Monochromatic Light & Information: Can It Convey?

[nomadreid]

I have read in several places that monochromatic light cannot convey information; e.g. in http://www.mathpages.com/home/kmath210/kmath210.htm. it is stated that "in order to actually convey information, a signal cannot be a simple periodic wave". But doesn't all light have the energy hf? And doesn't that mean that we can effect a change, for example by Schrödinger's equation? And isn't every change a piece of information? So why wouldn't a monochromatic pulse of light convey information?

 

[marcusl]

By definition a pure monochromatic wave is sinusoidal and infinite in duration, so must have no changes at all. But no information can be conveyed by a wave whose character is known exactly and never varies. In fact, change is required to convey information. Extending this to its logical conclusion yeilds Shannon's information theory, which states that maximum information transfer corresponds to maximum entropy or minimum a priori knowledge of what signal to expect. This is why a monochromatic wave cannot carry information.

 

[nomadreid]

Thank you. There seems to be a problem with semantics: your reply makes perfect sense if one knows in advance that the light is monochromatic. If one is however the observer receiving a signal, and not knowing what the signal is, then even 11111111111... will constitute information because it is different from, say, 1010101010101010...

 

[marcusl]

As the signal duration becomes infinite and the frequency content a delta function, the information content approaches zero. Try sending your paragraph (which contains little information by modern standards, only a few hundred bytes) over a monochromatic beam. You can't.

 

[nomadreid]

I understand that not much information can be transmitted on a monochromatic wave, but there is a difference between "not much" and "zero". Even if it transmits only a single bit of information, this would be different to the statement that it can transmit none. To take a silly example, it is agreed that as long as energy is being received from your planet, we will not destroy it. Monochromatic light is not no light, and it does carry energy; energy, by its very definition, is something which can cause a measurable change. No?

 

[marcusl]

No. If you allow it to turn off, to signal that we want our planet to be destroyed, the wave is not monochromatic--it's been modulated, and that modulation carries the information. True monochromaticity can only be accomplished by infinite time. Have you studied Fourier theory? Review the time-frequency uncertainty relations in Bracewell or Papoulis. A monochromatic wave existed before the Big Bang, before one planet could negotiate with another, and it exists unchanged after both planets die and the universe explodes or crunches or whatever it does. In the limit of infinite duration and delta function frequency, there's no information conveyed. Not "not much." None.

 

[nomadreid]

Thank you, Marcus. Yes, I have studied the basics of Fourier Theory, and I shall re-think the issue on the lines you recommended. Thanks again.