🚀 go-pugleaf

Trivial, but false, formal proof that quantum connection communication is
impossible.

Nonlocal joint probability that receiver photon b counted in + channel and
twin transmitter photon counted in phi channel is

p(b+|a,phi) = <a,b| |+><+| |phi><phi| |a,b>

where |a,b> is pure pair state.

Local receiver probability is

p(b+) = Integral{p(b+|a,phi) p(phi)d(phi)}

where p(phi) is classical probability density for exclusive noninterfering
alternatives for transmitter photon's detection.

But Integral{|phi><phi|p(phi)d(phi) = 1 by completeness or conservation of
probability.

Therefore,

p(b+) = <a,b| |+><+| |a,b> which will have neither a phi nor theta
dependence when the actual pair state is substituted. In fact this
p(+) = 1/2 period! Therefore, no FTL communication. QED!

However, there is something wrong with this proof.

1) |a,b> can have a parametric dependence on phi so that it is
mathematically incorrect to use the completeness relation in isolation. So
that it does not cancel out of the problem destroying the controllable
nonlocal coherence effect.

That is we have a problem in which we want to compute

Integral{f(phi)g(phi)d(phi)} in which Integral{g(phi)d(phi)} = 1. It does
not follow that Integral{f(phi)g(phi)d(phi)} = f(phi) unless f is not a
function of phi - which it is.  However, if

p(phi) = Dirac delta [phi - phi']

then

p(b+, phi',theta) = <a,b| |+><+| |phi'><'phi| |a,b>

and there is quantum connection signal.  In fact it is

S =p(b+, phi',theta) - p(b-, phi',theta) = sin(2theta)cos(phi')

One can replace the Dirac delta function by a sharply peaked function in
which the uncertainty in phase rms(phi) is small compared to 2pi. in which
case one will get

S = sin(2theta)cos(phi')sinc(rms(phi))