Re: Teleportation of photons using entanglement

From: George Jones (george_llew_jones_at_yahoo.com)
Date: 03/06/05 

Date: Sun, 6 Mar 2005 07:51:12 +0000 (UTC)

Sci~Girl wrote:
> On howstuffworks.com... article on photon teleportation...
> http://travel.howstuffworks.com/teleportation2.htm
>
> The article only goes into a basic description of how entanglement
> works, and I wasn't able to find a more detailed one anywhere else. I
> was wondering if anyone knew of any other (accurate) sites that give a
> detailed description of the method. I'm writing a research paper on it
> for an independent study project.
>
> The article also mentions teleportation of a laser beam, and
> information on that would be helpful too.

I'll give the theory behind an experiemnt described in the 11 December
1997 issue of Nature. Maybe what I say won't be very helpful. In this
issue, Nature has a technical but readable article describing the
experiment and a non-technical article describing quantum entanglement.

Consider photons that are either horizontally (h) or vertically (v)
linearly polarized, so that state space is 2-dimensional, and suppose
that an EPR apparatus creates photons A and B which are in the entangled
state

|s> = 2^(-1/2)(|Ah>|Bv>-|Av>|Bh>),

and that A is sent to some experimental apparatus, while B
propagates freely. Suppose further that an arbitary photon, call it P,
with arbitrary state

|P> = a|Ph> + b|Pv>, a^2 + b^2 = 1

arrives (from anywhere) at the experiment apparatus at the same time in
and the same place as photon A. The state of the 3-photon system is thus

|u> = |P>|s>
     = (a/2)|Ph>|Ah>|Bv> - (a/2)|Ph>|Av>|Bh> + (b/2)|Pv|Ah>|Bv>
                - (b/2)|Pv>|Av>|Bh>.

The experimental apparatus then effectively performs a quantum
measurement on the photons P and A in such that they end up in the
entangled state

|q> = 2^(-1/2)(|Ph>|Av>-|Pv>|Ah>).

But before the measurement, A and B were in an entangled state |s>, so
this can't be done without affecting photon B, even though B doesn't
interact physically with the apparatus.

The effect of the measurement is represented mathematically by acting on
the first two components (P an A) of each term of the initial 3 photon
state |u> with the projection operator formed from the final combined
(and entangled) state |q> of P and A

Proj = |q><q|

      = |Ph>|Av>(<Ph|<Av| - <Pv|<Ah|) - |Pv>|Ah>(<Ph|<Av| - <Pv|<Ah|),

while operating on the third component (the B part) of each term of |u> by
the identity operator 1, since no measurement was performed on B.

After some algebra, one arrives at

(Proj x 1)|u> = -(a/2)|Ph>|Av>|Bh> + (a/2)|Pv>|Ah>|Bh>
                     - (b/2)|Ph>|Av>|Bv> + (b/2)|Pv>|Ah>|Bv>,

which factors into

                 (1/2)(-|Ph>|Av> + |Pv>|Ah>)(a|Bh>+b|Bv>)

Shazzam! Photon B, which, after creation, never came anywhere near the
experimental apparatus, now has exactly the state that the arbitrary
photon P had before the measurement (the interactions of photons P and
A with the apparatus). In effect the identity of photon P has been
teleported to photon B! Photon P as single entity has been "destroyed",
as it is now completely entangled with A.

Unfortunately, the measurement that created state |s> has 3 other
equally probable outcomes, sot this teleportation only happens 25% of
the time.

Regards,
George

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