“It’s your stock of entangled particles,” by iabervon, Slashdot, 8 August 2005, http://science.slashdot.org/comments.pl?sid=158439&cid=13274823.
In most situations we have prior information. For example, if Alice wants to tell Bob her phone number (which is ten digits long), and Bob knows three of the numbers in her phone number (he might sometimes know the area code if he knows where she lives), then Alice only has to send Bob seven of the numbers. So we can divide the information
Total information: 10 numbers
Prior information: 3 numbers
Partial information: 7 numbers
Notice that the total information is equal to the prior information plus the partial information.
Another way of understanding negative information, is that in quantum mechanics, you can know too much. Remember the telephone number example? Well, in that case, the total information (the phone number), was ten letters and Bob’s prior information was three letters. Alice needed to send him seven letters for him to get the total information. It turns out that in quantum mechanics, Bob could know more than the total amount of information. So he might know fifteen letters of information even though the total amount of information is only ten. So Alice can tell him the quantum phone number by sending him negative five quantum letters of information, which basically means that Alice and Bob can perform some tricks on their quantum letters so that Bob will learn the quantum phone number, and also, Alice will be able to send him more quantum letters in the future. Essentially, they will be able to convert part of their quantum letters into a resource which can be used to teleport quantum information between them.
A commentator on Slashdot expounds
The trick is that you can use quantum entanglement to have excess unspecified knowledge, which can be converted into specific knowledge. It’s like being on a quiz show where you are given a certain number of times you can look up an answer. These bonuses have to count in your total knowledge (I know 100 facts, plus I can look up things twice). If someone tells you something, you get positive information. If you look something up, you get zero information (you trade a bonus lookup for a fact). If you look something up, and you already knew the answer, you get negative information.
Now think about it as if someone else controlled the book. They can tell you things over the phone, and they can cause answers to pop out of the book. If they waste the book on something you actually already knew, your total information goes down, so the information in the transaction is negative.