Umesh Vazirani - Bell Inequalities lyrics

UMESH V. VAZIRANI: In the last lecture, we spoke about entanglement, the EPR paradox, and Einstein's ideas about how quantum mechanics is an incomplete theory. In this lecture, we'll talk about John Bell's landmark paper from 1965, about 30 years later, showing that entanglement has testable effects, that you can actually test Einstein's beliefs in the lab. -------------------------------------------------------------------------------------- So let's start by examining the basis of Einstein's beliefs a little more closely. This was this notion of what's called local realism, the notion that physics must be local. This has a long tradition. It goes back a long way. For example, Isaac Newton spoke about this in the context of his theory of gravity, which he did not find quite acceptable because it posited action at a distance. So he found this very disturbing. And he only very reluctantly published his ideas about the theory. Here's a quote from Isaac Newton. It says it's "inconceivable that inanimate matter should, without the mediation of something else which is not material, operate upon and affect other matter without mutual contact." So proximity was essential to having an effect. A more modern way of describing it would have been that nature is described by local differential equations. The second concept is one of realism. Let's see what Einstein himself had to say about realism. So he says, "I think that matter must have a separate reality independent of the measurements. That is, an electron has spin, location, and so forth even when it's not being measured. I like to think that the moon is there even if I am not looking at it." So this is getting at the heart of the matter of quantum mechanics, where one speaks about a system being in superposition, the quantum state being the superposition state when one's not looking at it. And it's only when you measure the state that the physical quantity of interest actually appears. And this is what Einstein found very troubling and problematic. So in view of this, let's go back and reexamine the EPR paradox just very quickly. ------------------------------------------------------------------------------------- Remember in the EPR paradox we had two cubits in a Bell state. So the thought was when these two cubits were brought close to each other and they interacted with each other, they got into this entangled state. And then we separated these cubits by a very large distance. Now, we also saw last time that the Bell state we could equally describe as an equal superposition of 00 and 11 or as an equal superposition of plus plus and minus minus. So now here's where the locality and realism comes in. Locality says, since these two cubits are very far apart, what you do to one cubit will not affect the state of the cubit, because even light has not had time to go from one to the other. Realism says there were two quantities that we were interested in measuring about these cubits. One was the bit, 01, and other was the sign, plus or minus. And realism would say, well, this cubit has this property regardless of whether you measure it or not.And so now Einstein reasoned since you can make the measurement on the first cubit --and you get to decide. And if you measure the bit value to be 0, then the bit value of the other cubit, the second cubit, is also 0. If you measure the sign value and it turns out to be plus, the sign value of the other cubit is also plus. But since the measurement did not have a chance to disturb the second cubit, it's bit and sign values must be unchanged. And so in fact, it must be the case in principle you could have obtained either one without disturbing the other. -------------------------------------------------------------------------------------- So that was the EPR argument. Now in 1965, in this landmark paper, John Bell showed that there's an experiment that distinguishes between quantum mechanics and any theory that's consistent with local realism. This is a remarkable thing, so let's see what John Bell said. Well, what John Bell said is that there's a particular experiment that you can perform. And then, based on this experiment, you collect your data and your data will say either that a certain quantity that you estimate --let's call that estimate E. E is going to be less than or equal to 3/4 if nature behaves according to --if nature behaves consistently with some local realism theory. If nature is governed by some theory that's consistent with local realism, then this quantity that you'll estimate will be definitely smaller than 3/4. On the other hand, if nature is governed by quantum mechanics, then E should be --well, ideally it will be cosine squared pi by 8, which is about 0.85. So what this said is that what Einstein gave or what the EPR paradox gave is a thought experiment. It was not an experiment you could actually perform to tell the difference between quantum mechanics and local realism. 30 years later, Bell was able to actually give an experiment that could distinguish between the two. This experiment was since performed. And it's been repeated many, many times to increasing degrees of accuracy. And the results have always been consistent with quantum mechanics and inconsistent with local realism. So the inescapable conclusion is that we have to live with this so-called incompleteness of quantum mechanics. There are certain questions we just cannot ask about nature. We have to live with an uncertainty principle. We have to live with this idea that when you're not looking at it maybe the moon's not there, well, at least at the subatomic level. Here's another thing to think about. By the end of this lecture, you will actually understand the details of Bell's experiment, the Bell Inequalities. This is something that if Einstein had understood back in 1935, he would have saved two decades of his life that he spent looking for such a theory that was consistent with local realism. This may seem remarkable, given that many of you have no background in quantum mechanics and this is just our fourth lecture. The reason we can do this is, well, first because Bell's experiment has since been simplified a great deal. And secondly, because we're going to be speaking in this language of quantum computation and quantum information. And that's going to simplify our explanation, understanding the narrative about this experiment a great deal.