- 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.
The Uncertainty Principle Determines the Nonlocality of Quantum Mechanicspage 2 (back to page 1)
Likewise, nonlocality was traditionally thought of in terms of something called a Bell inequality, which also obscured the link between nonlocality and uncertainty. But if you think of nonlocality in terms of games, the link becomes more transparent. My co-author Stephanie claims that the following gamewhich is equivalent to something called the CHSH inequality is so simple that two randomly chosen teenagers and one 12 year old now understand nonlocality (yep, she actually goes to cafes and forces children to play her quantum games). The game is cooperative and goes like this: there are two players (Alice and Bob) who are far apart so they can't talk to each other. Alice has two cats and two boxes, and there is a referee who flips a coin. If heads, the referee tells Alice to put an even number of cats in the box (that means either a cat in each box, or no cats in either box), and if tails (odd), Alice must put a cat in either the left box or the right box. Then the referee flips the coin again, and if it's heads, Bob is challenged to guess whether there is a cat in the left box and if tails, he must guess whether there is a cat in the second box. They get a point if Bob guesses correctly, and get no points if Bob guesses wrong.
Now, they can agree beforehand on any strategy, but once they start playing they can't talk to each other. If they can write down their strategy on a piece of paper, then on average, the best strategyOne optimal strategy is for Alice to always put a cat in the left box. Because of the way the game is designed, this strategy means that whether their is a cat in the right box will be completely random, and will depend on the coin flip. If Bob is lucky enough to be asked whether there is a cat in the left box, Bob should always say that there is. He will have to guess if he is asked the right box. There are other optimal strategies (e.g. always putting a cat in the box on the right side. wins 75% of the games. However, if they don't write down their strategy, but instead base their actions and answers on the outcome of measurements they've performed on quantum particles (and in particular, two entangled particlesEntangled particles are two distant systems in some shared state which we sometimes write as |&Psi〉AB=(|01〉-|10〉)/√2 (for example). If Alice performs a measurement on such an entangled state, she will know what state Bob has.) they can do better -- they can win 85% percent of the time. For two parties who can't communicate with each other, quantum theory allows better coordination than any strategy which they could write down. If they were to be able to do as well with a strategy they could describe on a piece of paper, then it would have to involve action at a distance. This is nonlocality -- although in reality there is no action at a distance, in our description of reality, there is action at a distance. This is what bothered Einstein, and why he described it as "spooky action at a distance," writing in a famous paper with Boris Podolsky and Nathan Rosen that “no reasonable definition of reality could be expected to permit this."
Well, maybe we've finally just gotten used to it. Now, instead of fighting nonlocality, we ask why our description of quantum reality doesn't require even more nonlocality. We don't ask why quantum theory is so weird, but rather, why isn't it weirder? Indeed, there could exist states of matter which are better than entanglement, in the sense that Alice and Bob could use them to win the game more than 85% of the time. These hypothetical objects (called PR-boxes, after Sandu Popescu and Daniel Rohrlich), allow Alice and Bob to win the game all the time, and they do not appear to be pathological, at least in the sense that they don't allow action at a distance. Why does nature rule them out?
We don't yet know, but one possible reason is that the uncertainty principle limits nature. We found that the probability of Alice and Bob winning the game is directly related to the strength of the uncertainty relations in the theory. This result is not without its irony. When Einstein, Podolsky and Rosen first pointed out the spooky nature of quantum correlations, they were attacking the implications of the uncertainty principle. Now we see that not only does nonlocality not undermine the uncertainty principle, it is determined by it.
The uncertainty principle is a restriction on measurements made on single systems, while nonlocality is a restriction on measurements conducted on two systems. We find that these restrictions are related, by treating both nonlocality and uncertainty as a coding problem. Here, tools from information theory and computer science are central. Lets go back to the game, and get a bit more technical, by noting that effectively, the placement of the cats has two bits a bit is a 0 or 1. If the cat is in a box, we treat this as a 1, if there is no cat, we take this as a 0. So, a cat in the left box, and no cat in the right box, can be written as 01. of information because there are four possible ways for Alice to position the cats. Bob on the other hand, is only trying to retrieve one of the bits (this is called a random access code). He is either trying to determine if there is a cat in the left box (one bit of information), or he is trying to determine if the cat is in the right box (the other bit of information). From Bob's perspective, this is like the situation he encounters in Heisenberg's uncertainty experiment. There, the experimenter is either trying to determine the position, or trying to determine the momentum. Here, Bob is either trying to determine if there is a cat in the left box, or trying to determine if there is a cat in the right box. Now we see how the uncertainty principle might come into play. A quantum particle cannot have a well determined position and momentum at the same time. Likewise, encoding the information of whether there is a cat in the left and the right box, cannot be done at the same time. We can think of an uncertainty relation as being a restriction on how well on average Bob can retrieve one of two bits of information (with for example, one bit encoded in the position of the particle, and the other bit encoded in the momentum of the particle).
So in the game, Alice has effectively announced to the referee a two bit string (the placement of the cats in the boxes), and Bob is trying to retrieve part of this information. If Bob tries to guess the position of the cats by performing a measurement on some quantum system, then the stronger the uncertainty relations obeyed by Bob's measurements, the more difficult it will be for Bob to retrieve the bit that is encoded. Thus strong uncertainty relations lead to less nonlocality. This relationship can be made quantitative, and we find that the degree of nonlocality is given by the uncertainty relation inherent in Bob's measurements.
That might seem a bit strange, because the uncertainty principle says that for measurements on single systems, quantum theory is more restricted than classical theory, while nonlocality says that when it comes to measurements on two or more systems, quantum theory is less restricted. Indeed, another element enters into the picture, and it's something which Schrödinger called steering. Steering is Alice's ability to prepare various states of some system at Bob's site by making measurements on her share of their jointly held state. Steering doesn't result in action at a distance, because Alice doesn't have total control over what state is prepared at Bob's site, but Alice's knowledge of the state of Bob's system does change. Well, it turns out that in all possible theories (including a future theory of quantum gravity), the degree of nonlocality is determined by both the uncertainty principle, and by the degree to which the theory allows steering. Next, we found that in quantum theory, when Alice and Bob use the optimal strategy for the game, the degree of steering is only restricted by the fact that it can't allow action at a distance. This means that the degree of nonlocality is determined only by the uncertainty principle, and that there is no action at a distance. We can think of it as:
nonlocality = uncertainty + steering (general theories, including any
future theory of quantum gravity)
nonlocality = uncertainty + no action at a distance (quantum theory)
As a result, Alice and Bob cannot do better at the quantum board game with measurements which respect the uncertainty principle. There's still a lot of this we don't know, such as understanding what restricts Bob's ability to retrieve both bits of information. So, if you're interested, and have stuck with me to the end of this page, you might want to give the main part of the actual paper a read. And if you get to the end of that and are still keen, heck, why not try the full 44 pages! What I hope you find is that there are some very simple and fundamental questions about quantum theory that we don't know the answer to, and that applying techniques from information theory to our laws of nature, allow us to make some progress in answering these questions, and more importantly, suggests basic and important questions we wouldn't normally think of.
- The quantiki entry on Bell's theorem and nonlocality
- The quantiki entry on entanglement
- Popescu-Rohrlich correlations
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