From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: "Elementary, my dear Ramsay!"
Organization: Whole Earth 'Lectronic Link
Date: Fri, 20 Nov 1992 05:27:46 GMT


Ramsay writes:
From: ramsay@unixg.ubc.ca (Keith Ramsay)
Subject: Sarfatti and would-be FTL
Date: 19 Nov 1992 05:09:44 GMT
Organization: UBC, Vancouver, B.C., Canada, "Loudmouths' Anonymous"

In order to talk about the superpositions of two states, the states
themselves must be already defined. We can, of course, recycle our
variables and use |a,e,+> and |a,o,+> to refer to analogous states
within another experimental design-- but each joint reference to
|a,e,+> and |a,o,+> has to refer to a particular one.

[Ramsay, the states by themselves are rays inb a projective space so that
there aboslute phase is undefined. In fact they are defined modulo any
complex number not necessarily unimodular. Therefore, the relative phase
can very easily be dependent on coefficients in superposition.]

What the principle of superposition means is that we can vary our
choice of superposition, e.g. z1 |a,e,+> + z2 |a,o,+>, *without* also
varying the experimental design,

[I don't think that is true. Why is that true?]

and hence also without varying such
relationships among |a,e,+> and |a,o,+> as the value of <a,e,+|a,o,+>.
Each design will have its own value of <a,e,+|a,o,+>, not varying
depending upon z1 and z2.

[Prove this! It's not obvious. Is it in a standard quantum text book. If
so, which one? What page?]

I assumed that you implicitly fixed the experimental design. That is
why I "automatically assumed" the invariance of <a,e,+|a,o,+>. If you
never fix the experimental design, then it is easy to evade questions
about unitarity as you have just described: by positing a different
experimental design for each state!

*Fix*, then, any *one* experimental design such as you've described,
so that it is possible to speak of *the* fixed value of <a,e,+|a,o,+>,
and so on. Consider the family of possible initial states of the pair
of photons:

    |a,b> = |a,e,+> [ z1 |b,e,+> + z2 |b,o,-> ]
           +|a,o,-> [ z3 |b,e,+> + z4 |b,o,-> ],

where |z1|^2+|z2|^2+|z3|^2+|z4|^2=1. If |a,e,+> evolves in the
experiment to e^{i*phi} |a,e,+> and |a,o,-> to |a,o,+>, then we get a
state

    |a,b> = e^{i*phi} |a,e,+> [ z1 |b,e,+> + z2 |b,o,-> ]
           +          |a,o,+> [ z3 |b,e,+> + z4 |b,o,-> ].

In order for this to be a state, its bracket with itself has to be 1.
This has to hold true for *all* possible choices of z1,z2,z3,z4. We
already know that |b,e,+> and |b,o,-> are orthogonal and normal. How
can you possibly arrange all of this, without <a,e,+|a,o,+>=0? It just
doesn't work.

[Actually, this is all really beside the point and a nonproblem. Start from

|a,b>=|a,e,+>[ z1 |b,e,+> + z2|b,o,-> ]+|a,o,-> [ z3 |b,e,+> + z4|b,o,-> ],

|a,e,+> = |a,e>|a,+>

|a,o,-> = |a,o>|a,->

The half-wave plate on |a,o,-> causes

U(1/2)|a,-> = |a,+>

so that,

|a,o,-> = |a,o>|a,+>

OK assume for the space kets that

<a,e|a,o> = 0

Nevertheless, recombining both e and o paths at detector whose ket is |1>
is described by the unitary transformation

|a,b> -> |a,b>'=|a,+>|1>{[ (z1+z3)|b,e,+> + (z2+z4)|b,o,->]

Note, Ramsay, that your condition

|z1|^2+|z2|^2+|z3|^2+|z4|^2 = 1.

from <a,b|a,b> = 1

does not determine relative phases of the z's.

Unitarity of |a,b> -> |a,b>' requires

|z1|^2+|z2|^2+|z3|^2+|z4|^2 + z1*z3 + z2*z4 + c.c.

so that

z1*z3 + z2*z4 + c.c. = 0

In the actual state required by the physics  z2 = z3 = 0, so that this
extra condition for unitarity is automatically satisfied by conservation of
parity and angular mommentym in the J = 0->1->0 photon pair atomic cascade.

Thus, my dear Ramsay, (as I puff on my pipe and play my violin at 10 Baker
St.) there is nothing in the unitarity requirement of the  standard quantum
formalism that precludes both spin and space disentanglement of the initial
entangled pair state |a,b> into a disentangled one |a>|b> in which the
transmitter (a) phase (phi) information as transfered, on the connection
communication channel beyond space-time, to the receiver (b) along with @
the realtive angle between the detector calcites at the moments of the
respective scatterings of each photon in the same pair. QED!]

Thus writes Sarfatti.