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Indistinguishability At A Distance

Valerio Scarani

Published Jan 5, 2006• 0 citations
Publisher

Abstract

Abstract During my first lecture at Saint-Michel College, I had sidestepped the question about randomness that the student had asked. As the reader will recall, it was the experiment of Fig. 1.1, as simple as it was, in which the particles arrived one after the other at a beam splitter. I had said that we do not know how to describe the behaviour of each particle, we do not know why a given particle is reflected rather than transmitted. Is it really uncertain, ran- dom (which is the current opinion among physicists) or is there a mechanism yet to be discovered? I had deliberately turned attention from this problem, in order to concentrate on the experiment of Fig. 1.3, the Mach–Zehnder interferometer. We have seen, then, that quantum randomness is a curious randomness – if we assemble two ‘randomness generators’ (beam splitters) in a certain way, we rediscover the certainty! I briefly remind the assembled students of this, a week later, in the old granary of the College.

DOI
10.1093/oso/9780198570479.003.0006

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Abstract During my first lecture at Saint-Michel College, I had sidestepped the question about randomness that the student had asked. As the reader will recall, it was the experiment of Fig. 1.1, as simple as it was, in which the particles arrived one after the other at a beam splitter. I had said that we do not know how to describe the behaviour of each particle, we do not know why a given particle is reflected rather than transmitted. Is it really uncertain, ran- dom (which is the current opinion among physicists) or is there a mechanism yet to be discovered? I had deliberately turned attention from this problem, in order to concentrate on the experiment of Fig. 1.3, the Mach–Zehnder interferometer. We have seen, then, that quantum randomness is a curious randomness – if we assemble two ‘randomness generators’ (beam splitters) in a certain way, we rediscover the certainty! I briefly remind the assembled students of this, a week later, in the old granary of the College.
After eighty years of Quantum Mechanics (QM) we have learned to live with wave functions without worrying about their physical nature. This attitude is certainly justifled by the extraordinary success of the theory in predicting and explaining not only all the phenomena encountered in the domain of microphysics, but also some spectacular nonclassical macroscopic behaviors of matter. Nevertheless one cannot ignore that the wave{particle duality of quantum objects not only still raises conceptual problems among the members of the small community of physicists who are still interested in the foundations of our basic theory of matter, but also induces thousands and thousands of physics students all around the world to ask each year, at their flrst impact with Quantum Mechanics, embarrassing questions to their teachers without receiving really convincing answers. Remember that Feynman once said \It is fair to say that nobody understands
Abstract Now that we have explored qubits and the phenomenon of superposition, we can ask the question: how do we know that superposition actually happens? is the evidence that shows that a quantum particle really does exist in two different locations at this same time while in a quantum superposition? The nature of science means that experiments are constantly updating previous results, so are there other interpretations of the experimental results that can explain the data without the need for superposition? In this chapter we’ll explore the experimental evidence interpretations other than quantum superposition. Further, while a flipping coin is a simple model of a qubit, it is not very useful for building a quantum computer because it does not exhibit all of the properties of a true quantum superposition. For example, we cannot manipulate the superposition amplitudes. In this chapter, we will study some real physical examples of quantum particles in a superposition containing two states. These examples include a photon in a beam splitter and the Mach–Zehnder interferometer.
In the last two chapters, we talked about computational complexity up till the early 1970s. Here, we'll add a new ingredient to our already simmering stew – something that was thrown in around the mid-1970s, and that now pervades complexity to such an extent that it's hard to imagine doing anything without it. This new ingredient is randomness . Certainly, if you want to study quantum computing, then you first have to understand randomized computing. I mean, quantum amplitudes only become interesting when they exhibit some behavior that classical probabilities don't : contextuality, interference, entanglement (as opposed to correlation), etc. So we can't even begin to discuss quantum mechanics without first knowing what it is that we're comparing against. Alright, so what is randomness? Well, that’s a profound philosophical question, but I’m a simpleminded person. So, you’ve got some probability p , which is a real number in the unit interval [0, 1]. That’s randomness. But wasn’t it a big achievement when Kolmogorov put probability on an axiomatic basis in the 1930s? Yes, it was! But in this chapter, we’ll only care about probability distributions over finitely many events, so all the subtle questions of integrability, measurability, and so on won’t arise. In my view, probability theory is yet another example where mathematicians immediately go to infinite-dimensional spaces, in order to solve the problem of having a nontrivial problem to solve! And that’s fine – whatever floats your boat. I’m not criticizing that. But in theoretical computer science, we’ve already got our hands full with 2n choices. We need 2ℵ0 choices like we need a hole in the head.
Abstract Leaving Saint-Michel, I take my leave of Jean-Paul who is going home, and I turn left to walk down the College Steps, a covered lane that descends into the heart of Fribourg. There, in front of the cathedral, three busy roads meet to form a roundabout. I continue my walk along the south side of the cathedral. Cars continue to stream down the road. Now that the roundabout is no longer in view, I cannot know by which route each car has entered it. In order to arrive alongside me, as it does now, this car could have taken three possible paths. I have put myself, in relation to the cars, in a situation of indistinguishability. Well then, why is the road network not a quantum interferometer? The answer is a matter of common sense – the indistinguishability of cars is only apparent, it is due to an ignorance that can easily be overcome. In order to obtain the information about the path by which each car entered the roundabout, it would suffice for me to go back a few paces or to ask the driver – neither of these steps would modify what the destination of the car will be.
Abstract ‘When all is said and done, physics is only a description, not an explanation’. I have uttered similar statements many times myself in informal discussions, when my interlocutor happened to slide into an excessive praise of experimental science and an excessive contempt of other forms of human knowledge. In the mouth of Jean-Paul Fragnière, a friend and philosophy teacher who lives in the small Swiss town of Fribourg, this sentence sounds, however, the expression of a different and distant line of thought of a rivalry between human and natural sciences that is still latent in the French-speaking world, a relic of the arrogance of Positivism. We are in the massive building of Saint-Michel College – a sixteenth century Jesuit establishment, a public school for several decades; and though Jean-Paul’s classroom is a modernization of what was the granary, history weighs heavily here. For the first generations of young people educated in this school, heliocentrism was one of the hot topics; now Jean-Paul has invited me to speak to his students about quantum physics.