On 06-04-2022 08:29, Brent Meeker wrote:
On 4/5/2022 5:14 PM, smitra wrote:
On 06-04-2022 01:14, Bruce Kellett wrote:
On Wed, Apr 6, 2022 at 9:03 AM smitra <smi...@zonnet.nl> wrote:
On 06-04-2022 00:21, Bruce Kellett wrote:
On Wed, Apr 6, 2022 at 1:13 AM smitra <smi...@zonnet.nl> wrote:
The conclusion that local hidden variables are rules out does
depend on
an argument about what would have happened had different
polarizer
setting been used than the ones that were actually used.
That is false. Where in the Aspect experiments is reference made
to
non-performed measurements?
It's in the argument that local hidden variable theories must
satisfy
Bell's inequalities.
You are stretching things quite a bit. All Aspect needed to do was
show that his measured results violated the inequalities. No
counterfactual reasoning involved. If you then want to argue that
Aspect's results extend to all possible such experiments, then of
course, not all those experiments have been done. But that does not
impact Bell's theorem. Bell, himself, did not have to make any
measurements.
How does one conclude that a set of measurement results cannot be
explained by a local hidden variable theory if you don't invoke
counterfactual reasoning to the performed experiment? Of course,
Aspect could simply refer to Bell's theorem, but Bell's theorem relies
heavily on counterfactual reasoning. This is also why you can have
superderminisitic loopholes to the argument against local
deterministic theories.
Saibal
Attached are my lecture notes on Aspect's test of the Bell
inequalities. I don't see anywhere I invoke counterfactual reasoning,
but I'd be happy to have it pointed out to me if I have missed it.
The counterfactual reasoning is in the derivations of the inequalities.
Take e.g. the Bell inequality involving the angle and the double angle.
If you flip the sign of Bob's results relative to Alice's results then
for the same polarizer settings they get the same results. For a nonzero
relative angle we can consider the number of differences. Then we need
to invoke a counterfactual reasoning to argue that the number of
differences at twice the angle must be less than or equal to twice the
number if differences as the original angle. Suppose Bob keeps his
polarizer setting fixed and Alice changes it from theta to 2 theta. Then
the argument goes that her 2 theta results are what she would have
gotten at theta plus on average he same number of changes relative to
what she got relative to Bob at theta, because the relative angle
between 2 theta and theta is theta.
In general, if one has concluded via some reasoning that the value of a
correlation at some angle is constrained by the values at some other
angles (assuming local determinism) then that begs the question of how
the different experiments at the different angles can tell you something
about the results at that particular angle, if not for counterfactual
reasoning.
The GHZ experiment is another simple example, this involves 4 entangled
spins.
We have an entangled state:
|psi> = 1/sqrt(2) [|up, up, up> + |down, down, down>]
Alice, Bob and Charlie receive one spin and then choose to measure
either the x or y-component. Using:
sigma_x|up> = |down>
sigma_x|down> = |up>
sigma_y|up> = i |down>
sigma_y|down> = -i |up>
and with the notation: (A) (B) (C) |state> for tensor product of A, B,
and C so that A acts on the first component of |state> and B on the
second component, and C on the third component, we see:
(sigma_x) (sigma_x) (sigma_x) |psi> = |psi>
So, of Alice, Bob and Charlie measure the x-components, then despite
their results being random, the product of their results will always
equal 1.
We also have:
(sigma_x) (sigma_y) (sigma_y) |psi> = -|psi>
So, if one person measures the x-component and the others measure the
y-component then the product of the results will always equal minus 1.
Argument against local hidden variables: We assume that if the ith
person chooses to measure the X component then the result will be Xi,
while it will be Yi if the Y component is chosen. This does not depend
on the choices made by the others.
For the cases where two people choose to measure the Y-component, we
then have the results:
(X1, Y2, Y3): product = -1
(Y1, X2, Y3): product = -1
(Y1. Y2, X3): product = -1
Taking the product of these results then yields:
-1 = Y1^2 Y2^2 Y3^2 X1 X2 X3 = X1 X2 X3
But as shown above, if all 3 measure the x-components and multiply their
results, the result will always be equal to 1. This contradiction with
local hidden variables clearly follows from invoking counterfactual
measurement results.
Saibal
Brent
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