It is like _any_ paradox in terms of contradiction;
much in the same sense that 'solving' one NP-complete
problem, will 'solve' the rest.
Paradoxes form a class and any solution on one paradox
gives a clue for solving others simultaneously and
instantaneously.
> either A and B are lacking in
> concrete meaning, or the "givens" are themselves false. You are
> missing whatever part it has that makes it really paradoxical, if any.
>
> Quantum-mechanically, a particle can have a state such that "A has
> spin up" is neither true nor false, but subject to a probability
> distribution. But once A is observed, if B is observed later, B may
> have its own probability distribution, or it may correlate with A in
> some fashion. But it can't, after observation, be both spin up and
> spin down, either.
You're thrashing here abit.
There is no 'probability Distribution' (PD) after the state
is measured. It's only active while everything is dynamic
and not measured and in a very large sense it is only an
abstraction during that interlude between measurements.
A histogram or barchart is a set of possible states with relative
frequencies attached to each state, but as such it is not
interpreted probabilistically. It is just a bunch of
positive amplitudes distributed over the _space_ of states.
If we interpret this histogram or bar chart probabilistically,
then we get a "probability _distribution_". If we Fourier
transform this probabilistically interpreted histogram
or space-like _distribution_ (spectrum) into
the time-like domain, we get a "probability density _function_" (PDF)
or "wavefunction". This is just a time-like Function with a
probabilistic interpretation just as its complementary
space-like Distribution was given a probabilistic interpretation.
The Fourier transform (or more generally, an orthonormal transform):
turns functions into distributions, and vice versa.
Both the PD and the PDF "collapse" when _a single_ measurement
is made into _a single_ state of all the possible states.
You can take that single state measured and add it into
your PD (or its corresponding PDF) to increase its
forecasting ability in future measurements.
_Empirically_, we never really know for sure how large a
state space is, but experimentation can indicate it's size
probabilistically speaking. This empircal and so this
non-deterministic approach leaves open the possibilities of
measuring a state that was never before considered part of the
state space (a hidden variable).
_Theoretically_, we might try to do better and deterministically
define the state space size. But quantum mechanics does not use
this approach (as Einstein was wanting to say to Bohr)
So, measuring a single spin-up particle collapses
both its PD in the space-like domain and its PDF in the
time-like domain and all the other states are then "false"
(in this case there is only one other state in the binary
state-space of up and down spins, so that the spin-down
state is instantly "false" when the spin-up state is
measured as "true")
This is the usual consideration for superpositions of
states (or phasors in state space...) and their corresponding
wavefunction phases in time within the time-domain.
But, entanglement is an additional problem when you consider
not just the state-space of a single particle, but the
state space of two particles that interacted and so their
PD's and PDF's have some memory of that event as if they
were two bell's (or impulse response functions[1]) that
once clanged together and when separated, they maintained
a "memory" of that event in their separate sets of PDs and PDFs.
Those separate memorys are what allow the two particles
to be non-locally correlated, or "entangled".
Those memories however tend fade away (decohere) after a while.
But they should be maintainable, by a _local_ resonant
communications between the entangled particles.
Of what use that may be to quantum cryptography &c.,
I am not concerned with, as I think there are more significant
implications than that.
[1] have a look at the "perturbation" methods
and "Green's" functions in this context.
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