F52 - Non clonazione quantistica e fisici (un po') hippie

Why can't we clone the state of a system in quantum physics? Because if we could, it would be possible to communicate instantaneously even at astronomical distances. It would be like having a kind of magical telepathy. A kind of telepathy. It sounds like a hippie idea. In fact, it is an idea that originated from a group of physicists who were a bit hippie. If you want, I'll tell you. Go ahead. So, you're telling me that a group of hippie physicists discovered and rediscovered the theorem of non-quantum cloning? It's not that they discovered it. They gave a stimulus that allowed them to discover this theorem of non-quantum cloning. In English it's called the No-Cloning Theorem. The story is very funny and very interesting to tell. This group is called the Fundamental Physics Group of Berkeley, with physics written in a strange way. It was a group of people who loved to explore consciousness, even with psychedelic substances. They were interested in telepathy, in psy phenomena. They were very open to the possibility that quantum physics could somehow explain, for example, all these extrasensory perceptions, things of this kind. In this group there were characters of remarkable level. In particular, a certain Frank Herbert, who I'm going to tell you about now, because he was at the origin of the theorem in question. But there were also other physicists of the calibre of Fridtjof Capra, of John Clauser, who won the Nobel Prize in 2022, Henry Stapp, another name that comes to mind, etc. So there was a group of people, a bit strange, a bit hippie in fact, but who were deeply interested in physics. There were physicists, theorists of good level, excellent level, who, however, also explored a bit strange issues. For example, they were interested in the idea of being able to communicate in an instant, precisely because they considered the hypothesis of telepathy valid. But does quantum physics allow to explain an instant communication between Alice and Bob, for example? Alice and Bob being two experimenters who are arbitrarily far apart. This was the setting, so to speak. So the story begins in the early 1980s, because this Frank Herbert wrote an article and tried to publish this article, he submitted it to the magazine Foundation of Physics, and in this article he used the quantum entanglement through a somewhat particular protocol, and somehow demonstrated that it was possible to obtain instant communication, superluminal, much faster than light, at an arbitrarily high speed. And you can imagine that those who found this article in their hands initially said, oh God, what is this thing? Because in quantum physics it is well known that information cannot be sent faster than light. Very good. In fact, this is something that everyone knows, at least many already know. For a long time it was known, but not for so long. There were some works, for example Girardi, Rimini, Weber, or Eberhard, there were already physicists who had already written theorems, formal results, where it was evidenced that this type of communication could not take place. That is, that the entanglement, which was able to produce instant correlations, could not be used, these instant correlations, to communicate instantly. Because it's a bit like Alice and Bob, who are these two characters who act at the same time, or almost at the same time, on an entangled state, therefore the state of a system formed by two parts, one part is Alice's, one part is Bob's, they operate on these parts, they observe things, their results are correlated in a particular way, the correlations are the result of their observations, but they can't understand what the other is doing. So they can't communicate in any way, because if Alice knew what kind of experiment Bob was doing, or Bob knew what kind of experiment Alice was doing, they could use this fact, this knowledge, to send each other information. Of course, if I know what a person is doing, it's a bit like if I see her, and if I see her, this person, with her actions, can communicate with me. Exactly, I can do certain experiments, each experiment has a meaning, a code, a letter, a word, and so on the other hand, if I know every time what you're doing, it's as if you're communicating things to me. Yes. I agree. It's a language, and it can transmit any kind of information. Of course. But these results were already known, they had already been published, a while ago. So this article was wrong in itself. So this article couldn't be right, because it came about ten years after these results, but, but, obviously you know that in physics an article comes to a magazine, this is given to be reviewed by peer reviewers. Yes. One of these peer reviewers was Asher Peres, who is one of the pioneers of quantum computing, so even there, a top-notch. And this scientist looks at this article and says, ok, I know it's wrong, because he knows about these theorems that already existed, about these conditions that in quantum physics are called no-signaling, no-communication conditions. It was already known that they existed. Another name is marginal laws, in a more probabilistic context. However, it was known that the structure of standard formalism of quantum mechanics was such that it didn't allow to extract information from correlations in order to allow Bob to communicate with Alice or vice versa. Ok. But Asher Peres looks at Herbert's article and says, yes, it's wrong, but he doesn't immediately see why it's wrong. So he has that rare intellectual honesty and that courage to say, this article is wrong. I know, it can't be right because it conflicts with other results. Yes. But to understand how and why it's wrong will be interesting. Now I can't tell you if he's right or wrong. I don't think so. But he saw that it couldn't be right and said, let's publish it anyway. It must be published because even if it's wrong, this article will give rise to answers and comments and these comments will be instructive because they will tell us why it's wrong, which is not so evident. So it was a very nice move by another reviewer, I don't know if it was Ghirardi, but he certainly gave this input and then told this story. And so, this happens, that after some answers these answers contained the argument that confused Nick Herbert's results was the fact that there was a theory of impossibility related to cloning. Because Herbert had introduced in his communication protocol a mechanism that used the cloning of states. I'll briefly explain what it was. Explain to me the protocol that Herbert used to use entanglement with cloning to communicate at a distance. I won't go into too much detail or we'll get too confused. His protocol was this. He says, Bob wants to communicate with Alice. Bob and Alice share this entanglement state. His idea is, Bob makes a measurement, any measurement on his part of the system. The purpose is that Alice can know what measurement he made. Now, the idea is, Bob makes the measurement before Alice. A moment before Alice. They agree. He always does it a moment before. Their ROGs are synchronized. Bob makes his measurement. At this point, Alice makes hers. But, before making her measurement, what does Alice do? She takes the state of her subsystem, of her part of the system, and makes an immense number of identical copies. That is, she clones that state many times. At this point, she has this myriad of clones, of equivalent states, and on different groups of these clones, she does experiments. She tries to do many experiments. In reality, what she tries to do is to find out which experiment Bob made. Because there is a single experiment, which is what Bob did, which is such that Alice's results for that experiment will all be certain. That is, if, for example, she measures a spin, there will be a particular direction where she will find all the spins, 100% of the spins, always up, or always down. And that can only be the same direction that Bob used. But to be able to get it, she must have at her disposal many sets of states on which to calculate the experimental probabilities for a choice of experiment, of direction, and therefore she must have the ability to create these clones. To create these clones, what does it mean? It means that she must take her part of the system, she must create a clone of it, but without destroying it. Not only must she create the clone without destroying it, but without knowing what this state is. So, Herbert's idea was it is possible to create a machine that creates clones, and it is a universal machine because it takes the state, even if I don't know what it is, it clones it, and leaves the original intact. Ok. Herbert had a mechanism, in his case, the physical entities were photons, he used amplification mechanisms through lasers, it was about polarization, I won't go into these details, but the idea was this. This allowed, in the end, to obtain an instantaneous communication. So, the objection that came with some works, one by Zurek, another by Eck, there were two articles written in the same year of response, and both contained this theorem of non-cloning, which is really two lines to show, this theorem. The interesting thing is that it is easy, once you have the mathematical-quantum formalism, it is really a few steps, and, among other things, it is surprising that until those years, no one had ever derived it, or almost, now I'll tell you why. Would it be easy enough to prove it? It would be easy enough. You have enough knowledge of linear algebra, so you can follow the demonstration. If you want, we can do it later. You'll see that in five minutes the theorem is immediately demonstrated. Because you immediately arrive at a contradiction. Being a no-go theorem, as they say in English, an impossibility theorem, what you want to do is start from a hypothesis, and the hypothesis is this universal machine of cloning, and you immediately arrive at a contradiction. If it exists, it cannot exist. So you get the demonstration. Now, in reality, let's say, the hippies of physics, in a good way, I'm using the word hippie, because they were people who had a vision, how to say, holistic, of reality. They really tried to find the connections between the different levels of experience, both external and internal. So, let's say, they were valid people who proved to be very valid. With this, let's say, with this push towards finding certain things, in any case, they produced very interesting reactions from the point of view of physics as such. In reality, then, we must say one thing, that the no-cloning theorem had already been derived about ten years earlier, in 1970, by a certain James Park. Always on that same magazine, Foundation of Physics, where they published the article by Herbert, there was another article, ten years earlier, where this gentleman was interested in measurement processes and was trying to understand how a measurement tool could completely acquire the state of the measured system. So, in a certain sense, without destroying the measured entity. So, there were questions very similar in the field of, let's say, problems related to quantum measurement. So, there was already that result, it had already been derived by this Park, but no one had noticed it in that field. So, it was rediscovered. But this, we must say, is something that happens continuously in physics. There are always results that are rediscovered. Or, when they are rediscovered the importance is not recognized and only when they are rediscovered in the right context it is understood that they were important. So, this is a bit the strange story of the fact that you can't clone a state in a way that you don't know, a priori, without destroying the starting state. This is a theorem. Among other things, small parenthesis, there exists, however, what we could call a non-cloning theorem, which is a bit the opposite, and it is the observation that in physics there are, we will talk about this another time, what are called indiscernible entities. If you take electrons, when you let these electrons interact, they are together, where there is no possibility to distinguish them, because you have not created an experimental context created specifically to distinguish them, but you put them inside a system and let them interact. Electrons, photons, fermions or bosons, since these entities are indiscernible, they automatically enter states of entanglement called symmetric or antisymmetric, such that the states of individual individuals that are indiscernible are all the same. They are no longer described, now I use a technical word, they are no longer described by state vectors, but by so-called density operators, but what I mean is that there is a natural tendency of identical entities to self-clone spontaneously, without a cloning machine, as soon as you leave them together in some way. This is obviously a completely different context from the context of the no-cloning theorem, which instead has to do with entities that can be distinguished, because in order to have an original and a system that is the receiver of the original state, you must necessarily be able to distinguish these systems. So you are in a situation of discernible entities, this is just to say that there are mechanisms in quantum entities of automatic cloning in some way, which, however, in laboratories can be created conditions where certain systems are well separated, and at this point they are indeed discernible. Well, it seems to me that it was a nice excursus, if you want, let's see if in three seconds we can prove this damn theorem. Let's go! So... So, this demonstration of the theorem of non-quantum cloning is very simple. Enough for me to understand it. Certainly, apart from the details, you certainly understand the essence. We have to deal with a composite system made up of two parts. One part is the original system of which you want to clone the state. That we do not know. The other part is the receiver system of the state on which we want to obtain the copy. What will become identical. It will acquire the same state without destroying the starting state. Right. So, basically, what do we have? Let's assume that the original system is in a psi state, described by this strange notation, think of a vector, a psi vector, the state of the original system, and the other system is in a phi state. And this complex state, which describes the set of the original system and the receiver system, is a so-called produced state. Produced because it is written with this strange product, which now I write it, but then maybe I do not always write it, because it is useless. I can also remove it from the writing and simply write it in this way, without putting the symbol. There is a product between them, it is called a tensorial product, we are not interested in it, we can just see it. It is a state where it is clear in which state the first system is located and in which state the second system is located. Yes, a produced state. The entanglement states have not been produced, because it is no longer possible to correctly determine the state of the various subsystems. But this is the beginning. Now what happens? Now I use this cloning machine that makes this state evolve. It makes it evolve towards the state I want, where it must remain the same, but it must become equal to it. So, the evolution operators in quantum physics are unitary operators and are necessarily linear operators. We will see later what linear means in practice, in the demonstration. So what do I do? I apply this machinery that is this unitary operator of evolution and applying it to these vectors I want the result to be a new produced state where the original always remains in its psi state. But the receiving system also finds itself in a psi state. So this means that the cloning has taken place. Now I have two systems that are exactly in the same state. Now we have to see a sort of contradiction. Ok, I'll leave you some space and you show me. Ok, then. After you know the questions, I have to start from a hypothesis and see that from that hypothesis I get a sort of contradiction. Not a sort, a real contradiction. So I imagine that the psi state of the original system can be written as a superposition of two states. Let's call it a sigma state plus a rho state with a and b which are coefficients. Now these coefficients for hypothesis I suppose that they are different from zero. I want them different from zero. That is, that this superposition is a non-trivial superposition. Now, since the state of a system therefore when I do the scalar product of the vector with itself must always be equal to 1 because the state vectors are unitary. This implies conditions on a and b. I want to hypothesize here that these states, sigma and rho are states that are mutually orthogonal. That is, if I do the scalar product of this vector with this I get zero. They are orthogonal vectors. With this hypothesis that these two vectors are orthogonal that a and b are different from zero the normalization condition also imposes me that a squared plus b squared must be equal to 1. These are the hypotheses. I wrote the vector in this way. Now I have to build a contradiction. What I can do is to start from this state psi phi and now I replace psi with this expression in which I have just written. So I get a sigma plus b rho multiplied by phi and if I distribute I get a sigma phi plus b rho phi and so on. Now what I want to do is to apply the operator U to this state. If I apply it here I have to apply it here and if I apply it here I have to apply it here too. Now this U here is applied to a sum of two vectors and that's where linearity comes into play. The fact that U applied to this sum is the same as applying U to each of the terms of this sum. So what do I get here? I get A then U applied to sigma phi plus b with U applied to rho phi. And since U is an operator of cloning, what will it do? It will get the same result in this system of receivers. So I get by definition of U A sigma sigma plus b rho rho. Voila! I get this result. Now I also know that this object here which I also wrote here is also equal to psi psi. But if I write psi psi what is this psi psi? Well, I can use this expression again so I get A sigma plus b rho multiplied by A sigma plus b rho. I distribute and get A squared sigma sigma then I get the term rho rho and then I get the mixed terms A b sigma rho plus b a rho sigma. b a or a b is the same thing because a and b are not complex numbers or real, in this case they are the same. So I get this expression. Now, this expression is the same as this one so this expression that I have here must be the same as this expression that I have here. So I can take this expression minus this one must be zero. Let's call this triangle and this square. I can do triangle minus square must be zero. But if I do this expression minus this expression, what do I get? I get the terms where I have sigma sigma, so A squared minus A sigma sigma then I get the terms b squared minus b rho rho and then I get the terms obviously A b sigma rho plus A b rho sigma. And all this must be equal to zero. Now what can I do? If, for example, I do the scalar product of this vector which must be the null vector I do the scalar product with the vector sigma sigma since sigma and rho are mutually orthogonal when I do the scalar product with sigma sigma this term gives zero and also these terms give zero because they contain the vectors rho. So the scalar product will give zero. So there is only the scalar product of sigma sigma with sigma sigma which gives one. So if I do this scalar product I get the condition A squared minus A equal to zero. But I can do the scalar product of this vector also with the vector rho rho. At this point this term gives zero this gives zero, this gives zero only this will remain and I get the second condition B squared minus B equal to zero. Now, however, these expressions can be both equal to zero only if only if what? Only if A and B are equal to zero. Because among other things I can have A equal to one but if I have A equal to one then this condition here forces me to have B equal to zero but I have hypothesized that B is different from zero. Same thing for B equal to one does not go well. So if I do not have A equal to one B equal to one, etc. in the end both are easy to convince that they can only be verified if either A is equal to one or A is equal to zero but both these solutions contradict my initial hypotheses and so I have arrived exactly to the condition to the contradiction that I needed. Voila. All right? All right. It was very easy, I must say. There was only a little redistributive and some vectors. Basically you have to use the linearity of this U. This little calculation tells us that states cannot be cloned through a universal cloning machine without destroying or altering in any way the state of the original system. But if you alter the original system a little and if you give up having a perfect cloning, etc. then you can do a lot of things obviously. It is a theorem that tells us about an ideal and universal situation that is not possible to obtain. So for telepathy in quantum physics we still have to wait a few years. So telepathy in quantum physics who knows? Telepathy may not need quantum physics and quantum physics may not need telepathy. Thank you.