August 14, 2005

I'll take a vowel, Vanna, and then I'll know less--What if I gave you information that would result in your knowing less than you did before I gave it to you--negative information? Can there be such a thing? Well, yes, in the quantum world, anything is possible.

Jonathan Oppenheim at Cambridge has a wonderful webpage describing negative information, how you can actually reduce knowledge by adding to it. It's great fun--if you believe in quantum mechanics or existentialism.

Oppenheim points out that in information theory, “this dude Claude Shannon” pointed out that understanding information is less important than understanding how much there is. Give an information theorist a copy of War and Peace, and she would say, “gosh, that’s a lot of information.” [Woody Allen once said he took a speed-reading coure and read Moby Dick in 15 minutes. It’s about a whale.] A message with 10 letters doesn’t transmit a lot, so Shannon defined information as the minimum amount of letters needed to convey a message, irrespective of how it is delivered.

Think of the television show “Wheel of Fortune,” in which you have to guess a sentence as letters are added to a matrix. Once you have the meaning of the sentence, the rest of the letters are irrelevant. "Wheel of Fortune" uses information theory. Now consider there is something called prior information--knowing some of the letters before the game began--and partial information--the number of letters you need before you know the sentence. Got it?

OK, but the quantum world is different. It is, you should pardon the expression, existential. In Erwin Schroedinger’s wonderful quantum cat mind exercise, the answer to the question, is the cat alive or dead, is yes. Is an object here or there? Yes. It could also be neither here nor there. Einstein found that notion repulsive, but generally, that’s what physicists think is the nature of nature when dealing with very small objects. Feynman (stealing from von Neumann), is rumored to have said that "you don't understand quantum mechanics, you just get used to it." Also keep in mind, that quantum objects are fragile. Look at one, and it stops being quantum. An electron can be pointing up and down simultaneously. The moment you look at it, it is either one or the other, or with Schroedinger’s cat, it is either dead or alive. So, if you decide that an electron pointing up is a 1 and one pointing down is a 0, the first person who looks at the electron removes it from its quantum state and decides it is one or the other. That’s why quantum computing is so difficult.

OK, now negative information. A mythical Bob in Oppenheim’s paper, has some prior information so that when the equally mythical Alice wants to send him a message, we can figure out how much information she must send before he understands the message. That’s tough because Alice doesn’t know what Bob knows, and in the quantum world, Alice doesn’t even know what Alice knows. [Alice is a good choice of names for this exercise]. Oppenheim uses a new adjective to decribe this, "Rumsfeldeque." Love it.

He then points out that the equation used to quantify how much information Alice has to send--how much partial information is required for Bob to understand the message--can produce a negative number. Huh?

Well, you probably are going to have to read this more thoroughly on your own, but here is Oppenheim’s example from his web page.

Information is always information about something. This can be expressed in terms of correlations. If you have either a white or a black ball, and I claim to know the color of your ball, then there is an easy test to figure out whether I really know something. I write the color of your ball on a piece of paper, and you put your ball in a box. Then we give the piece of paper and the box to a referee, who checks whether the color I wrote on the paper is the same as the color of the ball in the box. The referee can determine that I know something about what you have. If we repeat this test over and over again, the fact that I have information about your ball, will manifest itself in the fact that the referee will always find that the color I wrote on the piece of paper will be correlated with the color of the ball you gave her.

Now let us imagine that you give me your ball. Now I still know something, because the color on my piece of paper (and the ball I have), is correlated with your brain, which remembers what the color of the ball was. So by giving me the ball, you haven't changed anything: a referee can still perform a test to determine if I know something. The referee will ask me for the piece of paper, and ask you to write down the color of the ball you sent me. If they match, we have proved to the referee that I know something about what you have (or had).

But if the balls were quantum particles, then it can sometimes be that there is no memory of what color they are. A quantum ball can be completely isolated from anything which might record a memory of its color....

Now let us imagine that you have a quantum ball, which is black or white, and I have a piece of paper where it is written what color the ball is. Then I have some information, because my piece of paper is correlated with your ball. We could convince the referee that I know something about what you have. But if you give me the ball, then I will no longer have any correlations with things external to me. You have no memory of the color of the ball, and so there is no test we can perform with the referee which will prove that I knew something. By giving me the ball (giving me information), I now know less!

And, now, I think, I do too. Well, not really.

While we're on the subject of the weird and fascinating, may I recommend this? Report back.

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