entanglement and information teleportation

Physicists:

I have a question regarding quantum computing, entanglement, wave
interference, and information teleportation. No doubt, you have had
this discussion before, and one of you will be able to set me straight
with some quick advice.

My question is inspired by the two-slit experiment (conventionally
used to introduce the peculiarities of quantum mechanics to students
of classical physics). Let me explain my reasoning so that I may be
clear:

Where my question comes from:
Experimental Control of Interference Pattern:
It is my understanding that an experimenter may 'turn-off' the
interference pattern produced by particles passing through the two-
slits and striking a photographic plate by measuring which of the two
slits the particle passed through to reach the plate from the source.
Imagine that the particle in question is an electron. The
experimenter might choose to perform this measurement by shining a
laser beam in front of one or both of the slits. When an electron
passes through the laser beam, it will diffract some of the laser's
photons, producing a flash of light. Something in the act of
performing this measurement or acquiring the information about the
location of the electron 'collapses' its wave state. The collapse
causes the electron to pass through only one slit, preventing it from
interfering with itself to produce the interference pattern.

Given a stream of electrons that would normally pass through both
slits and produce an interference pattern, two experimenters could
play a game by which an experimenter controlling the laser beam could
send a message to an experimenter watching the photographic plate
using a binary signaling scheme (like ASCII or Morse Code) where
"laser-on, no interference" corresponds to one signal and "laser-off,
interference pattern present" corresponds to the other signal.

Entanglement and the Two Slit Experiment:
Now imagine that an experimenter has contrived to send two entangled
electrons through two separate two-slit apparatuses such that knowing
which slit the electron has passed through in one apparatus allows the
experimenter to predict which slit the electron passed through in the
other apparatus. Maybe this is done by entangling the momentum of the
electrons (so that if one goes up, the other goes down). Or this is
done by producing electrons with entangled polarizations (so that the
magnetic-moment vectors of the two electrons sum to zero) and passing
the electrons through magnetic fields that sort the electrons (by
polarization) into the two slits.

In this setup, two experimenters could play the same communication
game that the experimenters played in the more mundane two-slit
experiment. Perhaps the two experimenters share a building, but have
offices on opposite ends. The experimenters could arrange for the
entangled electrons to be produced in the hall half-way between their
offices, and for one of the two-slit apparatuses to be placed in each
office. The experimenters could take turns sending each other
messages. On an experimenter's turn, the experimenter would turn a
laser (that allowed the experimenter to measure which slit the
electrons where passing through) on and off to represent symbols in a
binary signaling scheme. When it was the not an experimenter's turn,
the experimenter would watch the photographic plate in the apparatus
assigned to the experimenter's office to determine if an interference
pattern was present or not (in order to determine which symbol the
other experimenter was sending).

Where This Goes:
Now imagine that the entangled particles could be stored in their
entangled state. Some entangled particles could be accelerated (along
with the appropriate measuring apparatus and an experimenter) up to a
large fraction of light speed (if we ignore the obvious impracticality
of this proposition...). If an experimenter remains at rest with the
other half of the entangled states that were accelerated to tremendous
speed and the two experimenters try to repeat the game... well, they
won't be able to agree on who measured a particle from a given
entanglement set first. It seems like the experimenters have
uncovered a paradox. (If some solution to the simultaneity paradox
can be found, then the door may be opened to causality paradoxes....)

I suppose for small divergence in simultaneity-space, this does not
necessarily cause a paradox. If the two apparatuses are either close
together or in nearly the same relativistic frame, then the resulting
instantaneous communication is not likely to produce paradoxes.

Quantum Computer Science:
The Usual Approach:
In quantum computation theory, the question of instantaneous
communication is often dealt with by considering whether or not Bob
can detect Alice's measurement of the first qubit in

(1) phi = |01>/sqrt(2) + |10>/sqrt(2)

and in so doing, either measuring |0> (choosing the first state), and
leaving Bob's second qubit as |1>; or measuring |1> (choosing the
second state), and leaving Bob's second qubit as |0>.

The conclusion seems clear: if Alice has not measured q1, Bob has
equal probability of detecting q2 = 0 or 1. If Alice measures q1, she
has equal probability of detecting q1 = 0 or 1. Then Bob will detect
q2 = 1 or 0 respectively. Alice's measurement predicts Bob's
measurement, but does not change the probability and does not transfer
any information. No paradoxes.

Where I Start To Go Wrong:
What if Bob decides to measure the interference statistics of q2
instead of measuring q2 outright? Bob has

(2) q2 = (alpha)|0> + (beta)|1>

and where q2 is the second qubit from

(3) phi = (a)|00> + (b)|01> + (c)|10> + (d)|11>

and, when entangled as in the initial example, then

(4a) alpha = a + c = 0 + 1/sqrt(2) = 1/sqrt(2)
(4b) beta = b + d = 1/sqrt(2) + 0 = 1/sqrt(2)

If Bob has some way to detect interference, he can measure a statistic
akin to the difference between alpha and beta, that could be called
'I', such that

(5) K - I = mon1(abs(mon2(alpha) - mon2(beta)))

(where mon1(x) and mon2(x) are monotonically increasing functions and
'K' is some positive constant), or at least schematically

(6) I = 1 - abs(alpha - beta)

If Bob makes his measurement before Alice, alpha and beta are equal
(as in (4)) and Bob should acquire information suggesting a large I
(the presence of interaction between the two superimposed states of
q2). If Alice makes her measurement before Bob, then either she
measures q1 = |0> and 'causes'

(7a) alpha = 0
(7b) beta = 1

or she measures q1 = |1> and 'causes'

(8a) alpha = 1
(8b) beta = 0

In either case, Bob should obtain information suggesting a small I
(the absence of interaction between superimposed states in q2).

Where This One Goes:
Alice and Bob might recognize that Bob acquires a fairly miserable
certainty about the value of I from a single measurement. In this
case, Alice and Bob might agree to have Alice either measure or not
measure q1 in N distinct particle-pairs entangled as in (1). Bob will
compile statistics on the interference between alpha and beta in q2 of
the N pairs. By adjusting the value of N, Alice and Bob can set Bob's
certainty about the value of 'I' that Alice intends him to measure
arbitrarily to any finite probability.

So What Am I Missing?
What do the quantum computational theorists have to say about
measurements that reveal interference statistics like 'I'?

If I imagine that I have accounted for the best understanding of these
ideas in the above discussion, then I could make certain statements
about physics.

By this, I mean that either I am missing something, or entanglement
really does allow for instantaneous communication. If instantaneous
communication is possible using entanglement, then there must be
solutions for the simultaneity or causality paradoxes, or it will
prove impossible to maintain entanglement between relativistic
frames. That is, that the entangled pair-halves on board an
accelerating space-craft will quickly lose their entanglement.

Now that I am way out in speculative limbo, I might as well go on.
Could this mean that relativistic divergence is responsible for the
collapse of the wave function? (This as opposed to Roger Penrose's
gravitational energy divergence causing wave collapse.)

_james

.

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