[time 1009] [Fwd: Simpson's Paradox and Quantum Entanglement]

Stephen Paul King (stephenk1@home.com)
Sat, 20 Nov 1999 14:36:53 -0500

Hi All,

        Robert Fung is making some great points!



attached mail follows:

   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

> 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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