> What's worse, just *measuring* the photon will influence the probability
> that it gets to its target - even if it doesn't deflect the photon by so
> much as an inch. Because when you measure a system, all probability
> amplitudes are *immediately* converted to probabilities and the system
> "collapses" or is "reduced" to a single branch of reality. The photon
> is suddenly in only one place in only one branch of reality. This is
> known as "the collapse of the wavefunction" or "state-vector reduction".
While this is true, I don't think it is the most informative way to view
the situation. Collapse of the wave function is not purely a matter of
acquiring information. It requires interacting with the system.
Here is Eliezer's very useful diagram, for reference below:
> 1
>
> | Arrows indicate direction.
>
> A/--------/---2 A single photon leaves START,
> | D| and is reflected from mirrors.
>
> |C | What is the chance that the photon
> START---/--------/B will reach target 1, or target 2?
In the case of the hypothetical measuring device which collapses the wave
function, it may be OK for it not to "deflect the photon by an inch,"
but it nevertheless will interact with and affect the photon. If your
measuring device were somehow able to measure (some aspect of) the photon
without altering its quantum state in any way, it would not destroy the
interference, and the photons would continue to all take path 2.
Consider this hypothetical experiment. Let the beam splitter C work on
the basis of polarization. Horizontally polarized photons go through
path A, and vertically polarized photons go through B. This will work
the same as was described; the photons will be recombined and all will
take path 2. We can send in photons of any polarization and this will
happen. With a diagonally polarized photon it in effect has an equal
chance of choosing either path (or you could say that it goes equally
down both paths in some virtual sense).
Now, create a pair of photons in correlated states as in the EPR
experiment, both diagonally polarized, but such that they will both
produce the same result when their vertical/horizontal polarization
is measured. Send one into the experiment above, and send the other
into another copy of the experiment. In each case the photon passes
through its instance of the experiment and comes out path 2.
Now block path A in the second experiment. The photon in that experiment
comes out path 1 or is absorbed. We know which path it is following,
and there is no interference with its "other half". What happens in
the first experiment? We know which path the photon is taking from
the results of the second experiment. In effect, we are measuring the
photon path without disturbing it. What will happen? Will the photon
still always head out path 2, or will it sometimes head out path 1 as
when we measured it directly?
My understanding is that if you work out the math and/or do the
experiment, you will find that measuring the other photon doesn't
change how this one behaves (indeed, if it did, that would be FTL
communication). This photon continues to go out path 2, even though
you now know which path it is going down every time.
This is really very logical. Your second experiment is basically just
being used to measure the vertical/horizontal polarization of the
photon. Once you have done so, the paired photon can be thought of
simply as vertically or horizontally polarized. Since sending either
of these two kinds of photons into the experiment leads to them
coming out path 2, it should not be a surprise that measuring the remote
photon leads to this result.
However this shows that there is more going on here than a simple
model where the photon must take both branches in order to interfere
with itself and leave on path 2. Eliezer is right that if you
measure that photon within the experiment in order to determine
which path it is taking, you will destroy the interference and allow
it to leave on path 1. What is wrong is to say that this is because
the measurement collapses the wave function. Rather, the reason this
happens is because the measurement perturbs the state of the photon.
That is the point I want to get across, which for some reason the
authors of most articles on the strangeness of QM are reluctant to
discuss.
Hal