- Uncertainty and nonlocality: a description of my paper with Stephanie Wehner, Science 2010.
- Quantum information can be negative: an accessible description of my paper with Horodecki and Winter in Nature.
- Sending quantum information down channels which cannot convey quantum information: A perspective in Science. A copy is available here.
- Quantum computing as free falling: A Science perspective on quantum computation as geometry. A copy is available here.
But English is a very silly language -- it is very inefficient, since it uses many letters over and over again, and has all kinds of spaces in it. So what we can do is "compress" these 2500 words so that the same information is conveyed, but by fewer letters. Shannon defined the information to be the minimum number of letters needed to convey a message. Notice that we abstract the information from the method used to send it. We don't care if the message is conveyed by regular mail, email, carrier pigeon or smoke signal.
In Nature's News and Views section, Patrick Hayden explained our work in a way which reminded me of the game show Wheel of Fortune. I'm not sure how universal this game show is, but I will explain information in terms of it. In Wheel of Fortune, you are given a bunch of boxes where letters should go, and you have to guess the sentence. Then the players start guessing letters until they have enough letters to complete the sentence. Watching the game is just like watching people play the game hang man, but with bigger prizes. One way to think of the information content of the sentence, is to think of it as the minimum number of letters you need before you can guess the sentence. The rest are redundant -- you don't need them to know what the sentence means -- you can guess the rest of the letters from the ones which are already there. So, the number of letters required before you know the secret sentence in Wheel of Fortune is much less than the full number of letters we usually use to spell it out. This is why we say that the information can be compressed. A compressed version of this webpage can be found here, and is about a third the size of this text, yet it contains all the information. You can uncompress it using this program
Partial and Prior Information
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.
An important point to mention is that the quantum properties of objects are extremely fragile. If you look at the object, you make it classical, in that you destroy its quantum nature.. Because the quantum properties of particles are so delicate, they must be kept in isolation from any contact with external objects, or they will become less quantum. This is hard to do, and is one of the reasons that building a quantum computer is so difficult. So basically, the elctron can be pointing both up and down at the same time, but if you look at it, it will point either up or down, just like we are used to. But if you don't look at it, it actually behaves as if it were both up and down.
Now, in classical information theory, Alice might send Bob the first letter of her phone number by sending him a piece of paper with the number "1" on it. And if we want to explore quantum information, she might send a small particle which encodes the number 1. For example, she might send an electron to Bob, which is pointing up (this means it is a 1). If she wants to send Bob the number 0, she would send him an electron pointing down.
But in quantum mechanics, the electron can also be sent so that it is pointing both up and down, so in a sense she can send him both 0 and 1 at the same time! Or neither 0 or 1 if you are a pessimist. This is a quantum message, and the first person to really quantify what quantum information meant was this other dude, Ben Schumacher, who said that we can quantify the amount of quantum information by the minimum number of electrons (or some other quantum particle), needed to convey a quantum message.
This might appear rather silly, and even simple, but it is neither and has enabled us to gain deep insights into the theory of quantum mechanics, and is now an entire area of study which includes such exotic phenomena as quantum computation, quantum teleportation and quantum cryptography.
Partial and prior quantum information
Strangely, the equation which quantifies how much partial information Alice needs to send, sometimes gives amounts which are negative. In other words, the equation says that there can be situations where Alice can send the full message to Bob, and the amount of quantum particles she needs to send Bob is negative. What on earth can this mean?
I will mention three ways of understanding negative information. They are just rough analogies, but they kinda make sense.
Quantum Wheel of Fortune (Pat, I'd
like to sell a vowel)
Some people with a bit of
knowledge about quantum information might want to know how you will end
up getting the vowels for free. The way you get them is as
follows: first remember that you have some prior quantum information,
and the host of the game show has the rest. The host is Pat
and his assistant is Vanna
White. But this is not
In order to send you the full quantum message and give
you the potential to get quantum vowels for free, Pat and Vanna will
perform some very weak measurement on their quantum words. The
measurement is so weak and so random, that they won't learn anything
about the state (thus the measurement won't
destroy the quantumness of the
words). Yes amazingly, the measurement is helpful!
They tell you the result of their measurement (we don't care about
communication -- it is classical and we want to quantify the quantum
information, so we only
count how much quantum particles they need to send you).
Then you perform some operation on your prior
quantum letters. After you do this, you will end up with the full
quantum sentence, and
you will also share some special states with Pat and Vanna.
These special quantum states (called Bell states), can be used to
teleport quantum words. Teleportation sounds all Star Trekky, but
it isn't really -- it is just a way to send quantum states without
needing to actually physically send quantum particles. You will
however have to talk on the telephone to use the Bell states as
Knowing too much
If I tell you, you'll know less.
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 colour of your ball, then there is an easy test to
figure out whether I really know something. I write the colour 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
the colour I wrote on the paper is the same as the colour 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 colour I wrote
on the piece of paper will be correlated with the colour of the
ball you gave her.
Now let us imagine that you give me your ball. Now I still know something, because the colour on my piece of paper (and the ball I have), is correlated with your brain, which remembers what the colour 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 colour 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 colour they are. A quantum ball can be completely isolated from anything which might record a memory of its colour (it is not just you who might have a memory of their colour, a ray of light which touches them can have a record of their colour). Quantum balls can be kept completely isolated so that no light touches them, and there is no record of their colour.
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 colour 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 colour 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!
I could send both the ball, and my piece of paper to the referee, and she could see that they are correlated, but the referee is not stupid. She knows that I could just paint the ball some colour and write that colour on the piece of paper, and then send everything to her. There is no way for me to prove to the referee that I have correlations with something else.
Executive SummaryNow that all the pieces are here, it might be good to restate the chain of logic.
- In classical information theory, information is measured by the minimum number of classical letters (bits) I have to send you, so that you get to know the message. If you already know part of the message, then I have to send you less bits (but never less than zero bits!).
- In quantum information theory, we likewise define the quantum information to be the minimum number of quantum particles (qubits) I need to send you, so that you get my (unknown) quantum message. If you already have part of this message, then I need to send you less quantum particles.
- We derive a formula for how many quantum particles I need to send, and find that sometimes, the amount can be negative.
- When it is negative, it turns out that you can get my message without me having to send you any quantum particles, and in fact, we get the corresponding potential for future sending of quantum particles for free.
- Another way to think about negative quantum information, is that if I just send you the negative information, the amount of quantum information you have will decrease.
Back to page 1