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Communication across Spacelike Intervals
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This is the result of various musings, systematic and otherwise, over
the weekend.  One conclusion from those musings is that, contrary to
my previous opinion, the descriptions of this notion given by quantum
mechanics and relativity are not in contradiction, but yield mostly the
same predictions.

In the musings, the starting point was my sudden realisation, last week,
that the supposed "collapse" of the wave function is relativistically
invariant: it will appear spacelike from all reference frames.  The end
point, near enough, was a further realisation that the collapse cannot
be used to synchronize distant clocks more precisely that the theory of
relativity allows, even in the presence of "instantaneous" communication.

With which introduction, here is the full account of the matter.

Let's start with a wave function, |psi>, that is bimodal: it has a large
amplitude in two different places, and not much in between.  For instance,
the correlated photon pair emitted in the Friedman-Clauser experiment has
just such a wave function, representing two photons receding one from
another, and correlated because they are in the same arbitrary state of
polarisation.

Let's suppose that we observe this wave function in two places, called
optimistically Sender and Receiver, and with coordinates (xs,ys,zs,ts)
and (xr,yr,zr,tr) in the underlying coordinate system of the wave
function.  These observations are accomplished by measurements of some
kind, and hence can be represented as applying to the wave function some
variety of operator, and let's call them S and R, for "send action" and
"receive action".

Hence, the initial |psi> becomes, after the send action, S|psi>, and,
after the receive action, RS|psi>, and I will assume for now that the
operators are applied in that order.  What will be observed, then, is

	<psi|RS|psi>

and for this to be useful, something must be observed at both sites, the
sender and the receiver.

One obvious candidate is to let both S and R be a filter that transmits
photons in horizontal linear polarisation, so S=R=H.  What is observed
is 

	<psi|HrHs|psi>

which we've beaten to death already, and resolves to a prediction that
the sites will perceive either two H photons or two V photons, with
complete correlation between them.

Can this be used for communication?  Evidently not: if both sender and
receiver know what the other is going to do, they might just as well
turn off the apparatus and calculate the results.  This is no more
instantaneous communication than having both sites calculate pi is
an instantaneous transmission of "3.14159..." from one to the other.
The most we can say is that, if the calculation implies a collapse,
the apparatus will ensure that both sites read the same stream of random
numbers.  But this cannot be used to convey a fixed message between
sender and receiver. (Though it could be used, for instance, to coordinate
the tactical operations of a group of warships, by ensuring they all
adopted *the same* random tactic from a preset list.  So it can be
a source of useful information, in a certain sense.)

For communication to be possible, the sender must have a *choice*.  That
means, there must be two actions available to the sender - S and S', say
- that yield different results.  The sender can then make one of two
things happen

	<psi|RS|psi>
	<psi|RS'|psi>

and communication will occur if we can find a suitable R,S,S' that give
different results at the receiver site for these two actions.

Here, again, is a simple example.  Let both sides put the photon through
a standard H/V splitter based on linear polarisation.  let R be a standard
interference experiment on the two elements of the resulting superposition.
Let S be the same as R, on the other photon.  And let S' be a detector
experiment, that distinguishes between H and V photons.

The equations seem to me pretty clear.  In the case

	<psi|RS|psi>

nobody collapses the superposition, and both sites perceive interference
fringes between the H and V paths.  In the case

	<psi|RS'|psi>

the detector collapses the superposition, it must do so for both photons,
and *neither* site will perceive interference fringes.  The choice at the
send site has changed the result at the receive site.  And the S,S' setup
is also standard stuff - it's the "delayed choice" experiment of Wheeler.

Unfortunately, this is a miserable communication device.  The receiver
doesn't really see any interference, it sees only one photon, one
scintillation.  It's sometime possible to prove that this could not
be the result of interference, if it occurs in a trough, but not very
often.  Rather than refine the equipment, I'll apply a brute force
engineering solution: replicate the apparatus 100 fold, and use 100
pairs of photons at once.  A pattern of 100 scintillations can be
distinguished with high reliability, as either an interference or a
superimposition pattern.  So this expensive setup will transmit,
with pretty good accuracy, one bit from sender to receiver, for
every 200 photons consumed.

Given that photons travel at the speed of light, and that in this case
they are travelling away from each other, it's pretty clear that the
interval between (xs,ys,zs,ts) and (xr,yr,zr,tr) is spacelike - indeed,
you have to work pretty hard to make it other than spacelike.  So, if
the correlations are enforced, then in the collapse picture the collapse
is spacelike, and if this can indeed be used to communicate, then that
would be communication across a spacelike interval.  Yes, there is still
a distinction between tr,ts - the microscopic time coordinate of the
wave function - and Tr,Ts - the macroscopic time of the wall clocks.
And no, we cannot identify the two; the Principle of Indeterminacy
forbids it.  But realistically, everything macroscopic is in the same
reference frame, only the photons move, and it would be most perverse
to say other than that the T coordinate is correlated closely with the
t coordinate, within the limits imposed by delta-T >= h/2 pi delta-E.
So the macroscopic interval, too, is spacelike.

What are the bounds on this interval?  Here I recalled another key point
from the earlier discussion: the operators applied at send and receive
sites must commute.  This is sometimes given as a reason why communication
is impossible, so let's look at it.

Given RS = SR and RS' = S'R, then

	<psi|RS|psi>  = <psi|SR|psi>
	<psi|RS'|psi> = <psi|S'R|psi>

But this proves nothing.  The communication relies, not on some supposed
distinction between RS and SR, but on that between RS and RS'.  In fact,
what this does allow us to deduce is:

	<psi|SR|psi> ~= <psi|S'R|psi>

And this, I think, answers the question.  It is not necessary to assume
that the time of the send action is "the same as" that of the receive
action, nor even that ts<tr.  The effect works even if ts>tr, and hence,
I suppose, even if Ts>Tr, as measured by the macroscopic clocks.  So,
as an obvious corollary, any communication effect *cannot* be used to
synchronize remote clocks, since the effect will occur for any spacelike
separation, and not just for Tr=Ts or even Tr>Ts.

What this says, informally, is the following: if it is possible to send
information "instantaneously", then it is also possible to send information
across any spacelike interval: because the operators commute (both with
themselves and with the time evolution operator, though I skipped that
since it's standard stuff), either operator may be applied at any point
along the track of the photons; which means, the events may be separated
by almost any spacelike interval.

This was a satisfying conclusion, since it agrees exactly with what the
theory of relativity tells us.  The basic principle of relativity is that
the laws of physics are the same in all reference frames.  Put into
operational terms, this means that, if one observer in one reference
frame sees a set of macroscopic events, and can explain those events
in terms of certain underlying physical principles, then another observer,
in another reference frame, observing those same events, can explain them
equally well in term of *the same* principles.

As is well known, this implies that two distant events that are
simultaneous in one frame, may by a suitable choice of frame be placed
at the ends of any spacelike interval.  Hence, if any physical principle
allows instantaneous communication, that same principle allows communication
across any spacelike interval.  Which is precisely where the QM analysis
led.

Well, that's a big IF.  And, alas, neither relativity nor QM can help
us with it.  There are no synthetic a priori propositions: the existence
or non-existence of such physical principles cannot be decided by curling
up in a stove and thinking about it, but only by experiment.  Experiments
such as the rough one sketched out here, or the much more elegant and
well developed one of Jack Sarfatti.

To carry the musing a little further: this result has some important
engineering consequences.  It is not necessary for the source of the
correlated photons to be halfway between sender and receiver, not even
approximately.  It can be anywhere between them.

If we put the source very close to the sender, then the communication
path is very close to the light cone, and the device works much like
a telegraph.  On the other hand, if we put the source very close to
the receiver, we get the symmetric situation.  If the receiver were
receding from the sender, the messages would display a "transmission
lead" that is almost the inverse of the "transmission lag" of radio
communication, and the combination could be used to fix the distance
of the sender without the delay of a round-trip signal.

A pair of such devices could of course be used for round-trip
communication across a negative timelike interval, but this note
is already far too long for yet more musings.

Finally, then, what's my bet?  I believe pretty strongly that correlations
are indeed enforced across spacelike intervals.  Everybody agrees they are
enforced across timelike intervals; the equations that predict the effect
make no distinction between spacelike and timelike intervals; I trust the
equations.  Many people dissent, but as far as I know none of them has
proposed any alternative equations that yield their picture; and, as
Pythagoras said, you can't beat a horse with no horse.  So I'll stick
my neck out and say that, when Aspect's experiment is repeated without
the loopholes, it will give a positive result.

Can the correlations be used to transmit information?  That's a lot more
dubious, and of necessity, since we're building one unproven speculation
on top of another.  However, until now my main reason for doubt was the
assumption that such a phenomenon would contradict relativity.  Having
at last taken the time to think this through in detail, I no longer
believe that, and hence am a lot less doubtful.  Put it this way: if I
had a check box on my tax form, and a list of physics research
proposals to be funded, I'd check this one first.

Thanks for sticking with a very long post

Robert Firth