Recently I finally got a chance to read the new preprint arXiv:0905.2292 “A new physical principle: Information Causality” by M. Pawlowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Zukowski. It’s been a long time since I spent more than a few spare hours thinking about foundational issues in quantum theory. Personally I am very fond of approaches to foundational questions which have a information theoretic or computational bent (on my desktop I have a pdf of William Wootter’s thesis “The Acquisition of Information From Quantum Measurements” which I consider a classic in this line of interrogation.) This preprint is very much along these lines and presents a very intriguing result which clearly merits some deeper thinking.

(**Update:** see also Joe for details of the proof.)

The basic back story is as follows (readers who know what entanglement and Bell’s theorem are can skip the next three paragraphs.)

Quantum theory is the theory we use to predict probabilities of different outcomes in different experimental settings. But, while it predicts probabilities, it has some peculiar properties which separate it from how we normally think about systems which have a probabilistic description. Two of the defining characteristics are contextuality and quantum non-locality. The former tells us that we cannot think about a measurement on a quantum system as revealing a property which is independent of the measurement we chose to make. The later, and in my view more troubling, describes how measurements made by two separated observers can show strange correlations in quantum theory.

The setup to demonstrate quantum non-locality is as follows. Alice and Bob get together for lunch and bring with them their own quantum systems. Over hamburger’s they let their two quantum systems interact. They then put their quantum system’s into their backpacks, taking care not to measure them, and go back to their separate work places (Alice works at a restaurant and Bob works on a fishing boat.) Later, during their breaks, which happen to occur at exactly the same time (in the reference frame of their home), they both open backpacks and look at their respective quantum systems. Well they don’t quite just “look”: they choose to measure their system in a particular way. Quantum theory allows us to predict what the probabilities of these measurements are, given which measurement each party chooses to perform. For some manners in which Alice and Bob let their quantum systems interact, where they create so-called entangled quantum states, and for some measurements in this case, Alice and Bob will see correlations in the outcomes of their measurements. For example, when Alice sees a measurement result of “0”, it may be that Bob always sees a measurement result of “0” and when Alice sees a measurement result of “1”, it may be that Bob always sees a measurement result of “1”, and that these two cases occur with equal (fifty percent) probability.

Now one could ask, well this isn’t so strange! Maybe when they let their systems interact, the properties of their individual quantum systems where set, and all we are doing by measuring them later is identifying the value of this property. For example, above I described how Alice and Bob’s outcomes could be perfectly correlated. But we could easily explain this: if I flip a coin and secretly put either a “0” or a “1” in both of the backpacks, then Alice and Bob will see exactly the above correlation. This, however, is not the mystery of quantum non-locality. The mystery is that while we could explain the correlations between Alice and Bob’s experiments via properties of their individual systems in a purely classical way in this case, it turns out that this is not always the case. This result is the famous “Bell theorem.” This theorem says that there are certain setups like Alice and Bob performed above on quantum systems which *cannot* be explained by believing that the properties being revealed were just stored in the backpacks before hand. Quantum correlations behave in ways that we cannot explain by what is termed a local hidden variable theory. The “hidden variables” are the properties in the backpack and “local” refers to the fact that these bags are separated from each other and so not allowed to influence each other. Bell’s theorem tells us that there is a set of inequalities (Bell inequalities) which must be satisfied for a local hidden variable theory. This inequalities are something like EXPERIMENTAL QUALITIES FOR LOCAL HIDDEN VARIABLE MODELS <=2 (or something like that ;) ) If you run those experiments with entangled quantum states you get EXPERIMENTAL QUALITIES IN QUANTUM THEORY >2, and hence there is a contradiction between the assumption of local hidden variables and quantum theory.

No Signal |

So quantum theory produces these strange correlations that do not have a local hidden variable description. Thus if one wants to “explain” quantum theory using our normal intuitions about probabilities and classical information one would need to believe that there is some way in which information about one parties measurement is somehow transferred to the other party during a measurement on an entangled state. (Or one could just accept quantum theory ) But here is what is cool about quantum theory: even though measurements on entangled quantum states exhibit these correlations that could not be explained in classical terms without some sort of communication between the parties, there is no way to exploit this and allow Alice and Bob to communicate between each other. Quantum theory does allow correlations which can only be explained by communication without allowing communication itself.

This sounds kind of strange but really it’s not too odd. Just because a protocol uses communication does not mean that this communication reveals itself in the output of the protocol. In fact we witness this all the time: when we are using the internet, for example, not all of the communication that goes through our internet cable is revealed on your computer monitor. There are all sorts of behind the scenes information being transfered which does not reveal itself by changing any pixels on your monitor. Of course the cool thing about quantum theory is that it *never* allows for signaling. This is good because if it were not so then it would be hard to get quantum theory to mesh nicely with special relativity (because we could signal faster than light which causes issues with special relativity.)

Given that quantum theory doesn’t allow signaling, one can ask more generally about other “possible theories” of the universe that don’t allow signaling. Quantum theory doesn’t allow signaling, but are there other such theories? And is there some way in which quantum theory is special among no-signaling theories? This later question was asked by Popescu and Rohrlich in a nice form: is quantum theory the most general no-signaling theory which has the maximum violation of Bell inequalities? The answer, it turns out, is no. Popsecu and Rohrlich showed a way to have no-signaling correlations which violate Bell inequalities even more than quantum theory (in fact they achieve the maximum possible violation for all theories.) Thus thinking that quantum theory was somehow special in the space of no-signaling theories took a hit, indeed there has now been many studies showing how much of what we consider intrinsically quantum appears in many no-signaling theories.

Which is where the preprint arXiv:0905.2292 comes in. Here the authors notice something kind of cool: that there is a different way to talk about no-signaling. This is what they term “information causality.” Here is their definition:

Formulated as a principle, Information Causality states that the transmission of m classical bits can cause an information gain of at most m bits.

At first sight you may say: oh this doesn’t say anything interesting…of course if you transmit m classical bits then the other side can only gain m classical bits of information. That’s just standard information theory. But recall again the situation above where you make measurements on a quantum system, or more generally where you get correlations from some other no-signaling theory. In this case you have this extra resource and, while it doesn’t allow you to signal, it might be useful. The m=0 case is then the no-signaling condition: where the measurements and outcomes by themselves (with no classical side channel) can not send more than 0 bits of classical information. But the m=1 case, for example, is different. It says that if you use the correlations plus a single classical bit of information transmitted, then you can only gain a single bit of information on the other side.

This all seems, at first thought, to be just a small modification. It helps, however, to recall a place where something like this fails. If I try to send classical bits using quantum bits then I can only send a single classical bit per quantum bit (this is Holevo’s theorem.) But if you allow an entangled quantum system shared between the two parties, then by using only one qubit you can send two classical bits (this is superdense coding.) Thus if we were to replace the information in the definition of information causality by quantum information, it would not be true.

Great. So it’s a new condition. What can you do with it? Well here is where the coolness comes in. What the authors of the preprint show is that (a) both classical (local hidden variable theories) and quantum theory respect the principle of information causality and (b) that quantum theory achieves the maximal value for a certain class of Bell inequalities and (c) that any no-signaling theory can violate the Bell inequality by more than quantum theory. (a) is neat, but not to surprising. (b) and (c) are where all the coolness lies. It says that there is a possibility of thinking about quantum theory as the theory that maximally violates Bell inequalities among a no-signaling theories…if you replace no-signaling by the more general principle of information causality. A small modification of the no-signaling condition which at first glance doesn’t feel all that different gives very different results and indeed gives us…quantum theory.

So is it possible to derive quantum theory as the the theory which most maximally violates Bell inequalities but still respects information causality? This is not known because the above result only holds for one type of Bell inequality (or at least this is my understanding. They show that the principle of information causality yields Tsirelson’s bound which does not describe the most general experimental setup for Bell inequality experiments.) In other words it is only known to be the maximally violating theory in a particular experimental setup. Another issue is that while it is shown that quantum theory achieves the maximum violation, it’s not shown that there aren’t other ways to produce correlations which is not quantum theory. Another important issue is that it’s not clear that the “size of the violation” of a Bell inequality is the right quantity to be considering here. A better way to quantify the strength of a violation of a Bell inequality is probably through a procedure like that described in The statistical strength of nonlocality proofs by van Dam, Gill and Grünwald. Any way you slice it, however, this preprint cries out for follow up work: maybe there is another Bell inequality for which the principle of information causality is not enough. Or maybe there is a way to generalize the argument in the paper to a large set of Bell inequalities.

This preprint opens up the door to a possible method for “deriving” quantum theory from some basic information theoretic principles. This would be an awesome achievement, but of course it might not shut the door on the mysteries of quantum theory: such a derivation would beg the question “why is quantum theory the maximally violating theory consistent with the principle of information causality?” What I find particularly nice about this result is that the principle of information causality is an very simple modification of what we mean by no-signaling, something required to mesh quantum theory with special relativity, but one which, apparently had not been previously seriously considered. Small tweaks which lead to big results are by far my favorite and this is a classic along those lines.