The Faintest Signals

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Margaret Valmikki was my grandmother. You may have heard her name in
discussions about polar communication. She is credited with the fundamental
idea and with working out its mathematical background. But she never took
credit for the idea herself.

The recent discovery has thrust polar communication to the public spotlight. It
has been featured in the news every day for the past month in some way. I am
sure you are all captivated by the ongoing story. You may also find it
interesting to learn of how polar communication was invented.

Margaret has given me permission to read you some letters she wrote to my
grandfather during her groundbreaking work.

Dear Leo,

I thank you sincerely for your charming letter. We still have not been able to
find a viable solution, and the team is under much pressure as the next launch
window approaches. Your letters are much needed relief.

The answer to your physics question is yes. A photon’s polarization is an
angle, so it can take the value of any real number from 0° to 360°. It is a
quantum state, however, and cannot be measured exactly. We can only extract one
bit of information from it, such as whether it is vertical or horizontal.

On the contrary, you can extract more than one bit of information from me. I
would encourage you to do so. I am always looking forward to your letters.

(The letter continues with more personal topics.)

Dear Leo,

Thank you for your enthusiastic letter. I love being reminded of how exciting
my job is. Working on the mundane issues from day to day it is easy to forget.
I am very happy to tell you more about radio communication.

The challenge lies in that if we move twice as far from a radio source, it
becomes four times fainter. You phrased your question in terms of photon
counts, but I have to warn you it is a misleading view. Power is a more
intuitive measure. You may think millions of billions of photons would be easy
to detect, but when I say “nanowatt” it should become clear that this signal is
extremely weak.

But this is what you and me, and all humans, are great at. Looking for the
smallest signs and making sense of them, be they a distant radio source or the
hint of something more in a friend’s smile.

Dear Leo,

I do mean to write you at length soon, but work keeps getting in the way. This
is just a quick note to answer your two questions.

Firstly, you must visit me in July. It is the best month to visit. I will find
a way for us to break away for a weekend!

Secondly, the error correcting code I am working on operates on a sequence of
bits. Missing some part of the message is called an “erasure”. Recovery may or
may not be possible, depending on how much was lost. I always say it is best
not to miss anything!

Dear Leo,

Thank you for your inspiring letters. When I await your next letter it happens
that I give our older letters a fresh read. Sometimes I find bits that I had
missed earlier. Faint signals.

A thoughtful reading of our letters this year has given me a wonderful idea.
But you must have had this idea yourself earlier. Or were your questions just a
happy coincidence? I cannot believe so.

A digital message in the form of a sequence of bits is just a number. There are
infinitely many real numbers between 0° and 360° — we can map every possible
message to a unique angle. Suppose we can create a photon with a polarization
corresponding precisely to this angle. This one photon alone would then contain
the entire message, no matter how long.

Too bad we cannot measure the precise angle of the polarization of a single
photon. Only a single bit can be learned from each photon, and even that is
subject to probabilities. The transmitter would have to send a stream of
identically polarized photons, and the receiver would be playing twenty
questions with a very uncertain individual. How many photons do we need to
extract a single bit then? The mathematics of the answer are particularly
interesting. I already started writing them up and will send you a copy of the
article as soon as I am done. Thank you for leading me to this beautiful
puzzle!

It will not be for a long time that we can build such a receiver, let alone a
transmitter. This is my only regret as, if we could, it would solve all my
current problems. We would no longer need to worry about error correction. It
would not matter how many photons we miss as long as we receive enough.

A million billion photons no longer feel so few anymore. A fainter signal will
just take a longer time to receive. With enough patience we can always
understand each other, no matter how far.

Love,

Maggie

Margaret has also asked me to tell you how happy she is to have lived to see
the first polar photon receiver built. Since the receiver has found the
extrasolar signal she is her young self again. Like many of us, she thinks it
must be an advanced alien race that was able to build her theoretically optimal
transmitter. Together with Leo they wish the best of luck to everyone working
on deciphering the message.

What do you hope the message will say?

About the Author: 
A software engineer from Hungary.