Re: Quantum communication might be possible?

Greg Egan

I thought I'd offer a brief summary of some matters relevant to this
discussion. This is based on "Quantum Theory: Concepts and Methods" by
Asher Peres. I'll try to present things in an "interpretation-neutral"
and uncontroversial manner.

Bell's theorem
--------------

Suppose you have a source of two perfectly correlated photons
(anti-correlations can also be analysed, but I'll follow Peres's example)
that are sent to two observers. What we mean by "perfectly correlated"
(regardless of any underlying physical theory) is simply that if the two
observers measure polarisation *along exactly the same direction* then it
is 100% certain that they will get the same outcomes.

Now suppose that Alice can *either* measure polarisation along direction
A or along direction C, while Bob can *either* measure polarisation along
direction B or along direction C.

If both choose C, they will find that both photons passed through their
polarisation filters, or both did not. If they choose different
directions, any correlations between their results will depend on the
assumptions being made.

Now, Bell's theorem asks us to imagine that each photon carries with it
some local properties that determine what the result *would be* for *any*
choice of orientation of the filters. Under that assumption, it makes
sense to talk about all three results for each photon pair (would the
photon pass through the filters with directions A, B and C?) even though
only two measurements can actually be made. If we call the results
(whether actually measured, or merely guaranteed by the photon's
properties) a, b and c, with values +1 for passing through the filter and
-1 for being rejected, then for each pair we have:

a(b-c) = +/- (1-bc)

Why? Well if b and c happened to be equal, both sides give zero. If b
is not equal to c, both sides have magnitude 2.

If we take averages over many pairs of photons, it follows that:

|<ab> - <ac>| + <bc> <= 1

where <ab>, <ac>, <bc> are the correlations between the results. This is
Bell's inequality.

Now, QM can't predict individual results, but it can predict values for
the pairwise correlations, if the photons are prepared in a particular
entangled state:

(x_1 x_2 + y_1 y_2)/sqrt(2)

where x_1 is the state in which the first photon's polarisation is
aligned with the x axis, etc.

What QM predicts is that

<ab> = cos 2(A-B)
<ac> = cos 2(A-C)
<bc> = cos 2(B-C)

If we put A=0, B=pi/6, C=pi/3, this becomes:

<ab> = cos (-pi/3) = 1/2
<ac> = cos (-2pi/3) = -1/2
<bc> = cos (-pi/3) = 1/2

Then the LHS of Bell's inequality is 3/2, so the predictions of QM
violate the inequality.

Peres says that Aspect's experiment (in which the choices of direction
occurred at events with spacelike separation, i.e. they were causally
isolated from each other according to special relativity) violated a
related inequality by five standard deviations.

Quantum communication
---------------------

This is a big subject, but to summarise the basics without going into any
of the sophisticated refinements or detailed practicalities:

Alice and Bob receive sequences of photons which are correlated as above.
Alice and Bob randomly choose between two directions, V and W, which are
45 degrees from each other. After they have collected a large number of
measurements, they publish their choices of directions; this allows them
then to know when they made measurements along the same direction. Their
polarisation results when their directions coincided are known only by
them, and will agree, so they can use this sequence of bits as a secure
key.

Any eavesdropper/data-corrupter will not know the choices of direction
made by Alice and Bob, and so will be forced to make his own choices. If
Alice and Bob publish some further statistics about the data, it's
possible for them to verify that no eavesdropping or corruption of the
data has taken place.

Nick Maclaren

In article <20060118010542.E99924B58A@mail.netspace.net.au>,
Greg Egan <gregegan@netspace.zebra.net.au> wrote:
>I thought I'd offer a brief summary of some matters relevant to this
>discussion. This is based on "Quantum Theory: Concepts and Methods" by
>Asher Peres. I'll try to present things in an "interpretation-neutral"
>and uncontroversial manner.

Thanks for the posting. It doesn't quite match with what it is being
said in various talks and papers, but I started probing on this group
because I was unhappy at what those did say (and what they didn't!)
Of course, I may have misunderstood, but I don't think I have. What
I say below is what I was told and read.

>Quantum communication
>---------------------
>
>Alice and Bob receive sequences of photons which are correlated as above.

So this is either an entangled CHANNEL or the number of bits transferred
is fixed? I am not sure how an entangled channel of photons would
work, and I was definitely told that the number of bits was indefinite
(see below).

> Alice and Bob randomly choose between two directions, V and W, which are
>45 degrees from each other. After they have collected a large number of
>measurements, they publish their choices of directions; this allows them
>then to know when they made measurements along the same direction. Their
>polarisation results when their directions coincided are known only by
>them, and will agree, so they can use this sequence of bits as a secure
>key.

And it is safe from snooping because, if Cecil does snoop, he will
destroy the entanglement and Alice's and Bob's keys will not match.
That is a critical characteristic - without it, quantum communication
is no advance on distributing two copies of a one-time pad.

>Any eavesdropper/data-corrupter will not know the choices of direction
>made by Alice and Bob, and so will be forced to make his own choices. If
>Alice and Bob publish some further statistics about the data, it's
>possible for them to verify that no eavesdropping or corruption of the
>data has taken place.

Eh? At most, that will enable them to say that about the key (and
my previous remark addresses that). All of the papers and talks have
sworn that snooping on the DATA is also impossible, because at least
some of it is transferred by the entangled channel.

When I have asked the speakers about details, they have wittered on
and it has been clear that either they didn't understand me or didn't
understand what they were doing in detail. I wasn't placated ....

Regards,
Nick Maclaren.