The Measurement Problem

Alright, let's talk about the measurement problem! This is defined by different thinkers in different ways, so it isn't always clear how we should talk about it.

This is how I have chosen to state the measurement problem, and it is admittedly arbitrary. Every interpretation chooses a different gripe to have about the measurement problem, and will likely formulate the problem in a way that makes their gripe, and subsequent solution, most obvious.

I don't aim to avoid bias in my formulation, I just want to define the problem in the biggest possible strokes that make sense to me. That is to say, I want a formulation of the measurement problem that most clearly frames the issues being debated and allows interpretations to be categorized and their relationships understood quickly.

The Measurement Problem

Some variables:
Let's call the observing mind of the scientist "O",
the macroscopic-sized measuring apparatus "A",
and the quantum state of the system being studied "Q".

The measurement problem could then be stated as three propositions which cannot all be true. One of these must be false.

1. Under the right conditions, A can be in the same quantum state Q as a microscopic-sized system; the size of the apparatus does not matter. Q can be a superposition state.
2. O directly perceives the world. This means that if it perceives the state of A, it perceives its state as exactly what it is.
3. A superposition state represents multiple realities, or a reality that is indeterminate between multiple realities. Minds cannot exist in multiple realities, so O cannot perceive or otherwise interact with A if it is in state Q.

This particular way of formulating the measurement problem has the strength of relative generality, but has the flaw of being fairly abstract and difficult to grasp without some explanation. So, let's look at each part more closely.

'A can be in state Q'

This is the least controversial of the three. Basically, it says that the laws of quantum mechanics are true in the scientific sense. It is the one claim that is a posteriori, meaning it can be verified empirically. Specifically, this means two things: 

a) Large macroscopic sized objects can follow quantum laws under the right conditions.

This is a point that could be brought under debate. Could it be that a relatively large-sized object like a cat actually cannot be brought to a quantum state so that it would behave as quantum mechanics would predict? Or is there some sort of physical effect that prevents that from happening? Some interpretations argue that there is an effect like this.

b) Quantum states can be superpositions.

This is a fundamental part of quantum mechanics. It would be difficult for an interpretation to deny this. Since superposition states are so effective at explaining quantum phenomena, denying this would mean completely denying that quantum mechanics is true in even an approximate sense and that we need to start from scratch.

Accepting this statement does not mean that one believes quantum superposition states must be real things that exist in the world, which is a much more controversial statement. Nor does it mean that a quantum state must be interpreted as a literal physical state. Many interpretations would argue that quantum states should not be considered real physical states because there is some better notion of state out there that we have not yet discovered. These interpretations would still have to admit that the notion of quantum states and the notion that those quantum states can be in a superposition state, however those two things are to be understood, are essential in empirically successful quantum mechanics.

Like I said, this statement is the least controversial.

'O directly perceives the world'

This is a metaphysical statement. Essentially, this statement boils down to whether or not we perceive reality as it really is, or if reality is there is an aspect of reality we fundamentally cannot perceive.

The famous philosopher Immanual Kant introduced the notion of transcendent that is used in modern philosophy. It refers to something that is essentially beyond human experience. Does an interpretation allow for a transcendent reality? If so, that interpretation would deny this statement.

This is important to include because it establishes whether or not it would allow for there to be some kind of "hidden reality" of the metaphysical kind. For example, the multi-universe interpretation allows for possible worlds that are beyond our perception because we are trapped in our own possible universe.

I would think that many would find no problem in denying this proposition. Some interpretations accept this statement, and some don't, and it is a good proposition to use to divide and begin to categorize interpretations.

'Superpositions represent multiple realities'

This is where the problem comes into full form. Superpositions are the linear combination of two other quantum states. That is to say, add two quantum states and you get a superposition. Since superpositions are just this kind of mathematical combination of two other states, it seems natural to assume that they should be interpreted as representing both states in some way. 

Perhaps it is an indeterminate state that ceases to become one reality until collapsed. Perhaps it is a dual existence, one that represents both realities existing at the same time until it is required to be one or the other. Whatever the case, this statement says that it cannot represent just one determinate state.

And because of this dual existence, this is a problem for how we should understand what perception of a superposition should look like. Excluding weird and unhelpful ideas, we don't typically think of our minds as existing in multiple realities. Since this is the case, these extra realities have to be shed somewhere between the physical superposition and our mind's perception of it.

Many would lump the notion of superpositions representing multiple realities in with the baggage associated with quantum mechanics. I don't do this because it is useful to separate this a priori claim from the a posteriori claims of the first statement. It's a statement about how to understand the significance of the superposed quantum state, not a matter of the predictions made using it, though the latter can inform the former.

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So that is how I like to phrase the measurement problem. It is simple enough to grasp, while still showcasing many of the concerns interpretations try to address.

Quantum Superposition Paradox

Superposition is a weird thing, and there's definitely disagreement over how to think about it. Mostly, there is disagreement over whether or not it literally represents multiple realities. That is to say, when Schrödinger's cat is in a superposition, is it both alive and dead at the same time?

It's hard to talk about that without getting into metaphysical debates (see my entry on the measurement problem). While metaphysics is good to get into, it'd be nice to avoid that, at least initially.

So maybe a good starting place for talking about the interpretation of superpositions are logical sentences about them.

Multiple Realities

For example, if it's true that 'Schrödinger's cat is in a superposition state', does that logically imply the statement 'the cat is alive' as well as the statement 'the cat is dead'. More symbolically, is it the case that

'Superposition' -> ('alive' & 'dead') ?

I'd wager that this view of superpositions is widely held. Most will probably think of superpositions as implying multiple realities in the very meaning of the term. This would be an a priori assertion, and it would agree with the phrasing above.

Opposed to this view would be the belief that there is no true logical relation that looks like the one above. One could argue, for example, that superpositions could possibly imply multiple realities, but that it would have to be decided empirically. This would mean that the above logical statement wouldn't always be true, and that it would depend on the circumstances.

See how much cleaner that is than normal? I don't have to talk about metaphysical things, but still get to articulate something providing some insight.

Contradictory Realities

The second half of this is what relationship is between the states of life and death. Do they contradict? If so, the statement

~('alive' & 'dead')

is directly contradictory with the superpositions statement above. In other words, one of these two statements must be false.

This second statement certainly seems natural to accept as well. We certainly think of life and death as being mutually exclusive states. But what could show that they really are mutually exclusive states, and not just compatible states that appear incompatible to our mistaken intuition?

More formally, maybe we can say that mutually exclusive pairs of physical states will create two conflicting set of predictions, both of which can't be true. For example, a macroscopic object can be both falling and electrically charged. There is no reason why these two physical states should make conflicting predictions, so a physical object can be in both states at the same time.

(I think the full story of why this is the case looks like this: since a system is said to be in a physical state only when it fits a physical model, and physical models are what we need to make predictions, having a contradiction in the predictions of two models applied to the same system suggests an error somewhere further up in reasoning. Either a model was applied that should not have, or there was an error in how one or more of them were applied. Feynman describes using this very line of thought to call out the existence of errors in other physicist's reasoning and impress others without working through all of the math. It's in Surely You Must Be Joking Mr. Feynman p.244-5. The fact that the statement could be wrong either because an incorrect model was applied or a correct model was incorrectly applied means that this second statement has an a priori sense to it and an empirical sense to it.)

Two mutually exclusive states would give predictions that contradict. For example, having a macroscopic object (not currently under quantum effects) be both charged and not charged would create contradictions. There would be two sets of predictions created by the two possible situations: when the object is charged and when it is not, and those predictions could differ significantly. Therefore, only one of them can be true, and a well designed experiment will typically decide between the two.

So are quantum states different? Do quantum effects somehow make states that are normally mutually exclusive so that they are not in the right circumstances?

An interpretation that would argue that this statement is false is the many-worlds interpretation. While classical states like 'alive' and 'dead' are mutually exclusive, in the quantum realm they are not. The simple fact for this being that they exist in separate worlds. It's not a contradiction to say that the cat is both dead and alive because the dead cat exists separately from the alive cat, and it is this world-splitting aspect of quantum mechanics that keeps these two states from producing contradictory predictions.

So yeah, I dunno how helpful this approach will be, or if there is any literature on breaking it down this way. I'm calling it the quantum superposition paradox, which seems appropriate, but I don't know if there would be a better name for it.