...Those are completely different statements. The one logically follows from the other, yes - since an inability to clone a state combined with "destructive reads" logically implies an unknown state cannot be fully determined - but they're not two ways of saying the same thing at all.
When we specify the initial state of a quantum system, we do so by some measurement that disturbs it. Having done that, we then do so again. Why does this matter? Because the only way we can now say something about the system is because we have built-in a theoretical and statistical framework such that we can call "preparing" a system something it is not. We don't prepare a system, we set up an experiment and describe the system according to
a priori theory, not actual observations or knowledge of that system.
Which means that if we want to "
uniquely determine an unknown quantum state of an individual system by means of measurements performed on that system only", we can't do so. We have to introduce statistically derived mechanisms which go into both how we describe the system and measure observables. Because without these, once we prepare a system don't have the means we use to obtain observables.
Because the Born rule says so. Why is the Born rule there? I don't know. I don't think anyone does.
Everybody knows why: It worked. I don't think anybody knows why it works, but there are reasons why they know it work that include how he derived it in the first place and the assumptions that went into it.
I don't know what you mean by the "can" in that sentence. We can do algebra in the Hilbert space without any notion of probability.
italics in original, emphases added:
"the description of the state and its evolution
does not constitute the entire quantum formalism.
The wave function provides only a formal description and does not by itself make contact with the properties of the system. Using
only the wave function and its evolution,
we cannot make predictions about the typical systems in which we are interested, such as the electrons in an atom, the conduction electrons of a metal, and photons of light. The connection of the wave function to any physical properties is made through the rules of measurement"
We can do algebra all day long. If I have an algebraic model of a line, I can use a more advanced version to describe everything from climate dynamics to the ways in which human cognition routinely fails us. Or I could sit around plugging random numbers into some regression model.
I can "do the algebra" billions of times, but what's the point? The point is to describe a "something" that has specific values when transcribed. How are these values obtained?
So measure it and then track the evolution from the resulting eigenstate.
If I "measure" it, then this process is pointless. The entire thing is dead on arrival. The preparation of the system that allows us to get an initial state
to track can't be the measurement because then we have no initial state.
What sort of thing goes in the gap?
That depends upon lab specific notations and specific equipment. Basically, the preparation process is a "measurement", but we can't call it that or we have no initial state (the no-cloning theorem relates to this), which is why reduplication is impossible except under certain logical assumptions relating repeated measurements/results into a probabilistic formalism which allows us to take specific experimental designs combined with QM and call something a system such that when we measure it we will obtain particular values that relate to the system.
I'm not sure what that has to do with what I meant - we know the mass, charge and other properties of the things we're trying to measure
We don't (but we're pretty good with modern ion traps and such to get very close to predicted values). And most of the time they're the same thing just looked at differently:
"Six quarks, six leptons, together with the gluons of QCD and the photon and weak bosons, are enough to describe the tangible world and more, with remarkable economy"
Cahn, R. N., & Goldhaber, G. (2009).
The experimental foundations of particle physics. (2nd ed.). Cambridge: Cambridge University Press
But what about electrons and muons and fermions and leptons and all the other guys? And why can't we use one quark instead of 6? Because most of the particles we talk about are just different ways of saying something else.
But our descriptions are theories!
No, they aren't. Because the descriptions relates the experimental set-up to a theoretical and statistical framework so that we can have a system we never observed in an initial state or any state and end up with a final state.
Again, I'm not sure what you mean or how this differs from classical physics. There's no subjective criteria involved in measuring a quantum system, ala psychology.
There absolutely subjective criteria for measuring it without psychology because it is uniquely determined in terms of specific characteristics that were never observed but will be related to specific measurements.
A Hilbert space vector is an object
Fantastic. I can give you a vector in Euclidian space which is an object corresponding to the word "Hilbert" in which each component is assigned a value based upon the ordinal position of the letter. But what's the point?
What are the values in the vector and how are they obtained?
How we can we assign a 'where' to a space, except by sticking it inside a bigger one?
The null space. Or, alternatively, any by having the "where" occupy the entire dimensional space.
Is there a physical significance to an experiment concluding?
Of course.
You seem to be suggesting that the system somehow transition from Hilbert space to "real" space, which is never does.
Hilbert space is just a name for an abstract mathematical space with certain properties. It's as "real" as addition or a billionth dimensional Euclidean space. But because a wave function is a
function, we require it to obey particular rules. A Hilbert space has these.
Why, when the position operator is what controls dimensionality of the measurement?
Because in general terms the projection postulate is an central distinguishing feature of
different QM interpretations, as well as the ways in which the abstract system is projected (or not, in which case something else is required to explain the experimental results) such that we have measurements of actual values.
Why not? What precisely does that mean? That I will get different results than you, because I am not you? Or because I built the apparatus differently?
Depends on how you want to conceive it. But it is easiest to frame it in terms of the experimental procedure and what that entails: "preparing" a system such that it has "values" which, but as this process isn't actually related to the system, quantum indeterminacy ensures that the same experimental process repeated exactly in same way will by no means ensure that the results is at all the same.
There isn't any probability of the system interacting with the slits?
Of course there is. Hence the reasoning behind the path integral.
MWI doesn't prohibit the Born rule, it just doesn't include it, which is indeed a problem
It does include it (or some version of it). There is no way to obtain any information about any experiment ever without this or something like it, which is the basis for the
ad hoc criticisms of MWIs.
However, AFAIK, there's no derivation of the Born rule inside Copenhagen either.
He won a nobel prize for this derivation.
