Re: FTL by Down-converting
From: Ralph Hartley (hartley_at_aic.nrl.navy.mil)
Date: 10/27/04
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Date: Wed, 27 Oct 2004 15:55:33 +0000 (UTC)
Horace Heffner wrote:
> FTL by Down-converting
Figuring out why such schemes don't work can be an interesting puzzle,
similar to figuring out perpetual motion machines, but this one is way too
easy.
> [...] A down-converter splits a photon into two photons each having half
> the energy of the original photon.
>
> Suppose we have a sender Alice, a receiver Bob, [...] a beam splitter
> create[s] two beams of laser light: L the left beam and R, the right
> beam. Charlie then down-converts the L beam to create beams L1 and L2,
> and similarly creates beams R1 and R2 from the beam R L1 and R1 [go] to
> Alice and beams R2 and L2 to Bob. The corresponding photons arrive at
> both Bob and Alice at nearly the same time, but here assume Alice
> receives hers first, but just barely before Bob.
>
> Bob directs beams R2 and L2 such that they can create an interference
> pattern [...] The signal photon beams R2 and L2 can create such an
> interference pattern because they are the two paths from a beam
> splitter.
>
> Bob will in fact see such an interference pattern provided Alice does
> not put detectors in idler beams R1 and L1.[1] If Alice does place
> detectors in both her beams, then this is equivalent to knowing which
> path each of Bob's photons have traveled, and thus Bob can observe no
> interference pattern.
The existence of R1 and L1 prevent Bob from seeing an interference pattern
regardless of what Alice does.
This is related to the famous "no cloning" theorem.
To see that this must be so, consider the case where the photons reach
Alice *after* Bob. Alice can transmit the information back to Bob, who then
knows which path his photons followed. If Bob can determine the path he
doesn't see interference.
The error is in assuming that a part of the system that is never measured
can just be ignored. Quantum Mechanics has a procedure for getting a
description of a part of a system from a description of the whole system,
known as a "partial trace".
Basically, it amounts to averaging over the possible states of the excluded
parts. In this case the system consists of L1, R1, L2, and R2. If L1 and R1
escape to infinity without being measured, we need to trace over (a kind of
average) them to get the state of L2 and R2.
The result is a "mixed" state, in which there is no interference.
There are ways for Alice to "erase" the path information in L1 and R1, and
restore the interference, but because QM preserves information, they all
require her to have access to L2 and R2.
Ralph Hartley
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