastbound photons are always blocked. In either case, the photons will do opposite things 100% of the time, regardless of the filter orientation. This is consistent with Prediction A, but inconsistent with Prediction B. In order to get both predictions right, the behaviors of the photons will have to depend on the orientations of the filters.

Strangely, according to quantum mechanics, the behavior of each photon depends not only on the orientation of its own filter but also on the orientation of the filter encountered by its entangled partner at the opposite end of the room! The aim of our “tiny computer” theory is to produce the same predictions as quantum mechanics without involving such bizarre non-local causation. Could there be some way of ensuring that the photons will behave as predicted by quantum mechanics, if each photon’s behavior depends only on the orientation of the filter it encounters? In other words, can we ensure that each pair of photons behaves the right way, without letting each individual photon “know” (so to speak) what happened to its partner? Unfortunately not, as the following hypothetical experiment illustrates.

Imagine that we’ve set up an experimental apparatus that allows us to vary the angles of the filters in the following six ways:

  East filter West filter
option 1 120°
option 2 240°
option 3 120°
option 4 120° 240°
option 5 240°
option 6 240° 120°

We change the configuration of the filters randomly among these six options. (This randomization step is included to avoid certain kinds of objections that could otherwise be raised against the argument given below.) And we repeat the experiment many times, so that the six possible configurations occur equally often.

Quantum mechanics says that the photons will do opposite things ¼ of the time for each of these six configurations, and hence ¼ of the time overall. (That’s Prediction B.) But any deterministic, local hidden variable theory must imply that the photons will do opposite things at least ⅓ of the time overall! To see why this must be so, notice first that there are only a finite number of possible ways to “program” the photons to ensure that Prediction A will hold. Two of those possibilities have already been considered: the eastbound photons might be programmed to pass through a filter regardless of its orientation while the westbound photons are programed to be blocked regardless of orientation, or vice versa. But that would yield opposite behavior 100% of the time. And each of the remaining possibilities yields opposite behavior ⅓ of the time, as illustrated with the example below.

Suppose the photons are programmed to behave as shown in the following table:

Filter orientation Eastbound photon Westbound photon
pass do not pass
120° do not pass pass
240° do not pass pass

This will yield opposite behavior for two of the six possible filter configurations:

East filter West filter Eastbound photon Westbound photon Opposite behavior?
120° pass pass no
240° pass pass no
120° do not pass do not pass no
120° 240° do not pass pass yes
240° do not pass do not pass no
240° 120° do not pass pass yes

So this program yields opposite behavior ⅓ of the time. And so do all of the remaining possibilities. (If you don’t believe me, try creating a different set of instructions for the two photons. Make sure your photons do opposite things for each filter orientation, as required for Prediction A. Then create a table like the one above, and you’ll find that two of the six possible configurations yield opposite behavior.)

Therefore, even if we try to mix and match these various possibilities, programming some entangled photon pairs to behave in one way and other pairs to behave in another way, we’ll never be able to find a combination that yields opposite behavior only ¼ of the time in this experiment. Thus, it is impossible for our hidden variable theory to give both Prediction A and Prediction B.

Experiments similar to the one described above have been done, and the results show that the predictions of quantum mechanics are indeed correct. No theory of local hidden variables can account for this.