You can make an observation of a system whose state is undetermined. By interrogating the system for its state, a state becomes determined. Suppose, for example, that you have a flipped coin and sent it rotating in space, never hitting the floor. Is it heads or is it tails? Until it is looked at, the question doesn't really make sense. And for this case, we'll define "looking at it" to mean sticking out your hand and catching it. There is a probability that it's landed heads up in your hand, and a complimentary probability that it's landed tails up. Once it's in your hand though, you can confidently say which state it's in.
Now, you could definitely take issue with this example, because you could argue that the rotation of the coin is well described, so with initial conditions, you can predict its position at any given moment. But imagine a microscopic quantum system, and, for the sake of this simple explanation, believe that its "rotating through the air" state really does not have any precise heads or tails definition. Until something gets in the way of that system, creating an interaction that exchanges information about its observable state, it's not meaningful to say that it's in one of the observable states at all.
A superpositition of states, as such, is essentially the representation of a state in terms of a basis set of observables. In the case of the coin, heads and tails are the two observable states, they are orthogonal, and they fully represent the state space of the coin. You could flip the coin, and put its state vector into the form of sqrt(2)/2 * Heads + sqrt(2)/2 * Tails. This state isn't observable, but it can be described in terms of observable components, where the coefficients represent the probabilities that a given observable state will be measured upon observation.
> You can make an observation of a system whose state is undetermined. By interrogating the system for its state, a state becomes determined.
For QM, this is not correct, although it's a common misstatement. The correct statement is this: you can make an observation of a system which is not in an eigenstate of the measurement operator you are using. After the measurement, the system is now in an eigenstate of the measurement operator--i.e., the act of measurement changes the state.
Note that this is only true on a collapse interpretation, like Copenhagen. On a no-collapse interpretation, like MWI, the "observation" is just an interaction that entangles the state of the measuring device with the state of the system being measured--it's all just unitary evolution.
> You could flip the coin, and put its state vector into the form of sqrt(2)/2 Heads + sqrt(2)/2 * Tails. This state isn't observable*
Yes, it is; but it isn't observable by a simple method like looking to see if the coin is heads or tails. But according to QM, every state is an eigenstate of some operator, so there will be some observation that will distinguish sqrt(2)/2 * Heads + sqrt(2)/2 * Tails from the state that is exactly orthogonal to it, which is sqrt(2)/2 * Heads - sqrt(2)/2 * Tails.
>The correct statement is this: you can make an observation of a system which is not in an eigenstate of the measurement operator you are using.
I should have distinguished better, but what you're more rigorously calling an eigenstate of a measurement operator, I'm calling an observable state. There is something lost in translation to an audience unfamiliar with terms like eigenstate, but that was my attempt. Would you suggest a better one?
>Note that this is only true on a collapse interpretation, like Copenhagen. On a no-collapse interpretation, like MWI, the "observation" is just an interaction that entangles the state of the measuring device with the state of the system being measured
The greater point being addressed is that MWI is no more deterministic than Copenhagen.
>Yes, it is; but it isn't observable by a simple method like looking to see if the coin is heads or tails. But according to QM, every state is an eigenstate of some operator
Some hermitian operator? But more to the point, if looking at the coin is the only operator at our disposal in the simple example, then its eigenstates are the ones we care about.
> what you're more rigorously calling an eigenstate of a measurement operator, I'm calling an observable state.
Yes, but "observable" here is relative to the measurement you are making. If you make a different measurement (i.e., realize a different operator), then the set of "observable states" by your definition is different, because the set of eigenstates of the operator is different.
> The greater point being addressed is that MWI is no more deterministic than Copenhagen.
But this isn't true. The MWI is completely deterministic, because wave function collapse never occurs, and wave function collapse is the source of all the indeterminism in the Copenhagen interpretation.
> Some hermitian operator?
Yes.
> if looking at the coin is the only operator at our disposal in the simple example, then its eigenstates are the ones we care about.
If all you're interested in is that particular experiment, yes. But here we're discussing claims that must apply to all possible experiments and all possible measurements, not just the particular one in the example you chose. So we have to consider all possible operators and all possible sets of eigenstates, not just the ones in your example.
>But this isn't true. The MWI is completely deterministic, because wave function collapse never occurs, and wave function collapse is the source of all the indeterminism in the Copenhagen interpretation.
For what useful definition of deterministic? If a measurement comes with decoherence into multiple non-inteferring branches, then certainly the state evolves in a predictable way from "god's eye", but not from the perspective of the experiment occupying any given branch.
The definition that says the future state is entirely determined by the present state. That's the only definition I'm aware of.
>the state evolves in a predictable way from "god's eye", not from the perspective of the experiment occupying any given branch.
The entire "god's eye" state is the one that appears in the dynamical laws of QM (unitary evolution), so that's the one that's relevant for assessing determinism.
> the state evolves in a predictable way from "god's eye", but not from the perspective of the experiment occupying any given branch.
This "apparent randomness" of measurement results is equally true of chaotic classical systems; it's not something that only appears in QM. Basically, it's just a consequence of the fact that individual "observers" will in general not have complete knowledge of the state. That doesn't mean the state doesn't evolve deterministically; it just means the observers don't have complete knowledge.
>This "apparent randomness" of measurement results is equally true of chaotic classical systems; it's not something that only appears in QM.
Is that a fair comparison? Yes, in either case, the experimenter is limited in his predictive capability by the information available to him. But in a chaotic system, your predictive power can be improved arbitrarily by surveying more information with greater precision. As I understand it-- and hopefully you can clarify if this is accurate-- decoherence forbids a measurement from receiving information from a branched outcome, so even if you take a measurement with arbitrary access to information now and repeat the same measurement in the future, there becomes a set of information that is fundamentally off limits to the observer in a given branch.
I think so. Perhaps it will help if you look at it this way: you repeat some measurement multiple times, and get a sequence of results that looks random. Is the randomness because of classical chaos, or because of quantum "indeterminacy"? From the measurement results themselves, in many cases, there will be no way to tell. The only case in which there would be a way to tell would be if you specifically made measurements on entangled quantum systems in order to test the Bell inequalities; if those inequalities are violated, the measurements can't be due to classical chaos. But that just underscores my point: looking at "apparent randomness" of measurement results is not sufficient to tell whether they are due to "quantum indeterminacy.
> decoherence forbids a measurement from receiving information from a branched outcome
Once again, this is a misleading way of stating it. What is happening, again, is that the observer evolves into a superposition, corresponding to the superposition that the measured system is in. Decoherence just means the branches of the superposition don't interfere with each other. But the system is still in a single state; the "branches" are not separate states or separate entities, they're parts of a superposition.
(Note, also, that decoherence does not guarantee that the different branches will never interfere with each other. Decoherence is not a fundamental limitation; it's just a recognition of what happens in the usual case, where no special measures are taken to isolate the system or to facilitate interference. According to the MWI, there is in principle always a way to cause the different branches to interfere, i.e., decoherence is never absolute.)
> there becomes a set of information that is fundamentally off limits to the observer in a given branch
According to the MWI, the observers in different branches are not different observers; they're different terms in a superposition that the observer is in. Thinking of them as "different observers" with access to different information implicitly assumes something like the Copenhagen interpretation.
