Collapse and unitarity

Quantum collapse is often said to “violate unitarity”. Either I’m confused or this phrasing is misleading or both.

A bounded linear operator P on a Hilbert space H is said to be unitary iff it is surjective and preserves inner products. But as I understand it, quantum collapse is not even an operator. An operator on H is a function from H to H. But a function f, given a specific input |ψ⟩, yields a unique output f(|ψ⟩). Quantum collapse does no such thing. It is an indeterministic process. Sometimes given input 2^−1/2(|ψ₁⟩+|ψ₂⟩) (where |ψ₁⟩ and |ψ₂⟩ are eigenvectors corresponding to the measurable we are collapsing with respect to) it gives output |ψ₁⟩ and sometimes it gives output |ψ₂⟩.

While strictly speaking if some process is not modeled by an operator, it is not modeled by a unitary operator, to call that a violation of unitarity is misleading. It is better to say it’s a violation of operationality or functionality. We cannot even say what it would mean for a process not modeled by an operator to be unitary, just as we cannot say what it would mean for a frog to be unitary or a linear operator to be a vertebrate.

One might try to say what it would mean to have unitarity for a non-deterministic evolution. Suppose that |ψ⟩ would collapse to |ψ′⟩ and |ϕ⟩ would collapse to |ϕ′⟩ under some measurement. Then one could claim that unitarity would say that ⟨ϕ′|ψ′⟩=⟨ϕ|ψ⟩. But this assumes that there is a fact of the matter as to what |ψ⟩ and |ϕ⟩ would collapse to. Now, if |ψ⟩ in fact collapses to |ψ′⟩, it might make sense to say that |ψ⟩ would collapse to |ψ′⟩. But for unitarity we need the identity ⟨ϕ′|ψ′⟩=⟨ϕ|ψ⟩ for all inputs |ψ⟩ and |ϕ⟩, not just for the ones that actually occurred.

I suppose one could have a generalized Molinist thesis that there is always a fact of the matter as to what a given wavefunction would collapse to, so that we might be able to define a collapse operator. And then we could say that unitarity fails. But it would still likely be misleading to say that unitarity fails, since we would expect linearity to fail, not merely unitarity. And in any case, such a generalized Molinist thesis is quite dubious.

But I know very little about quantum mechanics, and so I may simply be confused.

Observation, collapse and circularity

The following four premises seem to be contradictory:

1. An observation of an event E is caused by the event E.

2. Observation causes collapse.

3. What is observed is the collapsed state.

4. There is no circular causation.

Here is my first attempt to get out of this, on behalf of those attracted to the observation-causes-collapse view. For concreteness, let’s suppose that we’re observing an electron in a mixed up-down spin state, and suppose that we observe that it’s in the up state. Distinguish these two events:

• O₁: Observing whether the electron is in an up or a down state.

• O₂: Observing the electron to be in an up state.

Then I think what the defender of observation-causes-collapse can say this: O₁ causes the collapsed state which in turn causes O₂. But this is rather strange. For O₁ and O₂ actually seem to be the same coarse-grained event, which makes that coarse-grained event be its own cause! Another way to see the problem is to note that O₁ is the disjunctive event of observing the electron to be in the up state or observing the electron to be in the down state. But then O₂ grounds O₁: disjunctions are grounded in their true disjunct(s). But then O₁ is causally prior to its grounds, which seems absurd.

A second attempt: deny (3). Compare Elizabeth Anscombe’s theory that an intention to ϕ in the successful case constitutes one’s knowledge that one will ϕ or Thomas Aquinas’s theory that God’s knowledge of the world is the cause of the world’s being as it is. On these cases, the direction of fit in the knowledge is reversed. Observation of quantum phenomena could be like that.

Third attempt: cut up an act of observation into two parts. Metaphorically speaking, we could imagine the mind querying the world: “Is the electron in an up or a down state?” In response, reality collapses, and the mind observes that reality is in an up state. Thus, we have a query event Q and an observation-proper O₂. It is Q that causes collapse, and P is then the observation of the collapse. This solves the circularity problem, but strictly speaking it’s incorrect to say that observation causes collapse. Rather, it is the pre-observation query event Q that causes collapse. And if simultaneous causation is possible, then Q and O₂ may be simultaneous.

I think the second and third attempts are the way to go, assuming we're keeping the basic idea behind observation-causes-collapse.

Consciousness causing collapse and temporally extended conscious states

I have recently been arguing that states of consciousness are not localized to specific times. Instead, during an interval of times of non-zero length, say from t₁ to t₂, there can be a fact of the matter how many states of a particular sort have occurred.

It’s now occurred to me that there is an interesting difficulty for conjoining this theory with the consciousness-causes-collapse interpretation of quantum mechanics. Let t₂ be the earliest time at which it is correct to say that a collapse-causing consciousness state Q has happened, and suppose that Q does not occur at t₂ but over an interval of times ending at t₂. Then when does collapse happen? Suppose that collapse happens at a time t < t₂. Then we have a problem for consciousness-causes-collapse. For by time t there has yet to have been a conscious state. If God were to annihilate the universe right after t₂, there would be no conscious state, and yet the collapse would presumably have already occurred. So the collapse wasn’t caused by consciousness—unless there is backwards causation, which is counterintuitive.

So collapse can only happen at a time t ≥ t₂. If collapse happens at a time t > t₂, then either there is backwards causation or else the conscious state cannot count as an observation of the collapsed state, since the collapsed state is occurring after the conscious state. Again dismissing backwards causation, and assuming that the conscious states that cause collapse are observations, it follows that the collapse must occur precisely at t = t₂.

But now we have something weird: The bulk of the conscious state that causes collapse occurs before t₂. Yet only what is happening at t₂ can be caused by the collapse. So the very last moment of the temporally-extended conscious state has to be what makes the difference as to the qualitative content of the conscious state—say, whether it is a consciousness of a red light or a green light. That’s a bit strange, but not impossible.

Superpositions of conscious states

Consider this thesis:

1. Reality is never in a superposition of two states that differ with respect to what, if anything, observers are conscious of.

This is one of the motivators for collapse interpretations of quantum mechanics. Now, suppose that S is an observable that describes some facet of conscious experience. Then according to (1), reality is always in some eigenstate of S.

Suppose that at the beginning t₀ of some interval I of times, reality is in eigenstate ψ₀. Now, suppose that collapse does not occur during I. By continuity considerations, then, over I reality cannot evolve to a state orthogonal to ψ₀ without passing through a state that is a superposition of ψ₀ and something else. In other words, over a collapse-free interval of time, the conscious experience that is described by S cannot change if (1) is true.

What if collapse happens? That doesn’t seem to help. There are two plausible options. Either collapses are temporally discrete or temporally dense. If they are temporally dense, then by the quantum Zeno effect with probability one we have no change with respect to S. If they are temporally discrete, then suppose that t₁ is the first time after t₀ at which collapse causes the system to enter a state ψ₁ orthogonal to ψ₀. But for collapse to be able to do that, the state would have had to have assigned some weight to ψ₁ prior to the collapse, while yet assigning some weight to ψ₀, and that would violate (1).

(There might also be some messy story where there are some temporally dense and some temporally isolated collapse. I haven’t figured out exactly what to say about that, other than that it is in danger of being ad hoc.)

So, whether collapse happens or not, it seems that (1) implies that there is no change with respect to conscious experience. But clearly the universe changes with respect to conscious experience. So, it seems we need to reject (1). And this rejection seems to force us into some kind of weird many-worlds interpretation on which we have superpositions of incompatible experiences.

There are, however, at least two places where this argument can be attacked.

First, the thesis that conscious experience is described by observables understood (implicitly) as Hermitian operators can be questioned. Instead, one might think that conscious states correspond to subsets of the Hilbert space, subsets that may not even be linear subspaces.

Second, one might say that (1) is false, but nothing weird happens. We get weirdness from the denial of (1) if we think that a superposition of, say, seeing a square and seeing a circle is some weird state that has a seeing-a-square aspect and a seeing-a-circle aspect (this is weird in different ways depending on whether you take a multiverse interpretation). But we need not think that. We need not think that if a quantum state ψ₁ corresponds to an experience E₁ and a state ψ₂ corresponds to an experience E₂, then ψ = a₁ψ₁ + a₂ψ₂ corresponds to some weird mix of E₁ and E₂. Perhaps the correspondence between physical and mental states in this case goes like this:

1. when |a₁| ≫ |a₂|, the state ψ still gives rise to E₁

2. when |a₁| ≪ |a₂|, the state ψ gives rise to E₂

3. when a₁ and a₂ are similar in magnitude, the state ψ gives rise to no conscious experience at all (or gives rise to some other experience, perhaps one related to E₁ and E₂, or perhaps one that is entirely unrelated).

After all, we know very little about which conscious states are correlated with which physical states. So, it could be that there is always a definite conscious state in the universe. I suppose, though, that this approach also ends up denying that we should think of conscious states as corresponding in the most natural way to the eigenvectors of a Hermitian operator.

Collapse and the continuity of consciousness

One version of the quantum Zeno effect is that if you collapse a system’s wavefunction with respect to a measurement often enough, the measurement is not going to change.

Thus, if observation causes collapse, and you look at a pot of water on the stove often enough, it won’t boil. In particular, if you are continuously (or just at a dense set of times) observing the pot of water, then it won’t boil.

But of course watched pots do boil. Hence:

• If observation causes collapse, consciousness is not temporally continuous (or temporally dense).

And the conclusion is what we would expect if causal finitism were true. :-)

A Copenhagen interpretation of classical mechanics

One can always take an indeterministic theory and turn it deterministic in some way or other while preserving empirical predictions. Bohmian mechanics is an example of doing that with quantum mechanics. It's mildly interesting that one can go the other way: take a deterministic theory and turn it indeterministic. I'm going to sketch how to do that.

Suppose we have classical physics with phase space S and a time evolution operator T_t. If the theory is formulated in terms of a constant finite number n of particles, then S will be a 6n-dimensional vector space (three position and three momentum variables for each particle). The time evolution operator takes a point in phrase space and says where the system will be after time t elapses if it starts at that point. I will assume that there is a beginning to time at time zero. The normal story then is that physical reality is modeled by a trajectory function s from times to points of S, such that T_t(s(u))=s(u+t).

Our indeterministic theory will instead say that physical reality is modeled by a (continuous) sequence of probability measures P_t on the phase space S for times t≥0. These probability measures should be thought of as something like a physical field, akin to the wavefunction of quantum mechanics--they represent physical reality, and not just our state of knowledge of it. Mirroring the consciousness-causes-collapse version of the Copenhagen interpretation of quantum mechanics, we now say this. If from time u (exclusive) to time t+u (inclusive) no observation of the system was made, then P_t+u(A)=P_t(T_u⁻¹[A]). I.e., the probability measure is just given by tracking forward by the time-evolution operator in that case.

On the other hand, suppose that at time t an observation is made. Assume that observations are binary, and correspond to measurable subsets of phase space. Intuitively, when we observe we are checking if reality is in some region A of phase space. (It's easy to generalize this to observations having any countable number of possible outcomes.) Suppose P_t^* is the value that P_t would have had there been no observation at t by the no-observation evolution rule. Then I suppose that with objective chance P_t^*(A) we observe A and with objective chance 1−P_t^*(A) we observe not-A, with the further supposition that if one of these numbers is zero, the corresponding observation physically cannot happen. Then the probability measure P_t equals the conditionalization of P_t^* on the observation that does in fact occur. In other words, if we observe A, then P_t(B)=P_t^*(B|A) and otherwise P_t(B)=P_t^*(B|not-A). And then the deterministic evolution continues as before until the next observation.

As far as I can see, this story generates the same empirical predictions as the original deterministic classical story. Also note that while in this story, collapse was triggered by observation, presumably one can also come up with stories on which collapse is triggered by some other kind of physical process.

So what? Well, here's one thought. Free will is (I and others have argued) incompatible with determinism. One thought experiment that people have raised is this. If you think free will incompatible with determinism, and suddenly the best physics turned out to deterministic, what would you do? Would you deny free will? Or would you become a compatibilist? Well, the above example shows that there is a third option: give an indeterministic but empirically adequate reinterpretation of the physics. (Well, to be honest, this might not entirely solve the problem. For it might be, depending on how the details work out, that past observations narrow down the options for brain states so much that they become deterministic. But at least there would be hope that one wouldn't need to give up on libertarianism.)

The above way of making free will compatible with physical determinism is functionally similar to Kant's idea that our free choices affect the initial conditions of the universe, but without the freaky backwards-like (not exactly backwards, since the noumenal isn't in time) causation.

Here's another thought. Any indeterministic theory can be reinterpreted as a deterministic multiverse theory with traveling minds, while maintaining empirical adequacy. The multiverse traveling minds theory allows for causal closure of a deterministic physics together with robust alternate-possibilities freedom. Combining the two reinterpretations, we could in principle start with a deterministic physics, then reinterpret it in a Copenhagen way, and then impose on top of that the traveling minds interpretation, thereby gaining an empirical equivalent theory with robust alternate-possibilities freedom and no mental-to-physical causation. I bet a lot of people thought this can't be done.

What happens when collapse doesn't happen in collapse theories?

On collapse theories like GRW, Quantum Mechanics proceeds deterministically according to the Schroedinger Equation until a random "hitting" event occurs, when collapse occurs. There is a frequency parameter f that controls how often hitting events tend to happen. They tend to happen much more frequently when there are large amounts of matter involved than when there are small.

Nonetheless, the hitting events are random. Thus the physics of collapse theories implies that it is physically possible, with a non-zero (but presumably tiny) probability, that no hitting event happen in the universe over the next year, and hence no collapse happens over a year. Since this is physically possible, it should make sense to ask: What would it be like if this happened? Indeed, if we live in an infinite multiverse governed by a collapse theory with the same frequency everywhere, we can be confident that such no-collapse years do occur.

So what would it be like if no collapse occurred? I can think of three plausible proposals:

• Nothing: A (nomic or metaphysical) precondition of consciousness is a brain in a pure, or at least close to pure, quantum state, so in a no-collapse year, once everybody's brain states came to have a sufficient superposition of states (from the preferred basis), nobody would be conscious. However, after that year passed, there would be a collapse, and there would be false but plausible memories corresponding to the outcome of the collapse.
• Everett: For that no-collapse year, we would be living as if in a branching Everett-style multiverse. Either we would be experiencing different things in different branches, or we would have counterparts in different branches experiencing different things. Then with the collapse at the end of the year, all the branches but one would disappear.
• Weird: We would be having strange superposed experiences, perhaps quite unlike anything we can imagine. We would have superposed neural memory states. Then at the end of the year, when collapse occurred, our memories would also collapse, and we would end up with an ordinary set of memories corresponding to one component of the superpositions.

Here's one curious feature of all three proposals: At the end of the year, we would be back to business as usual, seemingly with normal memories of the past year. We would have no way of telling after the fact that we had a year with no collapse. On the Nothing proposal, we would have no way of telling during the no-collapse year, either, since we wouldn't be conscious during it. On the Everett proposal, some of our counterparts or branched selves would be having strange, improbable experiences. On the Weird proposal, we would be having strange experiences, but then we would have no memory of them.

If we take the Everett proposal, then the GRW theorist does not avoid the metaphysical oddness of persons in a branching multiverse—her only special contribution is to say that this oddness is unlikely to occur. If we take the Weird proposal, then the collapse theorist still has to deal with the metaphysical and psychological oddness of Schroedinger's cat phenomena—again, her only contribution is to say that such phenomena are unlikely. If these difficulties are really serious metaphysical problems, then the GRW theorist does not avoid them. A die that turns into a square circle once it rolls a million heads is not any less metaphysically problematic than a plain and simple square circle.

I suspect the Nothing approach is the best one for the GRW theorist. For instance, Nothing combined with compact-support-collapse helps with the infamous tails problem for collapse theories (see this for a nice discussion of the tails problem; alas, my suggestion doesn't help with the relativistic problems the author points out). For maybe we are conscious only at those instants when the wavefunctions have compact support. This is good reason to opt for the Nothing proposal if one has a collapse theory.

But the Nothing approach leads to a strange sceptical hypothesis, namely that I have not been conscious over the past week, notwithstanding apparent memories from yesterday. For remember that the collapse theories have a free parameter, f, which governs the frequency of collapses. If that parameter is low enough, then collapses will be rare, say once a month in my vicinity. And what reason do I have on the Nothing proposal to suppose that f isn't that low? The apparent memories of continuous past consciousness are exactly what I would expect with a low parameter, since the apparent memories are induced by the collapse of superposed neural states. We do have some constraints on f. For instance, f had better not be so low that it's surprising why anybody is ever conscious. Maybe there are some stronger constraints than that, though this is not clear to me. But there is no reason given the Nothing proposal to deny a value of f that yields once-per-month collapses.

The Everett proposal may well lead to a sceptical worry about low values of f as well. For how do we know that we're not right now in a no-collapse period?

The Weird proposal does not lead to this sceptical worry that f might be low. For on the Weird proposal, given a low f it's surprising that my current conscious state is non-weird, and so that's evidence against Weird plus a very low value of f. But Weird is weird.

The above sceptical worries about low values of f are ameliorated if in addition to being collapse theorists we are theists. For God likely wouldn't want us to have too many misleading memories, and hence would likely make f high enough to prevent misleading memories.

When a non-observation is an observation

Consider a standard Stern-Gerlach setup. An electron with mixed up/down spin is sent through a magnetic field. Then there are electron detectors that detect whether it went up or down.

Now, the up detector is connected to a very loud bell. The down detector is connected to a dim light. On the standard consciousness-causes-collapse (ccc) theory, if there is an observer who can both hear and see, she collapses the wavefunction, with probabilities given by the Born rule.

But now suppose that our observer has fallen asleep. The bell would wake the observer. The light wouldn't. What happens?

I think that if we accept ccc, we should also accept that the wavefunction collapses in this case. Thus, sometimes the wavefunction collapses in favor of up and a bell, and sometimes it collapses in favor of down and a light. In the latter case, there is no observation made—the observer is unconscious. Thus, on this solution, while an unconscious observer is capable of collapsing a wavefunction.

Perhaps one disagrees that the wavefunction collapses here. Then the observer in the lab is in a superposition of awake and asleep states. This by itself seems unacceptable. It seems that the whole point of ccc was to ensure that we did not have to worry about the weirdness of superpositions between different conscious states. But a superposition between a conscious and a non-conscious state is just as weird. Moreover, supposing that our observer is in a superposition of awake and asleep states, we can imagine a second observer coming into the lab. As soon as the second observer notes whether the first is asleep, we will have collapse. If that happens, then suddenly the first observer comes to be in a pure state. Some of the time, that pure state will be one of remembering being woken up by a bell. But that memory is false: she was never woken up by a bell, but was in a superposition of woken and non-woken states. So this interpretation leads to us having to attribute false memories to observers. And we should avoid that.

This line of thought suggests that we should treat non-observation as a kind of observation. Collapse happens whether the bell is observed or not. Collapse occurs always to exclude superpositions between different observational states, where non-observation counts as an observational state.

Suppose we do this outside of quantum mechanics? Well, this could have some implications for Sleeping Beauty. It would suggest that the Sleeping Beauty problem where one is woken on Monday and Tuesday on tails (with amnesia induced in between) and only on Monday on heads is equivalent to the variant where on heads one is also woken on Tuesday and informed that there was heads. For the non-observation on heads on Tuesday in the first problem should be thought of as a kind of observation. I think thirding is quite plausible on the variant on standard Bayesian grounds, so this supports thirding in the original.

We also get a variant of that lighthearted answer to verificationist worries about Christianity. The lighthearted answer is that whether Christianity is true is verifiable. Just wait until you're dead, and you'll see. Well, not quite, goes the riposte: you will see if there is an afterlife, but if there is no afterlife, you won't. But if we treat non-observation as a kind of observation, then you in effect do "observe"—through genuine observation if there is an afterlife and without one otherwise.

How far do we take the principle that non-observation can count as observation? Do non-observations by non-existent persons count? Quite possibly. Suppose that a lab apparatus is set up so that a conscious being is produced when the up electron detector is triggered and not otherwise. Similar reasoning to the above suggests that we should have collapse here, even if no conscious being is produced. The mere possibility of producing one triggers collapse.

Of course, ccc may be false.