Quantum information can be negative

Based on work with M. Horodecki, and A. Winter

Even the most ignorant cannot know less than nothing. After all, negative information makes no sense. But, although this may be true in the everyday world we are accustomed to, negative information does exist in the quantum world. Small objects such as atoms, molecules and electrons behave radically different than larger objects -- they obey the laws of quantum mechanics.

What could negative information possibly mean? In short, after I send you negative information, you will know less. Such strange situations can occur because what it means to know something is very different in the quantum world. In the quantum world, we can know too much, and it is in these situations where one finds negative information. Negative information turns out to be precisely the right amount to cancel the fact that we know too much.

While all this might appear to be very mysterious (not to mention, an abuse of the word know!), negative information, can be put on a rigorous footing. I will try to explain how to do so here, in a manner which I hope is accessible to all. This description is intended for those who have an interest in this subject, but may not have a background in quantum information theory.  Most of this text should be understandable by anyone willing to put in a bit of effort, and the rest should be understandable by anyone with some knowledge of quantum mechanics (or by anyone willing to put in a lot of effort).  So if there are parts which continue to be unclear after some time, please let me know [email J.Oppenheim (at) damtp.cam.ac.uk], and I can modify this text to make things clearer. An executive summary of the result can be found at the end of this text here if you get impatient.

In order to concentrate on the main points, I have sacrificed some precision, and even some accuracy, so those with a modest background in physics should first try our recent article on negative information, which is available in Nature here.  It was written with Michal Horodecki, and Andreas Winter.   Patrick Hayden has written a commentary on it here, and a short description of the contents can be found in this piece by Andreas Trabesinger. The original version of the paper can be obtained at the pre-print arxiv here. It has a cartoon and George Orwell quotes, which were deemed inappropriate for Nature. We will shortly finish a more technical account which has the full proofs, calculations and details.

Lets first understand what information is (we'll get to the quantum part in a moment). Information was first defined rigorously by this dude Claude Shannon who essentially started the field of information theory.  In information theory, we are unable to deal with understanding the content of information, we just worry about how much information there is.  So, if Alice tells Bob something, we don't know how to quantify the value of what she is saying, just how much she says.  If you gave an information theorist a copy of War and Peace, she would say "Gosh that's a lot of information". Shannon said that the information is equal to how much communication is needed to convey it.  That makes sense -- if I send you a message using only ten letters, the message probably does't convey a whole lot.  This web page has about 2500 English words, so it is conveying more information (although I can't speak to the quality of that information).

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

Okay, now lets imagine that you already knew some of the letters during the game of Wheel of Fortune.  In other words, you have some prior information.  Then the partial information, is how many additional letters you need before you can guess the sentence.  In the game show Wheel of Fortune, you can buy vowels, but let us imagine that you have to buy all the letters.  Then the partial information is how many letters you will need to buy before you can complete the sentence.

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
as follows:
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.

Small objects such as atoms, molecules and electrons behave in a radically different manner than larger objects -- they obey the laws of quantum mechanics.  The laws of quantum mechanics are really strange.  In the everyday world, things appear to be in definite states -- a cat is either here, or there.  But in quantum mechanics, objects can be both here or there (or neither here nor there). At the same time!  For some reason (and we don't know exactly why), we only see such strange situations when things get really small.  So sometimes small particles like electrons behave as if they went through one slit and another slit at the same time.  Or the electron might be pointing up, and pointing down at the same time (electrons point in a direction!). You might find this rather confusing, and in fact, it is. Feynman (stealing from von Neumann), is rumoured to have said that "You don't understand quantum mechanics, you just get used to it." So, having accepted such ludicrous ideas (we must remove our prejudices which were developed in the world of large objects) lets move on...

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.

We were able to find out what the meaning of partial and prior quantum information is.  So if Bob already has some part of the quantum message, we were able to find out how many quantum particles Alice needs to send, before Bob gets the full message.  This is a bit tricky, because Bob doesn't really know what he knows (to keep the message quantum, he actually shouldn't read the message, or it will become classical).  To make matters worse, Alice doesn't know what Bob knows, or even what she knows.  While this sounds a bit Rumsfeldesque, it turns out that there is a way for Alice to send Bob some partial information so that he will learn everything.  

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. 

Remember Wheel of Fortune?  Well imagine that instead of trying to guess the title of books and movies and such, you are trying to gain possession of quantum messages (quantum titles of books and movies for example).  In the case when the partial information is positive, it would be like you having to buy some quantum vowels in order to get the full quantum sentence.  In the case where the partial information is negative, it would be like you can get the full quantum message, but you don't need to buy any quantum vowels.  In fact, you instead win the right to get quantum vowels in the future for free!  You won't need to buy vowels for some time, you can even sell a few of them if you want to get all capitalist. 

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 Sajak, and his assistant is Vanna White.  But this is not important.  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 this 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 teleportation devices.

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. 

It sometimes seems that we become more ignorant after talking to certain individuals. Perhaps they are saying things which are confusing or untrue. Well, after getting negative information, you know less. But not in the same sense as someone who tells you lies or tries to bamboozle you. Remember, that we don't worry about the quality of information (whether it is true or false for example). We just concern ourselves with how much there is. So, if we know less after receiving negative information, the amount of information we have must actually go down. This obviously cannot happen classically, but let me try to explain why it can happen quantumly.

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 whether 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 Summary

Now that all the pieces are here, it might be good to restate the chain of logic.

1. 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!).
2. 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.
3. We derive a formula for how many quantum particles I need to send, and find that sometimes, the amount can be negative.
4. 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.
5. 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.

Well, I hope what is here made some sense... If it did, consider trying the full article, it is also fairly non-technical, and I promise it won't contain any negative information.