Of Particular Significance

Quantum Interference 5: Coming Unglued

Picture of POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 03/31/2025

Now finally, we come to the heart of the matter of quantum interference, as seen from the perspective of in 1920’s quantum physics. (We’ll deal with quantum field theory later this year.)

Last time I looked at some cases of two particle states in which the particles’ behavior is independent — uncorrelated. In the jargon, the particles are said to be “unentangled”. In this situation, and only in this situation, the wave function of the two particles can be written as a product of two wave functions, one per particle. As a result, any quantum interference can be ascribed to one particle or the other, and is visible in measurements of either one particle or the other. (More precisely, it is observable in repeated experiments, in which we do the same measurement over and over.)

In this situation, because each particle’s position can be studied independent of the other’s, we can be led to think any interference associated with particle 1 happens near where particle 1 is located, and similarly for interference involving the second particle.

But this line of reasoning only works when the two particles are uncorrelated. Once this isn’t true — once the particles are entangled — it can easily break down. We saw indications of this in an example that appeared at the ends of my last two posts (here and here), which I’m about to review. The question for today is: what happens to interference in such a case?

Correlation: When “Where” Breaks Down

Let me now review the example of my recent posts. The pre-quantum system looks like this

Figure 1: An example of a superposition, in a pre-quantum view, where the two particles are correlated and where interference will occur that involves both particles together.

Notice the particles are correlated; either both particles are moving to the left OR both particles are moving to the right. (The two particles are said to be “entangled”, because the behavior of one depends upon the behavior of the other.) As a result, the wave function cannot be factored (in contrast to most examples in my last post) and we cannot understand the behavior of particle 1 without simultaneously considering the behavior of particle 2. Compare this to Fig. 2, an example from my last post in which the particles are independent; the behavior of particle 2 is the same in both parts of the superposition, independent of what particle 1 is doing.

Figure 2: Unlike Fig. 1, here the two particles are uncorrelated; the behavior of particle 2 is the same whether particle 1 is moving left OR right. As a result, interference can occur for particle 1 separately from any behavior of particle 2, as shown in this post.

Let’s return now to Fig. 1. The wave function for the corresponding quantum system, shown as a graph of its absolute value squared on the space of possibilities, behaves as in Fig. 3.

Figure 3: The absolute-value-squared of the wave function for the system in Fig, 1, showing interference as the peaks cross. Note the interference fringes are diagonal relative to the x1 and x2 axes.

But as shown last time in Fig. 19, at the moment where the interference in Fig. 3 is at its largest, if we measure particle 1 we see no interference effect. More precisely, if we do the experiment many times and measure particle 1 each time, as depicted in Fig. 4, we see no interference pattern.

Figure 4: The result of repeated experiments in which we measure particle 1, at the moment of maximal interference, in the system of Fig. 3. Each new experiment is shown as an orange dot; results of past experiments are shown in blue. No interference effect is seen.

We see something analogous if we measure particle 2.

Yet the interference is plain as day in Fig. 3. It’s obvious when we look at the full two-dimensional space of possibilities, even though it is invisible in Fig. 4 for particle 1 and in the analogous experiment for particle 2. So what measurements, if any, can we make that can reveal it?

The clue comes from the fact that the interference fringes lie at a 45 degree angle, perpendicular neither to the x1 axis nor to the x2 axis but instead to the axis for the variable 1/2(x1 + x2), the average of the positions of particle 1 and 2. It’s that average position that we need to measure if we are to observe the interference.

But doing so requires we that we measure both particles’ positions. We have to measure them both every time we repeat the experiment. Only then can we start making a plot of the average of their positions.

When we do this, we will find what is shown in Fig 5.

  • The top row shows measurements of particle 1.
  • The bottom row shows measurements of particle 2.
  • And the middle row shows a quantity that we infer from these measurements: their average.

For each measurement, I’ve drawn a straight orange line between the measurement of x1 and the measurement of x2; the center of this line lies at the average position 1/2(x1+x2). The actual averages are then recorded in a different color, to remind you that we don’t measure them directly; we infer them from the actual measurements of the two particles’ positions.

Figure 5: As in Fig. 4, the result of repeated experiments in which we measure both particles’ positions at the moment of maximal interference in Fig. 3. Top and bottom rows show the position measurements of particles 1 and 2; the middle row shows their average. Each new experiment is shown as two orange dots, they are connected by an orange line, at whose midpoint a new yellow dot is placed. Results of past experiments are shown in blue. No interference effect is seen in the individual particle positions, yet one appears in their average.

In short, the interference is not associated with either particle separately — none is seen in either the top or bottom rows. Instead, it is found within the correlation between the two particles’ positions. This is something that neither particle can tell us on its own.

And where is the interference? It certainly lies near 1/2(x1+x2)=0. But this should worry you. Is that really a point in physical space?

You could imagine a more extreme example of this experiment in which Fig. 5 shows particle 1 located in Boston and particle 2 located in New York City. This would put their average position within appropriately-named Middletown, Connecticut. (I kid you not; check for yourself.) Would we really want to say that the interference itself is located in Middletown, even though it’s a quiet bystander, unaware of the existence of two correlated particles that lie in opposite directions 90 miles (150 km) away?

After all, the interference appears in the relationship between the particles’ positions in physical space, not in the positions themselves. Its location in the space of possibilities (Fig. 3) is clear. Its location in physical space (Fig. 5) is anything but.

Still, I can imagine you pondering whether it might somehow make sense to assign the interference to poor, unsuspecting Middletown. For that reason, I’m going to make things even worse, and take Middletown out of the middle.

A Second System with No Where

Here’s another system with interference, whose pre-quantum version is shown in Figs. 6a and 6b:

Figure 6a: Another system in a superposition with entangled particles, shown in its pre-quantum version in physical space. In part A of the superposition both particles are stationary, while in part B they move oppositely.
Figure 6b: The same system as in Fig. 6a, depicted in the space of possibilities with its two initial possibilities labeled as stars. Possibility A remains where it is, while possibility B moves toward and intersects with possibility A, leading us to expect interference in the quantum wave function.

The corresponding wave function is shown in Fig. 7. Now the interference fringes are oriented diagonally the other way compared to Fig. 3. How are we to measure them this time?

Figure 7: The absolute-value-squared of the wave function for the system shown in Fig. 6. The interference fringes lie on the opposite diagonal from those of Fig. 3.

The average position 1/2(x1+x2) won’t do; we’ll see nothing interesting there. Instead the fringes are near (x1-x2)=4 — that is, they occur when the particles, no matter where they are in physical space, are at a distance of four units. We therefore expect interference near 1/2(x1-x2)=2. Is it there?

In Fig. 8 I’ve shown the analogue of Figs. 4 and 5, depicting

  • the measurements of the two particle positions x1 and x2, along with
  • their average 1/2(x1+x2) plotted between them (in yellow)
  • (half) their difference 1/2(x1-x2) plotted below them (in green).

That quantity 1/2(x1-x2) is half the horizontal length of the orange line. Hidden in its behavior over many measurements is an interference pattern, seen in the bottom row, where the 1/2(x1-x2) measurements are plotted. [Note also that there is no interference pattern in the measurements of 1/2(x1+x2), in contrast to Fig. 4.]

Figure 8: For the system of Figs. 6-7, repeated experiments in which the measurement of the position of particle 1 is plotted in the top row (upper blue points), that of particle 2 is plotted in the third row (lower blue points), their average is plotted between (yellow points), and half their difference is plotted below them (green points.) Each new set of measurements is shown as orange points connected by an orange line, as in Fig. 5. An interference pattern is seen only in the difference.

Now the question of the hour: where is the interference in this case? It is found near 1/2(x1-x2)=2 — but that certainly is not to be identified with a legitimate position in physical space, such as the point x=2.

First of all, making such an identification in Fig. 8 would be like saying that one particle is in New York and the other is in Boston, while the interference is 150 kilometers offshore in the ocean. But second and much worse, I could change Fig. 8 by moving both particles 10 units to the left and repeating the experiment. This would cause x1, x2, and 1/2(x1+x2) in Fig. 8 to all shift left by 10 units, moving them off your computer screen, while leaving 1/2(x1-x2) unchanged at 2. In short, all the orange and blue and yellow points would move out of your view, while the green points would remain exactly where they are. The difference of positions — a distance — is not a position.

If 10 units isn’t enough to convince you, let’s move the two particles to the other side of the Sun, or to the other side of the galaxy. The interference pattern stubbornly remains at 1/2(x1-x2)=2. The interference pattern is in a difference of positions, so it doesn’t care whether the two particles are in France, Antarctica, or Mars.

We can move the particles anywhere in the universe, as long as we take them together with their average distance remaining the same, and the interference pattern remains exactly the same. So there’s no way we can identify the interference as being located at a particular value of x, the coordinate of physical space. Trying to do so creates nonsense.

This is totally unlike interference in water waves and sound waves. That kind of interference happens in a someplace; we can say where the waves are, how big they are at a particular location, and where their peaks and valleys are in physical space. Quantum interference is not at all like this. It’s something more general, more subtle, and more troubling to our intuition.

[By the way, there’s nothing special about the two combinations 1/2(x1+x2) and 1/2(x1-x2), the average or the difference. It’s easy to find systems where the intereference arises in the combination x1+2x2, or 3x1-x2, or any other one you like. In none of these is there a natural way to say “where” the interference is located.]

The Profound Lesson

From these examples, we can begin to learn a central lesson of modern physics, one that a century of experimental and theoretical physics have been teaching us repeatedly, with ever greater subtlety. Imagining reality as many of us are inclined to do, as made of localized objects positioned in and moving through physical space — the one-dimensional x-axis in my simple examples, and the three-dimensional physical space that we take for granted when we specify our latitude, longitude and altitude — is simply not going to work in a quantum universe. The correlations among objects have observable consequences, and those correlations cannot simply be ascribed locations in physical space. To make sense of them, it seems we need to expand our conception of reality.

In the process of recognizing this challenge, we have had to confront the giant, unwieldy space of possibilities, which we can only visualize for a single particle moving in up to three dimensions, or for two or three particles moving in just one dimension. In realistic circumstances, especially those of quantum field theory, the space of possibilities has a huge number of dimensions, rendering it horrendously unimaginable. Whether this gargantuan space should be understood as real — perhaps even more real than physical space — continues to be debated.

Indeed, the lessons of quantum interference are ones that physicists and philosophers have been coping with for a hundred years, and their efforts to make sense of them continue to this day. I hope this series of posts has helped you understand these issues, and to appreciate their depth and difficulty.

Looking ahead, we’ll soon take these lessons, and other lessons from recent posts, back to the double-slit experiment. With fresher, better-informed eyes, we’ll examine its puzzles again.

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13 Responses

  1. Matt,
    Great job helping me to make picturable (anschaulich) what Heisenberg called unpicturable, or at least a useful analogy!
    1. But, going further, I am struck by the apparent coordinance between your “space of possibilities” and Kastner’s “pre-spacetime” in her Possibilist Transactional Interpretation of Quantum Mechanics (The Transactional Interpretation of Quantum Mechanics: A Relativistic Treatment: Kastner, Ruth E.: 9781108830447: Amazon.com: Books). (Not trying to veer into metaphysical interpretation, just observing as you push me to understanding.)
    2. More importantly, please point me to where you explain how the imaginary unit “i” comes into the particle wave function. I missed that and cannot readily find it.

  2. Hi Matt, great post as always. It has cleared up a lot of the fog that had built up in my understanding. I know we’ve just started taking a look at God’s cards (to quote a well-known book by G. Ghirardi, the G of the GRW theory), but already now I would like, if possible, to know your opinion on the metaphysics of what we’ve learned so far: what is the relationship between the space of possibilities and the physical world? Is it just an instrumental relationship in the sense that it only serves to calculate the probability of what happens in the physical world? I cannot believe that the solution is just ‘shut up and calculate’…Thanks.

    1. I really want to avoid any interpretation and metaphysics until I’ve completed the process of establishing clear understanding. Otherwise any conversation about the meaning of quantum physics will be drowned in misconceptions. We have to do quantum field theory (which, unlike quantum mechanics of the 1920s, agrees with all data) before it makes any sense to attempt interpretation. I cannot tell you how much pointless chatter I have read that centers on the metaphysics of 1920s quantum physics, all while ignoring the fact that the whole thing was superseded 75 years ago by a theory with a difference conception of what elementary “particles” are (specifically, that they are quantized waves, not quantized particles.)

  3. In the second paragraph under Figure 8, should “x1, x2, and 1/2(x1-x2) ” be “x1, x2, and 1/2(x1+x2)”? Thanks as always for these posts.

  4. “Space of probability.”?

    Albert Einstein must be rolling in his grave.

    Lost in the math, maybe?

    I believe reality is staring at us in the face we are just too blind to see it, yet. I believe gravity (and therefore spacetime) can be quantized and because it’s non-renormalizable doesn’t mean it cannot be done, it means that we don’t have the math, yet, or we cannot use math to characterize it.

    Explaining things in one dimension is straight forward, but the real world is multi-dimensional, one that we cannot even visualize the eigenspace let alone the eigenvalues of each “quantum state”.

    I believe that the two edges of spacetime have the same mechanism, a black hole. The huge black holes we see in the center of galaxies on one edge and tiny black holes that make up the nucleus of every atom another other edge. from one singularity to another.

  5. Matt,
    again extremely well done. Thanks.

    First thing that comes to my mind is that ‘location’ in space doesn’t (necessarily) seem to play an essential role in interference phenomena.
    Purely mathematically I could imagine two observables (one randomly varying in shape from square to round and another from red to blue, the first being small and growing, the latter being large and shrinking). Some mathematical description of them could show ‘interference’ only when they are approaching the same size : a square shape of one would then never coincide with a blue color of the other. No location, no linear time coordinate, only some correlated variables.

    But the physical reality seems to be restricted to :
    1./ cases where the 2 observables are ‘of the same type’ like positions on an x-axis
    2./ cases where a ‘configuration’ can be realized via more than one alternative route using ‘OR’.

    Are both conditions sufficient and necessary for interference to occur?

    1. Good question. Since it’s very precise as a question, I’d better think it through before answering. “Same type” may need better definition; there are a lot of cases I haven’t covered yet that add complexity, such as angular variables where things get mixed up in ways that positions do not. And the word “configuration” needs to be broadened; the space of possibilities doesn’t just refer to positions, of course, which is often what “configuration” means, though it’s open to definition.

  6. Hi Matt Strassler,

    I’ve not followed your series closely, but when I took a quick scan of this one, I was puzzled by these extremely cohesive wave… packets?… you show crossing the screen. For the fairly long wavelengths you show when they interfere, shouldn’t those packets look more like spherical waves emanating from the sides of the container?

    1. Sorry, Terry, the posts explain this very carefully. This is not a container; first of all, it is a portion of the space of possibilities, not a physical object, and second, it has no walls (as you can see from the packets that move away).

  7. If the particles were identical would the space of possibilities still have an X1 axis and an X2 axis, or would the geometry of the space of possibilities have to change ?

    1. There would still be an x1 and an x2 axis. The statement that the particles are identical is not that they have the same position but that if I were to exchange them, you couldn’t tell that I had done so. So we still need two positions, and therefore two axes in the space of possibilities. However, the wave function needs to change so that if I switch one particle with the other [i.e. if I switch the x1 axis with the x2 axis] the wave function remains the same.

      We’ll do identical particles very carefully soon.

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