Tag Archives: Quantum information

Stop whining and accept these axioms.

One of the stated goals of quantum foundations is to find a set of intuitive physical principles, that can be stated in plain language, from which the essential structure of quantum mechanics can be derived.

So what exactly is wrong with the axioms proposed by Chiribella et. al. in arXiv:1011.6451 ? Loosely speaking, the principles state that information should be localised in space and time, that systems should be able to encode information about each other, and that every process should in principle be reversible, so that information is conserved. The axioms can all be explained using ordinary language, as demonstrated in the sister paper arXiv:1209.5533. They all pertain directly to the elements of human experience, namely, what real experimenters ought to be able to do with the systems in their laboratories. And they all seem quite reasonable, so that it is easy to accept their truth. This is essential, because it means that the apparently counter intuitive behaviour of QM is directly derivable from intuitive principles, much as the counter intuitive aspects of special relativity follow as logical consequences of its two intuitive axioms, the constancy of the speed of light and the relativity principle. Given these features, maybe we can finally say that quantum mechanics makes sense: it is the only way that the laws of physics can lead to a sensible model of information storage and communication!

Let me run through the axioms briefly (note to the wise: I take the `causality’ axiom as implicit, and I’ve changed some of the names to make them sound nicer). I’ll assume the reader is familiar with the distinction between pure states and mixed states, but here is a brief summary. Roughly, a pure state describes a system about which you have maximum information, whereas a mixed state can be interpreted as uncertainty about which pure state the system is really in. Importantly, a pure state does not need to determine the outcomes to every measurement that could be performed on it: even though it contains maximal information about the state, it might only specify the probabilities of what will happen in any given experiment. This is what we mean when we say a theory is `probabilistic’.

First axiom (Distinguishability): if there is a mixed state, for which there is at least one pure state that it cannot possibly be with any probability, then the mixed state must be perfectly distinguishable from some other state (presumably, the aforementioned one). It is hard to imagine how this rule could fail: if I have a bag that contains either a spider or a fly with some probability, I should have no problem distinguishing it from a bag that contains a snake. On the other hand, I can’t so easily tell it apart from another bag that simply contains a fly (at least not in a single trial of the experiment).

Second axiom (Compression): If a system contains any redundant information or `extra space’, it should be possible to encode it in a smaller system such that the information can be perfectly retrieved. For example, suppose I have a badly edited book containing multiple copies of some pages, and a few blank pages at the end. I should be able to store all of the information written in the book in a much smaller book, without losing any information, just by removing the redundant copies and blank pages. Moreover, I should be able to recover the original book by copying pages and adding blank pages as needed. This seems like a pretty intuitive and essential feature of the way information is encoded in physical systems.

Third axiom (Locality of information): If I have a joint system (say, of two particles) that can be in one of two different states, then I should be able to distinguish the two different states over many trials, by performing only local measurements on each individual particle and using classical communication. For example, we allow the local measurements performed on one particle to depend on the outcomes of the local measurements on the other particle. On the other hand, we do not need to make use of any other shared resources (like a second set of correlated particles) in order to distinguish the states. I must admit, out of all the axioms, this one seems the hardest to justify intuitively. What indeed is so special about local operations and classical communication that it should be sufficient to tell different states apart? Why can’t we imagine a world in which the only way to distinguish two states of a joint system is to make use of some other joint system? But let us put this issue aside for the moment.

Fourth axiom (Locality of ignorance): If I have two particles in a joint state that is pure (i.e. I have maximal information about it) and if I measure one of them and find it in a pure state, the axiom states that the other particle must also be in a pure state. This makes sense: if I do a measurement on one subsystem of a pure state that results in still having maximal information about that subsystem, I should not lose any information about the other subsystems during the process. Learning new information about one part of a system should not make me more ignorant of the other parts.

So far, all of the axioms described above are satisfied by classical and quantum information theory. Therefore, at the very least, if any of these axioms do not seem intuitive, it is only because we have not sufficiently well developed our intuitions about classical physics, so it cannot really be taken as a fault of the axioms themselves (which is why I am not so concerned about the detailed justification for axiom 3). The interesting axiom is the last one, `purification’, which holds in quantum physics but not in probabilistic classical physics.

Fifth axiom (Conservation of information) [aka the purification postulate]: Every mixed state of a system can be obtained by starting with several systems in a joint pure state, and then discarding or ignoring all except for the system in question. Thus, the mixedness of any state can be interpreted as ignorance of some other correlated states. Furthermore, we require that the purification be essentially unique: all possible pure states of the total set of systems that do the job must be convertible into one another by reversible transformations.

As stated above, it is not so clear why this property should hold in the world. However, it makes more sense if we consider one of its consequences: every irreversible, probabilistic process can be obtained from a reversible process involving additional systems, which are then ignored. In the same way that statistical mechanics allows us to imagine that we could un-scramble an egg, if only we had complete information about its individual atoms and the power to re-arrange them, the purification postulate says that everything that occurs in nature can be un-done in principle, if we have sufficient resources and information. Another way of stating this is that the loss of information that occurs in a probabilistic process is only apparent: in principle the information is conserved somewhere in the universe and is never lost, even though we might not have direct access to it. The `missing information’ in a mixed state is never lost forever, but can always be accessed by some observer, at least in principle.

It is curious that probabilistic classical physics does not obey this property. Surely it seems reasonable to expect that one could construct a probabilistic classical theory in which information is ultimately conserved! In fact, if one attempts this, one arrives at a theory of deterministic classical physics. In such a theory, having maximal knowledge of a state (i.e. the state is pure) further implies that one can perfectly predict the outcome of any measurement on the state, but this means the theory is no longer probabilistic. Indeed, for a classical theory to be probabilistic in the sense that we have defined the term, it necessarily allows processes in which information is irretrievably lost, violating the spirit of the purification postulate.

In conclusion, I’d say this is pretty close to the mystical “Zing” that we were looking for: quantum mechanics is the only reasonable theory in which processes can be inherently probabilistic while at the same time conserving information.


Communication without substance: a cautionary (fairy)tale

“The main point is that sending the vacuum state is not nothing”.  –Nicolas Gisin

I recently stumbled upon the following delightful exchange in the literature. It began with a tantalizing paper by Salih et. al., published in Physical Review Letters, in which the authors claim to have found a way to communicate information from one place to another without physical particles being transmitted in between. At first this seems astounding: surely information must be communicated by means of physical systems? Okay, some of you wiseguys will complain that you can transmit information by sending a wave through a medium without any individual of the particles making the entire journey, but that is not the point — one molecule must carry the influence to the next molecule, who carries it to the next, and so on. But Salih et. al. claimed to have found a way to transmit a message using photons and linear optics such that ultimately there is no complete chain of relays connecting the source to the detector. The authors write:

“In summary, we strongly challenge the longstanding assumption that information transfer requires physical particles to travel between sender and receiver by proposing a direct quantum communication protocol whereby, in the ideal asymptotic limit, no photons pass through the transmission channel, thus achieving complete counterfactuality. In so doing we highlight the essential difference between classical and quantum information.”

At this point, you are in danger of falling into a trap. Assuming that you believe the result of the paper (you should — it is correct) then you might chalk it up to yet another example of `quantum weirdness’. After all, the protocol is based on the `counterfactual’ nature of quantum mechanics, whereby one can gather information about a system without collapsing its wavefunction. This is precisely the phenomenon at work in the fascinating quantum Zeno effect (merely watching an atom stops it from decaying) and the ultimate in quantum bizarreness, the Elitzur-Vaidman Bomb Test, where it is possible to detect the presence of a bomb using a photon that doesn’t go anywhere near the bomb (the point is that it might have done). So is it any wonder that we can now tack another feat to the list, namely the phenomenon of `quantum counter-factual communication’?

We must be careful, however! If it walks like a quantum and talks like a quantum, it might still turn out to be qwassical. Indeed, in this paper published in Rapid Communications, Nicolas Gisin showed that you can communicate by `sending nothing’, without needing quantum mechanics at all! Actually, you can do it quite easily using just office stationary and the person sitting next to you. Here’s how:

“Hey Bob, at 12:00 I’m going to check the results of the live football match. If my team won, I’ll let you know by throwing a pencil at your head. If my team lost, I won’t throw anything.”

Sure enough, at the appointed hour, Bob ducks reflexively, but to no purpose: there is no impending pencil. Nevertheless, he now knows that your team lost the game. Okay, perhaps this is cheating a little bit. After all, in the event that your team won, a physical object really would have to traverse the length of the office and strike Bob on the head. So in some sense, the information still depends on the transmission of the object.

But wait! Christine is sitting midway between yourself and Bob and she offers to help out. Instead of communicating directly with Bob, you establish the routine with Christine. If she receives nothing from you at 12:00, she throws her stapler at Bob. If she receives your pencil, she does nothing. Now, if your team loses, a stapler goes from Christine to Bob but nothing from you to Christine. And if your team wins, a pencil goes to Christine but nothing to Bob. Either way, no physical object traverses the distance from you to Bob, and yet by 12:01 he knows which team won. Astounding!

What has happened here? Have we demonstrated that Rolf Landauer was wrong — that information is not tied to physical systems after all? Hardly. We have merely neglected the one particular exchange of physical systems that makes this whole thing possible: the channel itself. In his tactfully amusing one-and-a-half page note, Gisin drives this point home using blue and red balls instead of pencils and staplers, but the gist is the same.

For those of you who are still mystified, I have composed the following fairy-tale to bring the physics directly to your inner child.

Once upon a time, in a Kingdom far, far away, there was a King with a very beautiful daughter. Unfortunately the King’s castle was located some distance from the nearest town, where all the most eligible bachelors resided. The only means of communication was the mail-delivery cart and the King’s personal messenger — a small carrier pigeon with a black mark under its eye, identifying it as the Royal emissary. While the mail cart dealt with regular mail, the King reserved his pigeon for gathering news about the stock market.

Day after day, the postman came from the village, his cart practically weighed down by petitions and gifts from the local men for the hand of the Princess. Eventually the King got sick of it, so he made an announcement: no more letters by mail. Each suitor was to apply sequentially, in an order determined by drawing names out of a hat, by writing a letter to the King deliverable by the Royal carrier pigeon. And the King said: “Whoever is able to write me a letter, brought to me by the Royal pigeon, such that reading the letter does not reveal to me its informational contents, shall have my daughter’s hand. Should you fail in the task, you will be beheaded. That is all.”

Naturally, the less brilliant and more cowardly portions of the would-be suitors gave up immediately. But a few brave souls placed their names in the hat. The first bloke was a lawyer and immediately sent off a piece of paper to the King that was completely blank, and spent a good amount of time bragging about his cleverness to his friends afterwards over beer. The King turned up in town the next day.

“Would you agree,” said the King, “that nobody has informed me what you wrote in the letter?”

“Quite,” said the lawyer, who knew the King well enough to know he would not so blatantly cheat on such matters.

“Well then. If I were to say that you wrote nothing at all in the letter, you would have to agree that I was able to deduce this fact by reading your letter. Not so?”

Seeing where things were headed, the lawyer immediately tried to flee, but they caught him and lopped off his head as promised. Nobody could deny that the informational content of the letter – namely, that it was blank – could be deduced immediately by reading it.

At this point, the remaining eligible bachelors swamped the town hall and demanded to have their names removed from the hat. Only one name remained: that of the King’s stockbroker, who was unhappily out of town that day. By the time he returned and realized the fate that undoubtedly awaited him, it was far too late to request his name be removed. He had no choice but to go through with it.

Luckily for him, it happened that the Princess had seen him at business lunches with her father on more than a few occasions, and found him both witty and attractive, despite his nervous demeanor, which was a hallmark of all stockbrokers in those troubled times. Being rather a bright spark, she sent him a secret letter by carrier-squirrel outlining a clever plan. When he received her letter, the stockbroker immediately grabbed a bottle of ink and ran to the baker to fetch a sack of breadcrumbs.

Finally, the appointed day arrived. The King had a replacement broker standing by, the executioner’s axe was sharpened and a hush was upon the town. The King stepped out onto his balcony in the fresh light of morning to receive the letter. Sure enough, with the familiar fluttering of wings the pigeon arrived with a parchment bearing the broker’s seal. In the letter was the usual information about stocks – the king noted some strange selling happening in the housing sector – and a polite how-do-you-do. How boring! He thought. But then he noticed two other pigeons that had been quietly waiting on the perch, which he hadn’t seen in his haste. Each of the also bore the royal mark below their right eye. Tearing open the letters, the King found similar contents, but with the stock information slightly different here or there, and the greeting rephrased one way or another. By this time more and more pigeons were arriving, all identical, all bearing letters from the stockbroker, but each with different contents.

“Damnation!” cried the King, amidst a growing swarm of pigeons. “Which one is the real Royal pigeon? I want to know what the stock market really did today!” Of course, one of the pigeons was the real pigeon and the letter it carried did contain the information the King desired. In fact he had even read it — it was the first one. But in the end he still had to go into town to ask the stockbroker what he had written. And that’s how the humble stockbroker avoided the stocks and remained on the market for the fair Princess.

The moral of the story is, if you have established a communication channel, you can’t help but send information along it. There is always one value that you designate as `vacuum’, which is conventionally the state that gets sent when the source is not being manipulated in any particular way, such as the blank piece of paper in the story, or the empty air that constantly blows from the window to your desk to Bob’s desk at all hours of the day, gently bombarding his head. But the blank page still needs to be sent, in precisely the manner arranged, in order for it to carry information. Only by destroying the channel itself (such as by drowning it in noise) can you prevent the transmission of information. `Vacuum’ means no action, but `nothing’ means no channel at all; the vacuum is not nothing.