Bell state measurement

One of the central illustrations of the strangeness of quantum mechanics arises when one carries out measurements on each particle in an entangled pair. No understanding of the quantum world can be complete without grappling with this result.

The strangeness was first pointed out by Einstein, Podolsky, and Rosen in a thought experiment in 1935. It was in this context that Einstein coined the phrase “spooky action at a distance.” The correct interpretation of these effects were part of a longstanding argument between Einstein and Niels Bohr. The situation was clarified somewhat (or perhaps thrown into greater confusion) in 1964 by John Stewart Bell who mathematically proved that an experiment could, in principle, be performed to demonstrate that the “spookiness” was unequivocally real. This experiment was first carried out in 1974 by Stuart Freedman and John Clauser, confirming the spooky interpretation of quantum mechanics. Clauser, along with others who strengthened the result, were awarded the 2022 Nobel prize in physics for these experiments.

Using the standard measurement axiom the story goes like this: Consider two particles that interact so as to become entangled in a certain way. The two particles travel away from each other (possibly very far away). A measurement can then be performed on one particle, so that the same measurement performed on the other particle yields the same result. So far, this is unsurprising. Somewhat more surprising is that the outcome of the first measurement is not deterministic — say, two outcomes may occur with 50/50 probability. Yet, the result of the measurement on the second particle always matches the result of the first. Still more surprising is that the same thing works if we choose to measure a complementary observable: the result of the first measurement is random, but the result of the second measurement will match it. And if we measure one observable on one particle, and the complementary observable on the other? Then both outcomes are random.

As just described, these measurements on entangled particles are real head-scratchers. How does the second particle “know” what to do? It seems like somehow the choice of the measurement of the first particle is affecting the behavior of the second particle, despite the fact that the two particles may be very distant. If one observable is measured on the first particle, somehow the second particle acts like it has a well-defined value of that quantity, revealing that quantity when measured or returning a random result when the complementary observable is measured. And it works whether we measure either of the complementary observables on the first particle! A possible resolution to this problem is that the outcomes of measurements are not actually decided at random upon measurement, but that they are predetermined when the particles interact locally. But this so-called “local hidden variable” idea was ruled out by the experimental verification of Bell’s theory.

What we will see here is that, for our wave packet example, the measurement axiom is not necessary. Instead, the correlation of results for different measurements is precisely what we find for the behavior of quantum waves following from the Schrödinger equation alone.

The examples shown below are quite similar to those in the previous section, Complementary observables, where we showed repeated measurements of a particle. Now however, we perform the measurements on different particles, instead of the same particle.

The initial entanglement will, as before, be created by the interaction of a particle in a superposition of two positions with a “mirror” particle. Here, however, we will stop calling the second particle a mirror, and instead refer to them as “Particle 1” and “Particle 2”. The situation is nice and symmetrical if we let both wave packet components of Particle 1 reflect off of Particle 2, one after the other, as was shown back in the original example of interaction leading to entanglement back in Fig. 2.6. For convenience, we reproduce the same figure here as Fig. 6.1.

Figure 6.1

Video example of two interacting wave packets moving along a line plotted in 2D as a single two-particle wave function. The initial state is not entangled. After the interaction, the state is entangled.

The final state in Fig. 6.1 can be thought of as two entangled particles moving apart from each other. Particle 1 is moving to larger values of \(x\) (blobs moving to the right on the plot), and Particle 2 is moving to smaller values of \(x\) (blobs moving down on the plot). Each particle is split into two packets, one moving ahead of the other. The two leading packets are correlated with each other, and the two trailing packets are correlated with each other. We could say that the state is in superposition with two parts: A. Leading packet 1 and leading packet 2; and B. trailing packet 1 and trailing packet 2.

Starting from the final entangled state of Fig. 6.1, we can perform our choice of micromeasurements on either particle: a position measurement where the recoil of a “mirror” particle distinguishes between the leading or trailing packet, or a phase measurement where we cause the two packets to overlap and distinguish between their interference patterns. One difficulty in this section is that each micromeasurement introduces another particle, which can make things difficult to visualize, but we’ll do our best.

First, let’s look at the most straightforward case: a position micromeasurement on each particle. Starting from the final entangled state of Fig. 6.1, we will first do a position micromeasurement of Particle 2. That means, we place a “mirror” particle in the path of particle 2, which is light enough to get an unambiguous recoil. The leading packet of Particle 2 will be first to hit the mirror and reflect. At that point, the state will be in an entangled state of three particles, with parts: A. leading packet 2 reflected, mirror recoiled, and leading packet 1 unaffected; and B. trailing packet 1, trailing packet 2, and mirror, all unaffected.

If we wish to visualize this scenario, we now need a 3D plot. This is shown in Fig 6.2. As before, each of the three particles is still moving in one dimension (\(x\)). Now the state must be specified as an amplitude for each triple of positions for Particle 1, Particle 2, and the mirror. The entangled components are now shown as 3D blobs (colored green in this case instead of yellow). Since it is a bit hard to tell the precise alignment of the blobs in 3D, we also plot the shadows of the blobs projected onto the three back planes. The \(x_1-x_2\) plane is the 2D plane that we were previously plotting for Particle 1 and Particle 2. We can see the shadows are not aligned with each other in that plane, as in the final state of Fig. 6.1. This is the initial entangled state of Particles 1 and 2. On the other hand, if we look at the \(x_1-x_3\) or \(x_2-x_3\) planes, we see that the blobs are both at the same \(x_3\) position. The \(x_3\) position is the position of the mirror particle. This indicates that before the collision, the mirror particle is not entangled with Particle 1 and 2.

Figure 6.2

Video example of two entangled particles interacting with a third mirror particle all moving along a line plotted in 3D as a single three-particle wave function. The two particles are entangled in the initial state. After the interaction, all three particles are entangled. The blobs are shown as surfaces of constant amplitude squared, while the projections reflect the amplitude squared integrated over one axis.

In the 3D three-particle plot, the short-range interaction between the particles occurs when any two particles come close to equal value. This happens at diagonal planes where two of the three \(x\) values approach each other. One such diagonal plane is indicated by red dashed lines. In particular, this plane is the one where \(x_2\) and \(x_3\) become equal. That is, this plane is where Particle 2 collides with the mirror. The resulting collision is nothing new. It is just the same as the first collision between Particle 1 and Particle 2. First, the leading packet of Particle 2 collides with the mirror. After that, the particles are in an entangled state now involving all three particles, but still with two parts as given above: A. leading packet 2 reflected, mirror recoiled, and leading packet 1 unaffected; and B. trailing packet 1, trailing packet 2, and mirror, all unaffected. In Fig 6.2, we can see the new entanglement by the fact that the shadows of the blobs are now misaligned in all three planes.

Starting from the final state of Fig 6.2, we finally need to perform a position micromeasurement on Particle 1. As always, we do this by placing a (light) mirror particle in the path of particle 1. Sadly, we can’t plot the whole four particle state, as we would have to add a fourth dimension to Fig 6.2. But that’s not too bad because the behavior we have seen so far is not too complicated, and we can just extrapolate to the next measurement. Starting with Fig. 6.1, we started with two unentangled, aligned blobs. They reflected off each other, and we ended up with two entangled, non-aligned blobs. We added a third (mirror) particle in the third dimension of Fig 6.2, with the two blobs aligned in this third dimension. When Particle 2 interacted with the new particle, the blobs became misaligned in all dimensions as the new particle entered into the entanglement. So adding a fourth particle means that we still have two blobs, initially aligned along the fourth dimension. Once Particle 1 collides with the mirror particle, the two blobs one after the other, the blobs will not be aligned along any of the four dimensions, indicating that all of the particles are now entangled.

The final state after this process can be thought of as two entangled components: A. both leading packets reflected, both mirrors recoiled earlier; and B. both trailing packets reflected, both mirrors recoiled later. Or we could look at the state before the trailing packets have reflected. Then we would have components: A. both leading packets reflected, both mirrors recoiled; and B. both trailing packets unaffected, both mirrors un-recoiled. The key point is that there is a correlation between the effects on both mirrors. In a case where one mirror recoils early, the other also recoils early. In a case where one recoils later, the other recoils later.

Once again, we have the same correlation of outcomes as when we use the standard measurement axiom, but with the difference that both outcomes still exist after the micromeasurements, whereas the measurement axiom would say that one entangled component (one blob) vanishes into thin air. This is Part One of the Bell state measurement picture. But what about the weirder part where we can choose amongst different complementary observables?

Now we’ll take a look at the case where we start from the same entangled state shown at the end of Fig. 6.1, then perform a phase micromeasurement of each particle. So again, the starting state is one with two entangled components: A. Leading packets of both particles, B. Trailing packets of both particles. Recall how our phase micromeasurement works: we cause the two packets of a particle to overlap where the state of an atom is altered depending on the resulting interference pattern.

Looking at final state in Fig. 6.1, you might think we have big problem. We can cause the packets of either particle to overlap by reflecting its leading packet back to cross the trailing packet. But just as we saw in the sequential complementary measurements section (see Fig. 5.8), there will be no interference in this case! Due to the entanglement with the other particle, the two blobs do not cross just because the packets of one particle are overlapping. We could try to reflect both leading particles at once, so as to have both packets overlap at the same moment in time. But this isn’t supposed to be necessary. We should be able to measure one particle first and the other later. But don’t worry — it’s going to work, I promise!

For the first phase micromeasurement indeed there is no interference. This is just as we already saw in Fig. 5.8, with basically the same thing reproduced here in Fig. 6.3. The atom is placed at the midpoint where the Particle 1 packets cross in the horizontal direction. But since the blobs are offset vertically, instead of interfering and deterministically affecting the atom state, they each split up into a pair of packets, one out-of-phase pair correlated with the atom state \(|+\rangle\) and the an in-phase pair correlated with \(|-\rangle\). The phases are labeled at the end of the video. Since we have equal weight in \(|+\rangle\) and \(|-\rangle\), this amounts to a 50/50 outcome as far as the phase micromeasurement is concerned.

Figure 6.3

Video example of a phase micromeasurement on Particle 1 in a two-particle entangled state.

But wait — looking at the final state of Fig. 6.3, we see that performing a phase measurement on Particle 2 now will feature interference! Each blob has a vertically aligned partner. We now reflect the Particle 2 leading packet (the bottom one) back up towards the trailing packet, and place a second atom sensor (atom 2) at the midpoint between the vertically separated blobs. And look what is going to happen: the vertically aligned packets correlated with the \(|+\rangle\) state of atom 1 are out-of-phase, and the those correlated with \(|-\rangle\) state of atom 1 are in-phase. So that means that when we overlap those packets on top of atom 2, we expect the \(|+\rangle\) state of atom 1 to be correlated with the \(|+\rangle\) state of atom 2, and the same of the \(|-\rangle\) states.

In order to visualize the second phase micromeasurement, we need four 2D plots. Each 2D plot again represents amplitudes of pairs of positions of Particle 1 and Particle 2. The four plots each represent those position amplitudes correlated with the four possible combinations of atom 1 and atom 2 states: \((|+\rangle,|+\rangle)\), \((|+\rangle,|-\rangle)\), \((|-\rangle,|+\rangle)\), and \((|-\rangle,|-\rangle)\). This is shown in Fig. 6.4. The initial state is copied over directly from the end of Fig. 6.3, and is shown in the top two panels where the atom 2 state is in its initial state \(|+\rangle\).

Figure 6.4

Video example of a second phase micromeasurement on Particle 2, following a first phase micromeasurement on Particle 1, starting from a two-particle entangled state. The white dotted line indicates the position of the second atom, and the green dotted line indicates the position of an external reflecting barrier.

As the state in Fig. 6.4 evolves, we first see the leading packet of particle 2 reflect off of an external barrier placed at \(x=-21\) (indicated by a green dotted line). Meanwhile, the particle 1 packets keep moving in the horizontal direction, but otherwise not doing anything. Then the leading and trailing packets of particle 2 cross each other at the atom 2 position, indicated by the white dotted line. After the interaction with the atom sensor, we find that the components initially correlated with the \(|+\rangle\) state of atom 1 leave the atom 2 state also in the \(|+\rangle\) state (in the top left panel). But the components initially correlated with the \(|-\rangle\) state of atom 1 rotate the state of atom 2 to also be in the \(|-\rangle\) state. This is seen as a disappearance from the top right panel and reappearance in the bottom right panel. (There is a lot going on in this example, and it becomes difficult to fit everything into the simulation area. Ignore a bit of messiness, such as particle 2 again starting to reflect off the barrier at the end.)

The key result of Fig. 6.4 is that the outcomes of the two phase micromeasurements are correlated. When the first one is \(|+\rangle\), the second is also \(|+\rangle\), and vice versa. As in many previous cases, we have realized the prediction of the measurement axiom, except that both outcomes still exist \((|+\rangle,|+\rangle)\) and \((|-\rangle,|-\rangle)\) instead of one of them being chosen at random.

In the interest of not belaboring the point, we will not dwell too much on the other two cases — measurements of different complementary observables on the two particles. From the examples we have already seen, we can try to imagine what happens. We have already seen the first steps, an initial position or phase micromeasurement on one of the particles (Fig. 6.2 and Fig. 6.3, respectively). In both cases, the first micromeasurement results in a 50/50 split. In the former case, we can next imagine introducing an atom sensor to measure the phase difference between the packets of the other particle. Because the blobs are not aligned along any axis, there will be no interference, and the atom will wind up in a 50/50 state. In the latter case, we introduce another light mirror particle to interact with Particle 2. The leading packet of that particle (the lower 4 blobs in Fig. 6.3) will first reflect off the mirror, resulting in the mirror particle becoming entangled with the other two particles with one component having the mirror recoiled, and the other component unrecoiled. Again, a 50/50 split of the two mirror states. The end results are uncorrelated 50/50 results for both micromeasurements.

So what have we learned about Bell state measurements? First of all, purely using the Schrödinger equation the strange behavior predicted by the measurement axiom has been reproduced, with the exception that both outcomes still exist as two components of an entangled state. Can we now provide a satisfying, intuitive explanation for how the particle “knew” which observable it was supposed to cooperate with? Not really.

The way that the second particle knew what to do, is that the measurement on the first particle actually did, in a sense, affect the second particle. It did so instantly, despite the fact that the two particles could be very far apart. This is implicit in the state as illustrated by a 2D (or 3D) plot, and the evolution of those states given by the Schrödinger equation.

We are now used to looking at the blobs moving, reflecting, interfering, or becoming correlated with different states of an atom. But keep in mind that a blob represents two (or three, or more) different particles that may be nowhere near each other. A single blob centered at coordinates (1000 lightyears, -1000 lightyears) represents two wave packets separated by 2000 lightyears. If one particle locally interacts with a third particle, the blob itself reacts, according to the Schrödinger equation.

The way in which a distant particle is affected is quite particular: the states with which it is correlated can be changed instantly. The form of a distant wave cannot be changed instantly, which rules out the possibility of faster-than-light communication. If a correlation is changed, the two particles must be brought together to find out. The example of phase micromeasurement on two entangled particles illustrates this point. In Fig. 6.3, Particle 1 interacts with an atom phase sensor. Initially, the leading packet of particle 2 was correlated with the single leading packet of particle 1, and trailing packet of particle 2 was correlated with the single trailing packet of particle 1. After the phase micromeasurement, the particle 1 packets are both split into pairs. Now the leading and trailing particle 2 packets (which are still in the same place) are correlated not with single packets anymore, but with in-phase or out-of-phase pairs of packets. This change has a real effect when we next bring the particle 2 packets together for the second phase micromeasurement. Namely, it allows the outcomes of that micromeasurement to become correlated with that of the first.

In regards to the prohibition on faster-than-light communication, note that in the example of two phase micromeasurements, when all is said and done we wind up with two atoms with correlated states, \(|+\rangle\) correlated with \(|+\rangle\), and \(|-\rangle\) correlated with \(|-\rangle\). But if any additional particle interacts locally with one atom or the other, its interaction won’t depend on what that atom is correlated with, potentially lightyears away. To have any subsequent effect depend on the fact that this correlation exists, that additional particle must interact with both atoms, either by having the additional particle travel from one to the other, or by bringing the two atoms together. And according to the theory of relativity, this travel cannot happen faster than the speed of light.

As mentioned above, this nonlocal behavior of correlations is not at all intuitive or easy to grasp, but it is what the Schrödinger equation plainly predicts. And we see here that it is capable of explaining some of the strange behavior of measurements on Bell states, without invoking a measurement axiom.