| Topic: |
Science > Physics |
| User: |
"Edward Green" |
| Date: |
08 Dec 2006 10:33:30 AM |
| Object: |
Bell's inequality vs. Kolmogorov inequality |
One unexamined belief I had was that (the/a) Bell's inequality is
equivalent to (the/a) Kolmogorov inequality -- an equivalence I hoped
to more fully explore one century or the other. (I heard this on a
street corner.) That century has come.
I also had the ancillary unexamined belief that (the/a) Bell's
inequality was a statement about limits on the correlation structure of
a set of jointly distributed random variables, and so, I expected, were
(the/a) Kolmogorov inequalities. However, at...
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BellsTheorem/BellsTheorem.html
we find a description of "Bell's inequality" as a statement essentially
about counts, or the areas of Venn diagrams; which to be sure, is a
statement about event probabilities, but not obviously about
correlations. OTOH, at...
http://en.wikipedia.org/wiki/Kolmogorov's_inequality
we have a statement of Kolmogorov's inequality describing a bound on
the cumulative distribution function of the maximum over k trials of an
iid (independent identically distributed) random variable:
having something to do with probability, but not obviously with
correlations.
Can anyone help disambiguate my thinking?
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| User: "Sorcerer" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 02:47:33 PM |
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"Edward Green" <spamspamspam3@netzero.com> wrote in message =
news:1165595610.141449.267490@80g2000cwy.googlegroups.com...
|=20
<...>
| Can anyone help disambiguate my thinking?
|
There is no cure, but you might get relief of symptoms by
a) psychiatric help
b) alcohol
c) illegal drugs (not recommended)
I suggest you try not to divide by zero, the seat of the problem,
http://www.androcles01.pwp.blueyonder.co.uk/DominoEffect.GIF
then you need not write a pageful of ***** bemoaning your dilemma.
Doubtless this advice will fall on deaf ears.
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| User: "" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 12:24:33 PM |
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Ed, realize that very few physicists give a widow's mite about Bell's
Inequality, although for some unknown reason it appears to be a subject
of great interest to math majors. So, perhaps your post would be better
suited to one of the math newsgroups.
If you notice a bit of hostility on this subject, it is because in
college my course in Advanced Calculus was severely compromized by my
misfortune to have a visiting professor from Harvard teaching the
course. He wasted the entire term focusing on formal proofs and things
like Bell's Inequality, at the expense of the many really useful thing
generally taught in advanced calculus.
After a lifetime in science and engineering, when I hear someone
pontificating of obscure things like Bell's Inequality and comparing it
to Kolmogorov's, my immediate impression is that I am in the presense
of a poster, trying to employ obscure and useless ***** to impress
those in his presence. Sorry if this offends anyone, but this Liberal
Arts form of mathematics simply doesn't pay the rent in physics.
Still, it could get you laid if in the company of of
pseudo-sophisticated, pseuo-intellectual shapely females to are half in
the bag already. :-) Sadly, most physicists are too in love with
their work to even consider this option. Then too, we do OK too, which
account's for all the little physicists running around. Never
underestimate the seductive powers of a Bongo Drum!!!!
Harry C.
Edward Green wrote:
One unexamined belief I had was that (the/a) Bell's inequality is
equivalent to (the/a) Kolmogorov inequality -- an equivalence I hoped
to more fully explore one century or the other. (I heard this on a
street corner.) That century has come.
I also had the ancillary unexamined belief that (the/a) Bell's
inequality was a statement about limits on the correlation structure of
a set of jointly distributed random variables, and so, I expected, were
(the/a) Kolmogorov inequalities. However, at...
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BellsTheorem/BellsTheorem.html
we find a description of "Bell's inequality" as a statement essentially
about counts, or the areas of Venn diagrams; which to be sure, is a
statement about event probabilities, but not obviously about
correlations. OTOH, at...
http://en.wikipedia.org/wiki/Kolmogorov's_inequality
we have a statement of Kolmogorov's inequality describing a bound on
the cumulative distribution function of the maximum over k trials of an
iid (independent identically distributed) random variable:
having something to do with probability, but not obviously with
correlations.
Can anyone help disambiguate my thinking?
.
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| User: "Edward Green" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 07:02:09 PM |
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wrote:
Ed, realize that very few physicists give a widow's mite about Bell's
Inequality, although for some unknown reason it appears to be a subject
of great interest to math majors. So, perhaps your post would be better
suited to one of the math newsgroups.
If you notice a bit of hostility on this subject, it is because in
college my course in Advanced Calculus was severely compromized by my
misfortune to have a visiting professor from Harvard teaching the
course. He wasted the entire term focusing on formal proofs and things
like Bell's Inequality, at the expense of the many really useful thing
generally taught in advanced calculus.
After a lifetime in science and engineering, when I hear someone
pontificating of obscure things like Bell's Inequality and comparing it
to Kolmogorov's, my immediate impression is that I am in the presense
of a poster, trying to employ obscure and useless ***** to impress
those in his presence.
I think you meant "poseur", or maybe "imposter". Interesting... put
them together, and you get "poster". Perfect!
Sorry if this offends anyone, but this Liberal
Arts form of mathematics simply doesn't pay the rent in physics.
Oddly, I am not offended, though I can't say why: maybe I can't really
think you mean anything personal. Let's just say I'm interesting in
the foundations of quantum mechanics, as kind of a hobby -- like
another man might be interested in solving chess problems. I don't
think I'm a poseur or an imposter; quite the opposite -- I'm an
anti-weirder. Instead of making things as weird as possible, I'm
interested in making them as simple as possible.
Still, it could get you laid if in the company of of
pseudo-sophisticated, pseuo-intellectual shapely females to are half in
the bag already. :-) Sadly, most physicists are too in love with
their work to even consider this option.
Feymann of course being a famous exception.
Then too, we do OK too, which
account's for all the little physicists running around. Never
underestimate the seductive powers of a Bongo Drum!!!!
As of course you know.
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| User: "Ben Rudiak-Gould" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 02:45:10 PM |
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Edward Green wrote:
However, at...
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BellsTheorem/BellsTheorem.html
we find a description of "Bell's inequality" as a statement essentially
about counts, or the areas of Venn diagrams; which to be sure, is a
statement about event probabilities, but not obviously about
correlations.
I don't think that what's described on that page is Bell's inequality at
all. The derivation explicitly assumes that A, B, and C are independently
measurable properties of the system, which is a much stronger assumption
than Bell made. Obviously a theory in which all measurements commute can't
agree with quantum mechanics, but people knew that long before Bell's paper.
-- Ben
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| User: "Sorcerer" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 03:15:38 PM |
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"Ben Rudiak-Gould" <br276deleteme@cam.ac.uk> wrote in message =
news:elcism$okg$1@gemini.csx.cam.ac.uk...
| Edward Green wrote:
| > However, at...
| >=20
| > =
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BellsTheorem/Bell=
sTheorem.html
| >=20
| > we find a description of "Bell's inequality" as a statement =
essentially
| > about counts, or the areas of Venn diagrams; which to be sure, is a
| > statement about event probabilities, but not obviously about
| > correlations.
|=20
| I don't think=20
No surprise there.
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| User: "Daryl McCullough" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 08:08:19 PM |
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Ben Rudiak-Gould says...
I don't think that what's described on that page is Bell's inequality at
all. The derivation explicitly assumes that A, B, and C are independently
measurable properties of the system, which is a much stronger assumption
than Bell made. Obviously a theory in which all measurements commute can't
agree with quantum mechanics, but people knew that long before Bell's paper.
I don't agree completely. Many people thought that the noncommutativity
of measurements were due to interference between the observer and the
thing being observed. That's one of the informal ways to motivate Heisenberg's
uncertainty principle: the act of measuring a particle's position very
accurately disturbs the particle, which means that there will be a large
uncertainty in the particle's momentum immediately afterwards. This
"disturbance" model of the uncertainty principle leaves open the possibility
that a particle may possess both a definite position and a definite
momentum at all times, but that it is impossible to measure them both
simultaneously. Bell's theorem rules out this possibility (under the
assumption that all influences are slower-than-light, together with
the assumption that probabilities of joint events satisfy the usual
laws of probability). Bell's theorem deals with noncommuting spins,
rather than spin versus momentum, but the same principle applies.
I really don't see how Bell's proof is any more general than
the "Venn Diagram" argument. The Venn Diagram argument seems
to me to rule out the same hidden-variables theories that Bell's
proof does.
--
Daryl McCullough
Ithaca, NY
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| User: "Edward Green" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 09:07:58 PM |
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Daryl McCullough wrote:
I really don't see how Bell's proof is any more general than
the "Venn Diagram" argument. The Venn Diagram argument seems
to me to rule out the same hidden-variables theories that Bell's
proof does.
Well, if the quoted object is indeed "Bell's Inequality", then I'd go
farther: the diagramatic argument is not simply "not less general", it
is a faithful graphic representation.
When I want to understand some theorem relating probabilities of events
and overlapping events, I invariably have recourse to some such
diagram: probability theory is after all based on measure theory, and
"area" is a good measure.
Here's a possibly even more "d'uh" producing version: label the
regions on the diagram p_1 through p_8, for the probabilities of the
given event overlaps. Then the "theorem" reduces to something like:
p_1 + p_2 + p_3 + p_4 >= p_1 + p_2
All together now: "d'uh"
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| User: "Ben Rudiak-Gould" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 10:15:20 PM |
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Edward Green wrote:
Well, if the quoted object is indeed "Bell's Inequality", then I'd go
farther: the diagramatic argument is not simply "not less general", it
is a faithful graphic representation.
If by the quoted object you mean
Number(A, not B) + Number (B, not C) >= Number (A, not C)
then that's not Bell's inequality. Bell's inequality (or what substitutes
for it here) is
Number(right up 0, left up 45) + Number(right up 45, left up 90)
= Number(right up 0, left up 90)
which (this is one of the parts that bothers me) is not obtained from the
first inequality by substituting things for A, B, and C. Instead he uses the
fact (or at least the claim) that "right up N" is equivalent to "not (left
up N)" as well as the fact that measurements of the right and left particles
commute with each other, which apparently makes the original derivation
applicable since you don't have to choose a measurement order. I don't think
that's necessary wrong, but... for one thing, if this is a special case of
the original result, how would you carry through the original argument for
this special case? The argument involves groups of three measurements, even
though the result doesn't, and we have only two particles to work with.
Also, the original derived result is supposed to hold with absolute
certainty even for finite sample sizes (and does, if you use the Venn
diagram interpretation), but Bell's inequality is probabilistic. E.g. it's
easy to see that a local hidden variable theory that just returns an
independent coin flip for every measurement will have a small but nonzero
chance of violating the inequality for finite sample sizes. Again it's not
clear that the assumptions that went into deriving the first inequality
apply to the way it's used. I can't help feeling that I'm watching a magic
trick instead of real magic.
-- Ben
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| User: "Edward Green" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
09 Dec 2006 10:45:11 PM |
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Ben Rudiak-Gould wrote:
Edward Green wrote:
Well, if the quoted object is indeed "Bell's Inequality", then I'd go
farther: the diagramatic argument is not simply "not less general", it
is a faithful graphic representation.
If by the quoted object you mean
Number(A, not B) + Number (B, not C) >= Number (A, not C)
then that's not Bell's inequality. Bell's inequality (or what substitutes
for it here) is
Number(right up 0, left up 45) + Number(right up 45, left up 90)
= Number(right up 0, left up 90)
which (this is one of the parts that bothers me) is not obtained from the
first inequality by substituting things for A, B, and C.
I didn't realize that. I owe you thanks.
Instead he uses the
fact (or at least the claim) that "right up N" is equivalent to "not (left
up N)" as well as the fact that measurements of the right and left particles
commute with each other, which apparently makes the original derivation
applicable since you don't have to choose a measurement order. I don't think
that's necessary wrong, but... for one thing, if this is a special case of
the original result, how would you carry through the original argument for
this special case? The argument involves groups of three measurements, even
though the result doesn't, and we have only two particles to work with.
Perhaps you will recognize the experience of something on the threshold
of understanding -- that we can build a logic tower ... pile up the
dominoes just so ... and then, if we squint at it another way, or we
don't think about it for a week, or somebody suggests a question which
seems wrong-headed but which we can't quite answer or dismiss: the
tower comes crashing down, and we must pick up our tiles and start
building again.
That is my experience here.
I can build a few of the models pictured on the back of the box. There
is the "core structure": there are limits (I was wont to say "on the
correlation structure") on joint distributions of random variables, the
predictions of quantum mechanics trespass such limits -- experiment may
also -- and this trespass limits in turn the class of alternative
theories which could model the data.
Well... the devil is in the details... I've gotten myself confused
about a number of things now. There is the problem with correlations.
Everyone speaks in terms of them, but nobody actually uses them.
Apparently what passes for a "correlation" is more like a conditional
probability. We may be encoding the same information as if we
explicitly measured sample correlations, but I'm not sure. Then there
is the issue of finite samples. When I naively imagined we were
directly measuring correlations, I simply assumed that in a long enough
experiment we would get a sufficiently close estimate of the real
correlations to simply act as if we had measured them directly. Maybe
the same idea applies to conditional probabililties. But what about
the counting arguments, which seem to apply equally well to finite
samples, however small? And are we estimating conditional
probabilities, or taking counts, or what?
Sigh. Return to chaos, where before I saw order.
Also, the original derived result is supposed to hold with absolute
certainty even for finite sample sizes (and does, if you use the Venn
diagram interpretation), but Bell's inequality is probabilistic. E.g. it's
easy to see that a local hidden variable theory that just returns an
independent coin flip for every measurement will have a small but nonzero
chance of violating the inequality for finite sample sizes. Again it's not
clear that the assumptions that went into deriving the first inequality
apply to the way it's used. I can't help feeling that I'm watching a magic
trick instead of real magic.
I agree with you there, for what that's worth. Which philosopher of
science said that our beliefs condition what we will look for or can
see? I see Bell's work as an opportunity to test whether any local
theory can model reality, so I would tend to be more skeptical of
arguments from within quantum mechanics (like the ones you cite) used
to tighten the hoop reality has to jump through: someone whose prior
belief is "We know QM is correct, this is a kind of formality" will be
less demanding. If we cannot design an experiment free of assumptions
of a given model in the analysis, and simply violate a Bell inequality
with flashes of LED's on well separated boxes, then we may wonder why
the universe is being so coy. It's like a spoon bender who can't
perform when you watch him too intently.
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| User: "Edward Green" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 06:49:51 PM |
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Ben Rudiak-Gould wrote:
Edward Green wrote:
However, at...
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BellsTheorem/BellsTheorem.html
we find a description of "Bell's inequality" as a statement essentially
about counts, or the areas of Venn diagrams; which to be sure, is a
statement about event probabilities, but not obviously about
correlations.
I don't think that what's described on that page is Bell's inequality at
all. The derivation explicitly assumes that A, B, and C are independently
measurable properties of the system, which is a much stronger assumption
than Bell made. Obviously a theory in which all measurements commute can't
agree with quantum mechanics, but people knew that long before Bell's paper.
The intended audience of that page seems to be undergraduate, but you
are saying you don't think the material represents even a bastardized
version of his Bell's results? Interesting.
I think the gist of Bell's argument goes like this: _assume_
(presumably contrary to fact) that (as you put it) "all measurements
commute" (or something equivalent), then we may deduce an inequality
concerning the statistics of these measurements. The predictions of
quantum mechanics violate this inequality. Or, to put it another way,
the areas of the implicit set diagram labeled A,B and C are related to
the famous "what we would have got" if we measured A,B or C on a given
run, even though we can't measure them all simultaneously.
Whether what is demonstrated is the/a Bell inequality or not, your
observation is actually consistent with the overall argument.
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| User: "Ben Rudiak-Gould" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 09:52:28 PM |
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Edward Green wrote:
but you are saying you don't think the material represents even
a bastardized version of his Bell's results? Interesting.
Having looked more carefully at the page, I think I was wrong when I thought
it was wrong. But it still makes me uneasy. In particular, I don't
understand what Number(...) is supposed to mean. It appears that Number(X,Y)
is supposed to mean the number of objects that *have the properties* X and
Y, which seems ridiculously strong. I'd much rather it had an operational
definition, involving, say, measuring the properties from left to right. But
with that definition, at least one of the identities that's assumed in the
proof, namely
Number(not A, B, not C) + Number(A, B, not C) = Number(B, not C)
is violated by any local hidden variable theory that makes even a token
attempt to imitate quantum mechanics. So that can't be it.
Basically, I don't quite understand what the assumptions of this proof are,
and why I should believe they hold. (After all, they *don't* hold!) I much
prefer an operational approach, where e.g. we define a system as a state
machine which, upon being asked about a measurable quantity, returns an
answer (a function of the state) and then enters a new state. Classical
locality appears as the assumption that a system of two separated parts can
be described by two state machines, one for each part. This matches my
intuitive idea of what a local hidden variable theory is, and it avoids
metaphysical assumptions that are next to impossible to reason about. The
purpose of these metaphysical derivations seems to be to look as though they
make no nontrivial assumptions at all, thereby increasing the spookiness
factor of the result. I don't think this is useful unless all of the
assumptions are written out explicitly, and the result derived purely formally.
As far as teaching Bell's theorem to undergraduates goes, I think the
gambling-game approach is good:
http://groups.google.com/group/sci.physics/msg/7cfb4a461145fd1e
Message-ID: <eit4ih$bib$1@gemini.csx.cam.ac.uk>
It's operationally well defined and pretty easy to understand.
-- Ben
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| User: "Daryl McCullough" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
10 Dec 2006 08:11:52 AM |
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Ben Rudiak-Gould says...
Basically, I don't quite understand what the assumptions of this proof are,
and why I should believe they hold. (After all, they *don't* hold!)
The assumptions of the "Venn Diagram" argument are particularly
strong, but simple: Every particle possesses a spin value for every
possible direction, and when one performs a spin measurement along
a particular axis, the result is just the corresponding "hidden variable".
Mathematically, we assume that the spin state of an electron or positron,
for example, is a function S(R) which returns +1 or -1 for each direction
vector R. If one performs a spin measurement along direction R, then
the result will be "spin-up" if S(R) = +1, and will be "spin-down" if
S(R) = -1. The assumption is that each particle has an associated
spin state S(R), and this spin state can only change as a result
of local interactions (the spin state of one particle cannot be
affected by experiments involving a second, distant particle).
The predictions of quantum mechanics don't actually rule out this
hidden variables model, as far as I know, but if we make the
additional (seemingly harmless) assumption that the function S(R)
is measurable (so that it's meaningful to talk about joint
probabilities) then such a hidden variables model is ruled out.
In particular, consider three directions: R_A, R_B, and
R_C all in the same plane with an angle of 120 degrees between
any two of them. Then the predictions of quantum mechanics for
electrons produced in twin-pair production are
Probability that S(R_B) = 1 given that S(R_A) = 1
= 1/4
If we assume that S(R) is measurable, (and that things
are perfectly symmetrical under rotations and reflections)
then we find that
Probability that S(R_C) = 1 given that S(R_A) = 1
and S(R_B) = 1
= -1/2
which is of course nonsense (unless somebody knows a way
to interpret negative probabilities?).
Bell's proof seemed to be more general, but I don't think
it actually was. Rather than assuming that the result of
a spin measurement is deterministic, he assumed that it
is probabilistic, and that the probabilities depended in
some unknown way on the state of the particle and the
local details of the experimental setup. That seems
more general; however, in the case of twin-pair experiments,
the perfect anti-correlation observed between the results
of distant spin measurements along the same axis implies
that the results *cannot* depend on anything other than
the state of the particle and the choice of an axis. So
Bell's more general assumption reduces to the assumption
of the existence of a spin state as was assumed in the
Venn Diagram argument.
--
Daryl McCullough
Ithaca, NY
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| User: "Ben Rudiak-Gould" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
10 Dec 2006 11:18:53 AM |
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Daryl McCullough wrote:
The assumptions of the "Venn Diagram" argument are particularly
strong, but simple: Every particle possesses a spin value for every
possible direction, and when one performs a spin measurement along
a particular axis, the result is just the corresponding "hidden variable".
Okay, I finally get it. First of all you prove the first inequality on the
page for boolean-valued functions on some domain, where A, B, C are elements
of the domain. Then you argue that, assuming local realism, the results of
commuting measurements of the two particles can necessarily be modeled by a
boolean-valued function on the measurement angle (which I certainly
understand, since I did the same thing in my gambling-game argument). Then
you substitute elements of that domain (i.e. measurement angles, not
measurements) into the first inequality. So it's valid, and very similar to
the gambling-game argument.
-- Ben
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| User: "Edward Green" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
10 Dec 2006 01:32:26 PM |
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Ben Rudiak-Gould wrote:
Edward Green wrote:
but you are saying you don't think the material represents even
a bastardized version of his Bell's results? Interesting.
Having looked more carefully at the page, I think I was wrong when I thought
it was wrong.
I read your posts out of order, but I see you qualify this statement
slightly later: additional assumptions are necessary to map the quoted
inequality (below) onto Bell's Inequality.
But it still makes me uneasy. In particular, I don't
understand what Number(...) is supposed to mean. It appears that Number(X,Y)
is supposed to mean the number of objects that *have the properties* X and
Y, which seems ridiculously strong.
I admit I'm confused at this point (so I can't reply in my usual
know-it-all tone), but let me focus on what I _do_ think I understand:
the logic of the exercise calls on us first to deduce some properties
which must be held by any joint probability distribution. Therefore,
the assumptions related to "Number(...)" are not related to a physical
system, but to a sample distributed according to some probability
distribution.
[Doubt: in fact such counts are apparently related to finite samples,
and I'm not entirely clear how this relates to the eventual
consideration of probability distributions.]
I'd much rather it had an operational
definition, involving, say, measuring the properties from left to right. But
with that definition, at least one of the identities that's assumed in the
proof, namely
Number(not A, B, not C) + Number(A, B, not C) = Number(B, not C)
is violated by any local hidden variable theory that makes even a token
attempt to imitate quantum mechanics. So that can't be it.
I didn't actually follow the derivation of this inequality, since it's
obviously true simply by counting regions on a Venn diagram, and I
wouldn't be concerned about anything supposedly leading up to it: as a
statement about counts in a finite sample or probabilities (when
normalized), it stands by itself.
Basically, I don't quite understand what the assumptions of this proof are,
and why I should believe they hold. (After all, they *don't* hold!) I much
prefer an operational approach, where e.g. we define a system as a state
machine which, upon being asked about a measurable quantity, returns an
answer (a function of the state) and then enters a new state.
Yes! I fully agree with you here -- apart from anything connected with
this proof -- and I don't think this view is sufficiently appreciated,
though often described ambiguously as "things don't exist before we
measure them". Ordinary objects can act like this: e.g., behavioral
reponse is a function of the subject's prior state, modifies the state,
and "doesn't exist before we measure it" -- though this hardly sounds
spooky in this case.
Classical
locality appears as the assumption that a system of two separated parts can
be described by two state machines, one for each part.
Yes again.
This matches my
intuitive idea of what a local hidden variable theory is, and it avoids
metaphysical assumptions that are next to impossible to reason about.
Yes.
[My way of expressing this idea was that some packets of "information"
were presumed to be transmitted to both legs of the apparatus,
whereupon the outcomes of local measurements are determined by the
interaction of this information with the local environment.]
To carry our ideas through and repeat myself slightly, the two state
machines picture can be translated (I claim) into the requirement that
all measurable pairs of variables (one on each end) are matchable in
distribution by a single joint distribution on all variables. The
existence of this presumptive joint distribution places limits on
statistical relationships between the variables. We can check whether
quantum mechanics respects these limits, and apparently it doesn't. We
can also check whether experiment respects these limits.
The form these mysterious "relationships" take is essentially
correlations, although what is typically stated is not correlations,
but joint probabilities. Ideally, we would experimentally test the
correlation structure of operationally well-defined variables, and
observe a violation (or not). Real execution and analysis seems to
lack this simplicity.
At this point, I think I'll stop and wait for a return ping.
The
purpose of these metaphysical derivations seems to be to look as though they
make no nontrivial assumptions at all, thereby increasing the spookiness
factor of the result. I don't think this is useful unless all of the
assumptions are written out explicitly, and the result derived purely formally.
Not sure about that.
As far as teaching Bell's theorem to undergraduates goes, I think the
gambling-game approach is good:
http://groups.google.com/group/sci.physics/msg/7cfb4a461145fd1e
Message-ID: <eit4ih$bib$1@gemini.csx.cam.ac.uk>
It's operationally well defined and pretty easy to understand.
Very nice.
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| User: "" |
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| Title: Re: Bell's inequality vs. Kerr Rotation |
31 Dec 2006 12:27:25 AM |
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I have a question, after having read about a non-destructive spin
measurement experiment, which was cited as one of the top science
stories of 2006:
http://physicsweb.org/articles/news/10/12/15/1#11
http://optics.org/cws/Articles/ViewArticle.do;jsessionid=D7E47731913829DB57F86B4716735268?articleId=26434&channel=technology&page=1
So that announcement immediately makes me wonder about Bell's
Inequality:
http://en.wikipedia.org/wiki/Bell%27s_inequality#Description_of_Bell.27s_theorem
They say that you can't use the "spooky action at a distance"
(correlation violation) to communicate with, since you can't
predict/measure in advance what an entangled particle's state will be.
But the non-destructive measurement experiment shows that you can
indeed measure it in advance, without significantly disturbing/altering
that particle's state (or its entanglement?)
Wouldn't this Kerr rotation measurement method then allow for the
pre-screening of entangled pairs, based on measurement in advance of
state properties like spin?
Couldn't this then be used to exploit the correlation violation (aka
"spooky action at a distance") in such a way as to permit its use for
communication?
For instance, using the Alice & Bob example, wouldn't it be possible to
use pre-measured entangled electron pairs of known spin state, and use
the orientation of the apparatus on one end as a way to modulate an
information signal, which would then be detected with the other party
through the correlation violation?
To me, it would seem intuitive that the answer is yes. Why shouldn't
this be able to work?
Please, someone kindly take the time to give me a reasoned reply, even
if my post sounds ignorant.
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| User: "Sue..." |
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| Title: Re: Bell's inequality vs. Kerr Rotation |
31 Dec 2006 01:16:03 AM |
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wrote:
I have a question, after having read about a non-destructive spin
measurement experiment, which was cited as one of the top science
stories of 2006:
http://physicsweb.org/articles/news/10/12/15/1#11
http://optics.org/cws/Articles/ViewArticle.do;jsessionid=D7E47731913829DB57F86B4716735268?articleId=26434&channel=technology&page=1
So that announcement immediately makes me wonder about Bell's
Inequality:
http://en.wikipedia.org/wiki/Bell%27s_inequality#Description_of_Bell.27s_theorem
They say that you can't use the "spooky action at a distance"
(correlation violation) to communicate with, since you can't
predict/measure in advance what an entangled particle's state will be.
But the non-destructive measurement experiment shows that you can
indeed measure it in advance, without significantly disturbing/altering
that particle's state (or its entanglement?)
Wouldn't this Kerr rotation measurement method then allow for the
pre-screening of entangled pairs, based on measurement in advance of
state properties like spin?
Couldn't this then be used to exploit the correlation violation (aka
"spooky action at a distance") in such a way as to permit its use for
communication?
For instance, using the Alice & Bob example, wouldn't it be possible to
use pre-measured entangled electron pairs of known spin state, and use
the orientation of the apparatus on one end as a way to modulate an
information signal, which would then be detected with the other party
through the correlation violation?
To me, it would seem intuitive that the answer is yes. Why shouldn't
this be able to work?
Please, someone kindly take the time to give me a reasoned reply, even
if my post sounds ignorant.
slot+antenna
http://images.google.com/images?svnum=10&hl=en&lr=&safe=off&q=slot+antenna&btnG=Search
circular+polarization
http://images.google.com/images?svnum=10&hl=en&lr=&safe=off&q=circular+polarization+&btnG=Search
http://www.ee.surrey.ac.uk/Personal/D.Jefferies/antennas.html
Sue...
http://farside.ph.utexas.edu/teaching.html
http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/light/index.htm
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| User: "Sorcerer" |
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| Title: Re: Bell's inequality vs. Kerr Rotation |
31 Dec 2006 03:34:42 AM |
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"Sue..." <suzysewnshow@yahoo.com.au> wrote in message =
news:1167549363.777618.62630@n51g2000cwc.googlegroups.com...
[...]
I've never seen an aether, Dennis. Have you?=20
http://tinyurl.com/yndvwx
http://www.quackwatch.org/01QuackeryRelatedTopics/pseudo.html
http://www.youtube.com/watch?v=3DPf3z935R37E
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| User: "Timo A. Nieminen" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 03:20:15 PM |
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On Sat, 8 Dec 2006, Edward Green wrote:
[stuff cut]
I am amazed! Edward, you are an eternal optimist! To expect replies on
usenet groups such as s.p with subject lines like that. Given that only
the professional minority are likely to be familiar with Kolmogorov's work
(any non-pro Kolmogorov fans out there?), and most of the professional
minority here are unlikely to be familiar with Kolmogorov's work, just who
are you expecting to give a substantive reply?
I once did most of my work on a PC called kolmogorov, but that was a by
product of having done hydrodynamics at Re=10^9, which is the
Kolmogorov=god regime of fluid flow. Offhand, I can't tell you anything
about Kolmogorov inequality.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
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| User: "Sorcerer" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 05:29:35 PM |
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"Timo A. Nieminen" <timo@physics.uq.edu.au> wrote in message =
news:Pine.WNT.4.64.0612090715410.520@serene.st...
| On Sat, 8 Dec 2006, Edward Green wrote:
|=20
| [stuff cut]
The fuckhead "Timo A. Nieminen" does that. Oh well.
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| User: "Ben Newsam" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 04:16:55 PM |
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On Sat, 9 Dec 2006 07:20:15 +1000, "Timo A. Nieminen"
<timo@physics.uq.edu.au> wrote:
I am amazed! Edward, you are an eternal optimist! To expect replies on
usenet groups such as s.p with subject lines like that. Given that only
the professional minority are likely to be familiar with Kolmogorov's work
(any non-pro Kolmogorov fans out there?), and most of the professional
minority here are unlikely to be familiar with Kolmogorov's work, just who
are you expecting to give a substantive reply?
I once did most of my work on a PC called kolmogorov, but that was a by
product of having done hydrodynamics at Re=10^9, which is the
Kolmogorov=god regime of fluid flow. Offhand, I can't tell you anything
about Kolmogorov inequality.
That's one of the longest synonyms for "No" I have seen for quite a
while.
--
Posted via a free Usenet account from http://www.teranews.com
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| User: "Edward Green" |
|
| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 06:23:32 PM |
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Timo A. Nieminen wrote:
On Sat, 8 Dec 2006, Edward Green wrote:
[stuff cut]
I am amazed! Edward, you are an eternal optimist! To expect replies on
usenet groups such as s.p with subject lines like that. Given that only
the professional minority are likely to be familiar with Kolmogorov's work
(any non-pro Kolmogorov fans out there?), and most of the professional
minority here are unlikely to be familiar with Kolmogorov's work, just who
are you expecting to give a substantive reply?
Why, you, Timo! :-)
Besides, it's not as if I had to choose among (1) the mathematician and
theoretician who work down the hall from me, and into whose offices I
can pop (2) the moderated newsgroup, in which I am ever welcome to
publish my questions, not, and ... I don't know what (3) would be.
Possible the robotic companion named "Copernicus", which unfortunately
hasn't been invented yet.
It's like buying a lottery ticket -- you can't win if you don't play
(and part of my question was simply about Bell's work). This sucks. I
think I'll just devote my time to long-winded rants about the closed
minded scientific establishment. Better?
I once did most of my work on a PC called kolmogorov, but that was a by
product of having done hydrodynamics at Re=10^9, which is the
Kolmogorov=god regime of fluid flow. Offhand, I can't tell you anything
about Kolmogorov inequality.
I actually know nothing at all about Kolmogorov, except, as I said, my
vague idea that his name might be associated with inequalities which
generalized the rather ad-hoc Bell inequalities -- which ought to be
linear inequalities describing a polytope in correlation space (oh
shoot... I see I forgot to include the other raped commons, sci.math).
.
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| User: "Timo A. Nieminen" |
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| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 06:59:48 PM |
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On Sat, 8 Dec 2006, Edward Green wrote:
Timo A. Nieminen wrote:
On Sat, 8 Dec 2006, Edward Green wrote:
[stuff cut]
I am amazed! Edward, you are an eternal optimist! To expect replies on
usenet groups such as s.p with subject lines like that. Given that only
the professional minority are likely to be familiar with Kolmogorov's work
(any non-pro Kolmogorov fans out there?), and most of the professional
minority here are unlikely to be familiar with Kolmogorov's work, just who
are you expecting to give a substantive reply?
Why, you, Timo! :-)
Besides, it's not as if I had to choose among (1) the mathematician and
theoretician who work down the hall from me, and into whose offices I
can pop (2) the moderated newsgroup, in which I am ever welcome to
publish my questions, not, and ... I don't know what (3) would be.
Possible the robotic companion named "Copernicus", which unfortunately
hasn't been invented yet.
It's like buying a lottery ticket -- you can't win if you don't play
(and part of my question was simply about Bell's work). This sucks. I
think I'll just devote my time to long-winded rants about the closed
minded scientific establishment. Better?
Well, buy all the lottery tickets you want. Lotteries are, on expectation
value, a losing proposition, but this ignores the practical effect of
actually winning vs the few dollars that you'd otherwise waste anyway (if
you had an infinite life expectancy, it might be different), as well as
the entertainment value.
I once did most of my work on a PC called kolmogorov, but that was a by
product of having done hydrodynamics at Re=10^9, which is the
Kolmogorov=god regime of fluid flow. Offhand, I can't tell you anything
about Kolmogorov inequality.
I actually know nothing at all about Kolmogorov, except, as I said, my
vague idea that his name might be associated with inequalities which
generalized the rather ad-hoc Bell inequalities -- which ought to be
linear inequalities describing a polytope in correlation space (oh
shoot... I see I forgot to include the other raped commons, sci.math).
Kolmogorov did lots of very good stuff, but since he was a mathematician
(by and large) and I am a physicist, I am most influenced by his physicy
stuff, which, AFAIAC, is his stuff on statistical turbulence.
IIRC, his turbulence papers are well-written. It's been over a decade
since I read them, but I still remember being impressed by them.
Kolmogorov was a sufficiently-big shot that you might be able to find a
"collected works" or "selected works" in a nearby library.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
.
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| User: "Edward Green" |
|
| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 07:10:35 PM |
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Timo A. Nieminen wrote:
On Sat, 8 Dec 2006, Edward Green wrote:
It's like buying a lottery ticket ...
Well, buy all the lottery tickets you want. Lotteries are, on expectation
value, a losing proposition, but this ignores the practical effect of
actually winning vs the few dollars that you'd otherwise waste anyway (if
you had an infinite life expectancy, it might be different), as well as
the entertainment value.
I forgot to mention that this lobby actually has a negative cost to
play -- I enjoy writing the question, I usually learn something just
formulating it, and if I get any good answers, so much the better. It
might just have a postive opportunity cost, though.
I'm happy to hear something from you, but I'm just a _tiny_ bit
offended by your dismissive tone. Just thought you might like to know
that.
Ed Green
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| User: "Timo A. Nieminen" |
|
| Title: Re: Bell's inequality vs. Kolmogorov inequality |
08 Dec 2006 09:52:27 PM |
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On Sat, 8 Dec 2006, Edward Green wrote:
Timo A. Nieminen wrote:
On Sat, 8 Dec 2006, Edward Green wrote:
It's like buying a lottery ticket ...
Well, buy all the lottery tickets you want. Lotteries are, on expectation
value, a losing proposition, but this ignores the practical effect of
actually winning vs the few dollars that you'd otherwise waste anyway (if
you had an infinite life expectancy, it might be different), as well as
the entertainment value.
I forgot to mention that this lobby actually has a negative cost to
play -- I enjoy writing the question, I usually learn something just
formulating it, and if I get any good answers, so much the better. It
might just have a postive opportunity cost, though.
That's exactly the "entertainment" value I refer to above. If you play the
lottery (ie a real lottery), and enjoy it, you don't need to win to
benefit.
If you enjoy or learn from posting, how can you lose?
I'm happy to hear something from you, but I'm just a _tiny_ bit
offended by your dismissive tone. Just thought you might like to know
that.
I didn't intend "dissmissive"; initially I was simply truly amazed at your
optimism, but you already got more serious replies than I expected. Well
done, Ed, and well done, responders!
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
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