# Decoherence is not wave function collapse

In the double slit experiment, particles travelling through a pair of thin slits exhibit wave-like behavior, forming an interference pattern where they land that indicates that the particles in some sense travelled through both slits.

Now, suppose that you place a single spin bit at the top slit, which starts off in the state |↑⟩ and flips to |↓⟩ iff a particle travels through the top slit. We fire off a single particle at a time, and then each time swap out that spin bit for a new spin bit that also starts off in the state |↑⟩. This serves as an extremely simple measuring device which encodes the information about which slit each particle went through.

Now what will you observe on the screen? It turns out that you’ll observe the classically expected distribution, which is a simple average over the two individual possibilities without any interference.

Okay, so what happened? Remember that the first pattern we observed was the result of the particles being in a superposition over the two possible paths, and then interfering with each other on the way to the detector screen. So it looks like simply having one bit of information recording the path of the particle was sufficient to collapse the superposition! But wait! Doesn’t this mean that the “consciousness causes collapse” theory is wrong? The spin bit was apparently able to cause collapse all by itself, so assuming that it isn’t a conscious system, it looks like consciousness isn’t necessary for collapse! Theory disproved!

No. As you might be expecting, things are not this simple. For one thing, notice that this ALSO would prove as false any other theory of wave function collapse that doesn’t allow single bits to cause collapse (including anything about complex systems or macroscopic systems or complex information processing). We should be suspicious of any simple argument that claims to conclusively prove a significant proportion of experts wrong.

To see what’s going on here, let’s look at what happens if we don’t assume that the spin bit causes the wave function to collapse. Instead, we’ll just model it as becoming fully entangled with the path of the particle, so that the state evolution over time looks like the following:

Now if we observe the particle’s position on the screen, the probability distribution we’ll observe is given by the Born rule. Assuming that we don’t observe the states of the spin bits, there are now two qualitatively indistinguishable branches of the wave function for each possible position on the screen. This means that the total probability for any given landing position will be given by the sum of the probabilities of each branch:

But hold on! Our final result is identical to the classically expected result! We just get the probability of the particle getting to |j⟩ from |A⟩, multiplied by the probability of being at |A⟩ in the first place (50%), plus the probability of the particle going from |B⟩ to |j⟩ times the same 50% for the particle getting to |B⟩.

In other words, our prediction is that we’d observe the classical pattern of a bunch of individual particles, each going through exactly one slit, with 50% going through the top slit and 50% through the bottom. The interference has vanished, even though we never assumed that the wave function collapsed!

What this shows is that wave function collapse is not required to get particle-like behavior. All that’s necessary is that the different branches of the superposition end up not interfering with each other. And all that’s necessary for that is environmental decoherence, which is exactly what we had with the single spin bit!

In other words, environmental decoherence is sufficient to produce the same type of behavior that we’d expect from wave function collapse. This is because interference will only occur between non-orthogonal branches of the wave function, and the branches become orthogonal upon decoherence (by definition). A particle can be in a superposition of multiple states but still act as if it has collapsed!

Now, maybe we want to say that the particle’s wave function is collapsed when its position is measured by the screen. But this isn’t necessary either! You could just say that the detector enters into a superposition and quickly decoheres, such that the different branches of the wave function (one for each possible detector state) very suddenly become orthogonal and can no longer interact. And then you could say that the collapse only really happens once a conscious being observes the detector! Or you could be a Many-Worlder and say that the collapse never happens (although then you’d have to figure out where the probabilities are coming from in the first place).

You might be tempted to say at this point: “Well, then all the different theories of wave function collapse are empirically equivalent! At least, the set of theories that say ‘wave function collapse = total decoherence + other necessary conditions possibly’. Since total decoherence removes all interference effects, the results of all experiments will be indistinguishable from the results predicted by saying that the wave function collapsed at some point!”

But hold on! This is forgetting a crucial fact: decoherence is reversible, while wave function collapse is not!!!

Pretty picture from doi: 10.1038/srep15330

Let’s say that you run the same setup before with the spin bit recording the information about which slit the particle went through, but then we destroy that information before it interacts with the environment in any way, therefore removing any traces of the measurement. Now the two branches of the wave function have “recohered,” meaning that what we’ll observe is back to the interference pattern! (There’s a VERY IMPORTANT caveat, which is that the time period during which we’re destroying the information stored in the spin bit must be before the particle hits the detector screen and the state of the screen couples to its environment, thus decohering with the record of which slit the particle went through).

If you’re a collapse purist that says that wave function collapse = total decoherence (i.e. orthogonality of the relevant branches of the wave function), then you’ll end up making the wrong prediction! Why? Well, because according to you, the wave function collapsed as soon as the information was recorded, so there was no “other branch of the wave function” to recohere with once the information was destroyed!

This has some pretty fantastic implications. Since IN PRINCIPLE even the type of decoherence that occurs when your brain registers an observation is reversible (after all, the Schrodinger equation is reversible), you could IN PRINCIPLE recohere after an observation, allowing the branches of the wave function to interfere with each other again. These are big “in principle”s, which is why I wrote them big. But if you could somehow do this, then the “Consciousness Causes Collapse” theory would give different predictions from Many-Worlds! If your final observation shows evidence of interference, then “consciousness causes collapse” is wrong, since apparently conscious observation is not sufficient to cause the other branches of the wave function to vanish. Otherwise, if you observe the classical pattern, then Many Worlds is wrong, since the observation indicates that the other branches of the wave function were gone for good and couldn’t come back to recohere.

This suggests a general way to IN PRINCIPLE test any theory of wave function collapse: Look at processes right beyond the threshold where the theory says wave functions collapse. Then implement whatever is required to reverse the physical process that you say causes collapse, thus recohering the branches of the wave function (if they still exist). Now look to see if any evidence of interference exists. If it does, then the theory is proven wrong. If it doesn’t, then it might be correct, and any theory of wave function collapse that demands a more stringent standard for collapse (including Many-Worlds, the most stringent of them all) is proven wrong.