From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Newsgroups: sci.physics
Subject: Everything is unitary!
Date: 20 Aug 92 11:00:06 GMT
Organization: Whole Earth 'Lectronic Link


Quantum Spin Connection Communication is Unitary
by Jack Sarfatti

#1 The alleged Achilles Heel of faster-than-light (FTL) and backward-in-
time (BIT) communicator designs is not causal paradox but unitarity. In
the particular case I have put forward the consensus of opinion of Brian
Josephson, Carlton Caves, Aephraim Steinberg and Michael Gallis is that
my error is in the apparent non-unitarity of the local transformation at
the transmitter beam recombiner. I will show that this transformation is
one-sided unitary for pair-correlated light and that the response of the
recombiner for pair correlated light is not the same as the response of
the same recombiner for ordinary uncorrelated light.

#2 Let us recall the configuration for the transmitter. A doubly-
refracting calcite rhomb provides two alternative (i.e., extraordinary E
and ordinary O) paths for the transmitter photon 1. The linear
polarization of the photon in the E path is perpendicular to that photon
in the O path. The absolute orientation of this polarization frame
(i.e., spin basis) is determined by a plane in the anisotropic crystal
at the moment that the photon locally interacts with the crystal. The E
output from the calcite passes through a phase plate whose position can
be changed between successive photons to provide a variable relative
phase shift phi(1) which will be the modulation variable on which to
encode the message. The E output from the phase plate P is externally
reflected by a non-absorbing mirror M (i.e. reflection phase shift of pi
for M) to the input port I(M) of a 50-50 beam recombiner B. The O output
from the calcite passes through a half-wave plate H placed at 45 degrees
to the plane of O polarization. The half-wave plate, therefore, rotates
the O polarization by 90 degrees so that it coincides with the E
polarization. The output from the half-wave plate is fed into the second
input port I(H) of the recombiner. The two outputs I(M) and I(H) from
the recombiner are each fed to separate transmitter counters C(1+) and
C(1-).

#3 The receiver placed on the opposite side of a photon pair source is
an identical calcite rhomb oriented at angle x relative to the
transmitter rhomb. The local polarization frame of the receiver rhomb is
denoted by e and o. The e and o outputs from the receiver calcite are
each fed into separate receiver counters C(2+) and C(2-). The message
encoded on the photon pair spin connection by varying phi(1) in time for
different pairs is locally decoded by measuring the time variation of
C(2+)-C(2-). According to the "unitarity" arguments this difference is
absolutely zero under all conditions. I dispute this conclusion.

#4 Note that in the "delayed choice" mode where the arrival of
transmitter photon 1 is delayed relative to the arrival of its twin
receiver photon 2, the message can be decoded before it is encoded! The
usual time sequence between cause and effect is reversed.

#5 Furthermore, it can be shown that the receiver can sense the
difference between a left and right-handed rotation of the far-away
transmitter calcite so that this causality-violating process also
violates the mirror symmetry of parity like the weak force.

#6 Such a causality-violating superluminal universe need not lead to
paradoxes as shown in recent papers by Kip Thorne et-al. Indeed, Stephen
Hawking has invoked superluminal effects in quantum gravity to explain
why the universe started off: a) spatially homogeneous and isotropic
"even though different regions were not in causal contact according to
the classical metric"; b) in thermal equilibrium; c) why there are no
magnetic monopoles; d) why the universe is spatially flat; e) why the
cosmological constant is zero.

#7 The low energy mechanism I propose here has no obvious connection to
ultra-high energy quantum gravity at the Planck scale of 10-43 seconds,
but it can provide the mechanism for the intelligent creation of the
universe from the future in a kind of super-strong Anthropic
Cosmological Principle.

#8 (For the record, my old high school peer Professor Robert Solovay of
the Department of Mathematics at U.C. Berkeley calls this "drivel".)

#9 First consider ordinary uncorrelated light incident on the
transmitter. Suppose the single photon state is

|1> = cosx|E1> + sinx|O1>  (1)

The phase plate P induces the transformation

P|1> = e^iphi(1)cosx|E1> + sinx|O1>  (2)

The mirror M induces

MP|1> = -e^iphi(1)cosx|E1> + sinx|O1>  (3)

The half-wave plate induces

HMP|1> = -e^iphi(1)cosx|E1> + sinx|E1>

       = [-e^iphi(1)cosx + sinx]|E1>   (4)


What does the beam recombiner B do? Let the relative phase shift between
the reflected and the transmitted beams be b. Let the transmission
amplitude transfer function be T = |T|e^iphi(T) (5) and the reflection
transfer function be R = |R|e^iphi(R) (6) so that b = phi(R) - phi(T)
(7). Let the input ket to transmitter counter C(1+) [C(1-)] be |1+>
[|1->] where <1+|1-> = 0 (8). Therefore

|1>' = BHMP|1>

= [-Re^iphi(1)cosx + Tsinx]|1+> + [-Te^iphi(1)cosx + Rsinx]|1->  (9)

Unitarity means conservation of probability. Therefore,

|-Re^iphi(1)cosx + Tsinx|^2 + |-Te^iphi(1)cosx + Rsinx|^2 = 1      (10a)


|R|^2 (cosx)^2 + |T|^2 (sinx)^2 - 2sinxcosx|R||T|cos(phi+b)

+ |T|^2 (cosx)^2 + |R|^2 (sinx)^2 - 2sinxcosx|R||T|cos(phi-b) = 1  (10b)


Use |R|^2 + |T|^2 = 1  (10c)

Therefore,

1 -2sinxcosx|R||T|cos(phi+b)- 2sinxcosx|R||T|cos(phi-b) = 1        (10d)


2sinxcosxcos(phi)cos(b) = 0                                        (10e)

cos(b) = 0 for uncorrelated light if no absorption  (10f)

If there is absorption

|R|^2 + |T|^2 = 1 - |A|^2      (10g)

1-2sinxcosx|R||T|cos(phi+b)-2sinxcosx|R||T|cos(phi-b)+|A|^2 = 1 (10f)

so that b is shifted by absorption to

cos(b) = |A|^2/2sinxcosx|R||T|cos(phi)     (10g)

Clearly b is not an absolute constant independent of the input to the
recombiner.



#10 The effect of absorption or stimulated emission for an active beam
recombiner will be treated in another paper. Notice, if b could be
forced to be other than 90 degrees it would signal the opening of new
channels like resonant absorption or even, perhaps, the cohering of
vacuum fluctuations.

#11 The transmitter for ordinary light will show first order coherence
(local fringe visibility) for the interferogram at each counter. For
example the response probability of C(1+) is

p(1+) = |<1+|1>'|^2

= |R|^2 (cosx)^2 + |T|^2 (sinx)^2 - 2sinxcosx|R||T|cos(phi+pi/2)

= |R|^2 (cosx)^2 + |T|^2 (sinx)^2 + 2sinxcosx|R||T|sin(phi)        (11a)

and p(1-) = 1 - p(1+)       (11b)

#12 Now consider pair correlated light of the kind used in actual
experiments by John Clauser et-al from a double atomic cascade of an
electron in a calcium atom.

In terms of the respective local bases of the two calcites, initial
photon pair state emitted by the source is

|1,2> = [cosx/sqrt2]{|E1>|e2> + |O1>|o2>}

      + [sinx/sqrt2]{|E1>|o2> - |O1>|e2>}    (12)

The final pair state is given by

|1,2>' = BHMP|1,2>

= [cosx/sqrt2]{-e^iphi(1)R|1+>|e2> + T|1+>|o2>}

+ [sinx/sqrt2]{-e^iphi(1)R|1+>|o2> - T|1+>|e2>}

+ [cosx/sqrt2]{-e^iphi(1)T|1->|e2> + R|1->|o2>}

+ [sinx/sqrt2]{-e^iphi(1)T|1->|o2> - R|1->|e2>}

= [cosx/sqrt2]{-e^iphi(1)R|e2> + T|o2>}|1+>

+ [sinx/sqrt2]{-e^iphi(1)R|o2> - T|e2>}|1+>

+ [cosx/sqrt2]{-e^iphi(1)T|e2> + R|o2>}|1->

+ [sinx/sqrt2]{-e^iphi(1)T|o2> - R|e2>}|1->

= {-e^iphi(1)[cosx/sqrt2]R - [sinx/sqrt2]T}|e2>|1+>

+ {[cosx/sqrt2]T - [sinx/sqrt2]e^iphi(1)R}|o2>|1+>

+ {-e^iphi(1)[cosx/sqrt2]T - R[sinx/sqrt2]}|e2>|1->

+ {[cosx/sqrt2]R -e^iphi(1)[sinx/sqrt2]T|}|o2>}|1->   (13)

Conservation of probability in the final pair state requires (in analogy
to eq.(10) above for ordinary uncorrelated light) that the sum of the
squared moduli of the pair basis add to unity. Thus,


  |-e^iphi(1)[cosx/sqrt2]R - [sinx/sqrt2]T|^2

+ |[cosx/sqrt2]T - [sinx/sqrt2]e^iphi(1)R|^2

+ |-e^iphi(1)[cosx/sqrt2]T - R[sinx/sqrt2]|^2

+ |[cosx/sqrt2]R -e^iphi(1)[sinx/sqrt2]T||^2 = 1       (14a)


  2[(cosx)^2/2]|R|^2 + [(sinx)^2/2]|T|^2

+ 2[(cosx)^2/2]|T|^2 + [(sinx)^2/2]|R|^2

+ 2sinxcosx|R||T|cos(phi+b)/2 - 2sinxcosx|R||T|cos(phi+b)/2

+ 2sinxcosx|R||T|cos(phi-b)/2 - 2sinxcosx|R||T|cos(phi-b)/2  = 1

(14b)

But this is an identity because

2[(cosx)^2/2]|R|^2 + [(sinx)^2/2]|T|^2

+ 2[(cosx)^2/2]|T|^2 + [(sinx)^2/2]|R|^2 = 1        (14c)

Therefore, in the case of pair correlated light the transformation is
unitary at the pair level automatically for any value of the beam
splitter net reflection phase shift b. It is not forced to be 90 degrees
as it is in the case of uncorrelated light. There is no apriori reason
to suppose that the reflection phase shift b of the beam recombiner must
be independent of the nature of the input state. Indeed, one can imagine
designing a beam recombiner with paths of different optical length in
the multiple reflections inside the glass plate. The effective shift
would not generally be 90 degrees.

#13 Aephraim Steinberg's objection is that cosb = 90 degrees because
that is what it is for uncorrelated light in the absence of absorption.
Absorption will shift b away from 90 degrees in order to conserve energy
(e.g., eqs. 10 above). If, in fact, cosb is zero then there will be no
communication on the connection by this method. My point is that energy
conservation and probability conservation do not require cosb to be zero
for pair correlated light. In fact, to get cosb = 0 even for ordianry
light we also need time-reversal symmetry (e.g., no irreversible
absorption).


#14 The receiver counter responses are:

p(2+) = |<e2 1+|1,2>'|^2 + |<e2 1-|1,2>'|^2

= {1 + |T||R|sin2xcos[phi(1)]cos(b)}/2

p(2-) = |<e2 1+|1,2>'|^2 + |<e2 1-|1,2>'|^2

= {1 - |T||R|sin2xcos[phi(1)]cos(b)}/2

p(2+) + p(2-) = 1


The transmitter counter responses are

p(1+) = |<e2 1+|1,2>'|^2 + |<o2 1+|1,2>'|^2 = 1/2

similarly p(1-) = 1/2. Note that the phi dependence has disappeared from
the transmitter for pair correlated light.  It has "teleported" over to
the receiver provided that cosb is not zero. The quantum spin connection
signal is phase (phi) information. If we also vary x we get parity
violation.
Newsgroups: sci.physics