Quantum Random Number Generators

0:00 Before we stop, I would like to make a very important announcement. Yes, I'd like to warn you, tomorrow is Sunday, and Sunday the university is closed, so the game... The conference is canceled. No. We are not that lucky. We are not that lucky. So the gate will be locked. There is a guard 24 hours at the gate, but he would have to let you in. please, read your badges, the list of participants is given to the guard, and if you were kind enough to arrive between 8.45 and 9, you can be sure to get in because Gabor will be also at the gate, so he can let you in if there is any problem. Also, if you want to leave during the day, you would have to ask the car to open the gate. And again, if you want to come back, it might be problematic. So you'd better stick with the crowd tomorrow. But in any event, you have mobile phone numbers on this information sheet, so if you are in trouble, please... If you are in trouble, just call us and we will help you tomorrow. Is that clear then? So I... Can you just ask me, maybe I just did it in the beginning, but you said there will be a gate. Which gate will be... At both gates there are guards 24 hours a day. front door, yeah, before you enter the building. I'm not sure whether they're there. So, is that clear? I'm trying to be here between 8.45 and 9. We'll be there to assist you if there's any trouble. If you want too late during the day, then it might be problematic. Expect some difficulties, especially if you want to get back. That will be helpful, yes, please hear it. Yeah, we try to explain the situation.

2:30 Okay, next speaker, Professor Shek, quantum random numbers. Thank you. Um, this is, um, part of a larger program, um, to understand and formulate quantum mechanics in terms of Bayesian probabilities. And the program, I, um, when I collaborated in this program on, uh, who is also, um, uh, collaborating with me on this specific topic, and crystals. And the topic of this talk is to understand quantum quantum nanogenerations in this Asian subject which is very much. Quantum nanogenerals, I see reality. Here, this is too little. You can see it. I tried to make a thing about it. I couldn't. And this is taken from a white paper of the company eBay Quantique, Geneva. So this is Nicola Leeds' first company. And they actually make one of them with another generator. It's a device like this. They also make a card so you can apply it to your computer and start it in. using a quantum optical process. So it's sending close to the mirror and single photon detectives and single photon source and it uses that to make an anonymous does some post-processing to eliminate bias, etc. So these things exist. And so you can see, so I'm going to talk about this in three parts in this talk. I'm just going to talk about the particular problem of testing quantum quantum generators. Now I'm going to give you a pretty modern view of our approach of quantum mechanics and the radical phasing approach. So its main feature is, and this is as I am concerned, that we take the idea that probabilities are subjective degrees of beliefs.

5:00 of belief, working with the net-ease radical basing programs, and also digital quantum mechanics. So I just briefly, to briefly go over to this, but this is not going to be about this in the main topics. And then I'm going to give a base in the analysis of quantum-bending numbers and just tell you how I think one can understand the word set of quantum-bending numbers of the Bayesian perspectives, and we demand a very small, theorem-reliating, um, uh, characterization. Um, so, let's see. First of all, I want to get out of the way of possible ambiguity in the meaning of the word random. Random can refer to a single bit string. An infinite bit string can be algorithmically random, that means that it's incompressible. You can also define randomness for single finite strings. There's many possible definitions, many definitions are actually used. One approach is to talk about sequence of tests that these strings have to pass. So randomness can be a problem with a single string. So for the purpose of this and the next slide, I will talk about this. This is random sub 1. So you say a bit string x is random sub 1 if random you are not using it. The other possible usage of random is more common. It's a property of a probability distribution. So, you say, for instance, the string X is chosen at random from the set of all ended strings. And, of course, they're saying usually one means that X is chosen according to the uniform distribution of strings. So, which of these two meanings... If one says that the output of one of these devices here is random, which of these meanings does what they infer to. If you look at the papers of people building these devices, there is quite a long discussion

7:30 and confusion on these issues in the sense that it is straightforward and clear. And it is actually the second concept. So, in a quantum information way of providing authenticity, if you have n qubits, instead of n qubits, and this is the build of the experiment, and to prepare you n qubits in such a product state here, so each single qubit is in this position 0 plus 1, then we'll make the measurement on a computational basis, and then of course the outcome x is random, it's distributed according to this uniform distribution, So, it's just this random thing that's meant. And the close connection, if this is the case, then the statement, the probability that x is actually random one, the x is a random string in the algorithmic or any other sense, actually equates close to one. So, this is probably my connection. So, for no other one, I'm always going to talk about this as the second set of randomness. The question I'm going to ask is, how are quantum random numbers different? The vectors of quantum random numbers always have a place where these are truly random numbers. They are really random, as opposed to some other quantum randoms. Now, um, so I'm asking what does it mean. Now here is another example of a process which presumably is not truly random, tossing a fair coin at times. Now, I mean, usually if you look at the textbook, they write on this product distribution is the same. The sample redistribution, but in one case, one of the claims, it's truly random, the quantity is another case you claim it's not something else. Well, of course, it's rather clear what this means. Here, it's an anti-deterministic, if you knew the initial conditions exactly, et cetera, et cetera, But you could predict it, so this, we could say this is an ignorance probability, or it's not a real, not a true probability, whereas in the quantum case it's a true and an objective probability, something like this. I want to talk about these things in a more precise way. Um, so, but one thing you see already here, true randomness or more does not lie in the property distribution, because it's the same in both cases.

10:00 Here, the kind of super-randomness, if you make distinction between deterministic and indeterministic, truly random and just ignorance random, are much more dramatic. Because here is a very theoretical possibility, you can see a classic of processes, it's random for practical, there's no way of predicting anything. In a pseudo-random calculator, I'm going to use this definition again in the most general setting. It's a sequence of functions fn, which maps c to a pseudo-random string. So the seed is a string of n bits, and the pseudorandom string, basically the image of this function is a longer string. So L is supposed to be longer, bigger than n, otherwise you wouldn't have gained anything. And clearly, if you knew the seed, you could believe the pseudorandom string. So, it seems to be a very clear difference between quantum random numbers and these pseudo random numbers. And that's important, that's right, because nobody uses coin tosses in a practical context, but a lot of people use pseudo random numbers in their computers. So what makers of quantum nanomaginators propose is to replace these devices by these carbon devices. So here is something I would have said so far. So the question is, what does it mean to say that this distribution here is a true distribution for a given process, This is not a question of probability theory. Probability theory starts from the probability measure. It doesn't ask the question, is it a true probability distribution or is it not a true one? It's just, it's a mathematical theory, it starts from this thing, and the question is a true distribution, it's just not a question of mathematical probability theory. It is, though, a question of statistics deals with a lot.

12:30 Statisticians very often have just this problem. They have some hypothesis about, sorry, they ask, hypothesis testing, they postulate the distribution, and then they test the hypothesis that this is a true distribution. a lot of this is very hot, but this is the area of testing. And this of course, if a real physics experiment is listening to yourself by et cetera, and stop listening, and you just go to the lab and do an experiment. And you just test the thing and you prove this is a good random number. So, we do statistic tests. So the question, is the true distribution this or something else, is the question of statistics, you can run tests. And people have tested this extensively. The biggest such, I mean, it's a huge effort, and like, for the, I haven't got the viewer, I haven't got the viewer, I haven't got the viewer, I haven't got the video yet. After in 2003, enormously extensive tests, and these numbers passed them all, and all those, those were very useful. It is computationally impossible to distinguish the output of a strong super-renomar generator from a new context. This statement on the one hand says the experimenters could just, they wouldn't, they wouldn't need to start, they would never be able to establish that they deal with a quantum-renomar generator and not with a strong super-renomar generator. The second remark is a topology, because this is a definition of a strong-stilberonanism, right? So, I'm going to give you a quick background about the super-stilberonanism. So, we start with the same thing again, we have a sequence of Tauksen's effect. Each maps an n-bit C to an L of n-bit pseudo-renome string, and each L of n is supposed to be bigger than n, otherwise it wouldn't be doing anything. So such a sequence. And this is, now this is an informal, it's not quite, I'm going to say what this means precisely . So, such a sequence of functions, the strongest human genome generator, if its output is indistinguishable from uniform distribution.

15:00 So, this is . And this is work by Bloom, Yao, Astor. They make contributions to the theory. And this is these definitions. So, what do we mean by indistinguishable? But in order to formalize this, I have to formalize the concept of an adversary. An adversary is basically somebody who finds out, who can distinguish, who actually tells you, look at the string of the author and say, ah, this is stupid, not really bad. It's a function. And the disfunction is just supposed to tell me just one thing. It's a decision. It's a hypothesis testing. It gives me an author 0-1, and arbitrarily I say, well, let's say 1 means that the adversary judges the string that I give to him is 2-1. So if the adversary says 1, that means it's 2-1. So it's not really random. So, what is the adversary? It's again, for each m, we need to see the function, a m, which maps, well this is not the pseudomenon string, or the string that it's supposed to check, right, of length L of m, and it maps this to a single bit. And now that a m of s is to 1 is just supposed to mean that this adversary judges the string of the pseudomenon. So, um, now, here's the definition of what it means for the adversary to be unable to distinguish the output of the pseudonym of the term, the fm, from uniform distribution. If this difference is negative, then am is unable to distinguish the other one. Now this depends on two random variables. X is a scene chosen at random from uniform distribution, and Y is a full-length string chosen at random from uniform distribution. So, an of y equals 1 means that an says y is pseudo-fm. An of n of x equals 1 says an concludes that the output of fn, of nc, is pseudo-fm.

17:30 Now since X and Y are many variables in different distributors, of course it might not be morbidity for these things to happen, and if the morbidity for the adversary to say 1 is essentially the same for the uniform distribution as the pseudonym and non-determinator model, it's indistinguishable. So, and now we come to the definition, the common definition now. Fn is a strong stibular number generator. Actually, we practice standards also. Pseudo-random number generator. If no efficiently computable adversary, am, can distinguish its output particularly from distribution. Now, of course, it can always happen inefficient. And basically, since fn is not, I mean the function, it's actually even, it's the problem we can make as easy as possible for a n, because we can tell the a n what fn is, what the function is, what the function is, doesn't have to just look for a solution to another function. So, since we know what FN is, we can only construct a name by just trying everything. We just try all seeds and see if it all fits. And then we know in the end. Of course, it takes very, very, very long to do that. The seeds are typically unknown. You know, it takes longer, so you take forever. I mean, literally forever. So that's not possible, so the important question is here, can it be done efficiently? And so the question is, is there a sequence of adversarial AM that can do this job, this distinguishing job, polynomial ENA? The big question is, do those things exist? Do strong-stool-eatomachinaria exist? On the other side, yes and no. Any one-way function could be used to construct functions. And this was the reason to construct a discrete law of a factoring. So, factoring, one of the functions in factoring is multiplying together to a large interface,

20:00 and then undoing it as impossible. So, based on the difficulty of factoring, you can actually build a strong number generator. Now we know factoring is only supposedly hard, but there's no proof. So we can build such a thing based on the product of factoring, or the discrete law, but there's no proof that it actually is a strong number generator. And I should make you stay very clear to how the proof is done. We try to perform organizer proofs. The proof goes like this. It's conceptually very simple. It's a proof that shows that if such an adversary existed that could efficiently distinguish the distinguishing thing that I just talked about, they factoring the multi-generations. So this process could be used to actually do factoring efficiently. Similarly, if it's been long, then you can just turn this around and say factoring is a no. To be easy there is no, no, no, we're not doing this. It's amusing that the quantum system, I'm going to talk about the quantum, a quantum computer that actually writes these things in the principle but that's really the sense of what I'm going to say, because you could build one way function of NP-complete problems because nobody knows that NP-complete problems can meet hackers of the common computer. Most people don't need it. If you want other people to do that, you can find them. I don't know, in fact, if you want a skew-to-nanoma generator, then nobody, now, or in the next 50 years, is like if you ever distinguish on a quantum computer output, where you just use KDS or triple desks to construct simple functions that takes about 128-bit C to a very, very, very long output. I can put on this machine that you just use to see the built-in PGP program, so it gives you a string that nobody, even not the American government with all its resources on the building to develop limited resources. and even they couldn't possibly do that.

22:30 So that's it. What does all this tell us about, many of the quantum and the non-generators have better pay attention for the quantum mechanics. Because just the usual approach, the usual physics approach as well, all this is really irrelevant for what we are, all the data to ignore, because we just go to the lab and test things. All this tells us, and it's true, If we really want to sell these devices to sell people well, they're better. Well, just testing isn't good enough. Now, very briefly, this is now, okay, in a nutshell, what one of our final points is that quantum states represent subjective degrees of believable measurement outcomes, of the same kind as ordinary, uh, uh, in, in, in, in, in, in, in, in, in, in. It's not about references. It's a color of us, uh, um, um, the, um, when I say subjective, I mean, I mean something very specific. I mean that, um, um, objective, is not enough for objective. The objective would be that the quantum state is determined by facts. By facts, very classical facts, experimental facts, technology, or all of these facts. Whereas subjective means really that two physicists with access to the same experimental facts may legitimately assign different quantum states to the same. So it's a rather strong statement. Now, classically, it's a very well-established and consistent and truthful theory, although of course, not everybody subscribes to it, but at least it's fully developed and good. So, here's the trailer. The Finetti in 1931, Sandwich Foundation of Statistics, a marvellous books, the latest books of 7.2. The Humanity is a new edition of his Dioperability book. Van Allen Smith, The Bayesian Bible, 1994. Smith, I should have said, A.M. Smith, he recently headed the Smith inquiry into British higher education. So, there is a very strong

25:00 Bayesian influence now on mass education in secondary schools and the mechanism in Britain. If only the school children could condition on what the teacher says. And then a very recent book, The Physical Ability Review of Sting, which one should even person could call the title. It's a very nice, very famous book, I think. And from this book, I take this quote, but it is a mode of judgment, probabilities are so better to be the belief which require a probation definition, decision-making. So decision-making is the important thing. And since I'm going to come back to this a little bit, decision-making, decision-theory is complicated. Betting can be probably a very easy way to attempt betting, making bets as a natural approach to decision-theory. So now, um, yes, yes, again, I'm just repeating the same thing. So, the, the, the important thing here is that probabilities are assigned to the outcomes of single trials. So there aren't, there aren't people by nature, they're actually assigned by the experiment evaluation by the physicists. And there's a literature with a belief, and it's important that any probability argument starts from a judgment in the form of a prior probability assignment, very much like a mathematical probability theory. They always start from a measuring assignment, and that isn't the kind of drive from any facts, isn't it? So it's just to say, and ultimately it's a subjective prior judgment, and the prior can have to do some of their facts. Now, quantum mechanically, we just use the same set of standard points. So, any quantum argument starts from adjustment and prior quantum status standards. And the prior standards cannot be used from a regular practice. I'm not going to extend this now, because I've done it before, and my colleagues have done it, let's just summarize. And in order to actually analyze experiments of physicists who do repeated trials in this context, we need to be disconnected from us.

27:30 So, the quantitative theorems are important for this because otherwise there would be no simple, I mean, conceptually it wouldn't be a good negative difference, but there would be no convincing way of actually explaining what businesses do when they analyze the experiment. The divinitive theorems allow you to do that, and so there's two which are important here. One is the classical quantum divinitive theorem, Dorma, Absinthe, Nulli, and elementary proof by Abs. And then there's the extension, which wasn't mentioned the other day, which is divinitive theorem for exchangeable sequences of quantum operations. The collaborations are called computer-positive instruments. It's not history, I don't know. Okay, so these are used in products that are actually used in finance. Now let's go wrong. So, for a Bayesian true randomness is a even that we don't, it could be a real problem. Because it's a classical Bayesian random again. Let's look at these two statements. an agent is standing at 0.5 to buy a Newton between the next Bundesliga. Okay? One statement. And this is another one is, um, we say an agent is standing at 0.5 to the outcome of a certain part of measurement. So, everybody feels that there is something different about these two things. One, you would normally say, well, this is given by physical facts, this is something to confirm. This is, perhaps, I mean, what else can we say? But, for a radical Bayesian, there is no difference between these two statements. They are, they determine decisions, they determine whether you buy a certain login ticket or not, whether you make decisions, take a certain point of action or not, using decision theory, facility analysis, and that's all there is to it. And if you want, if you want to talk, can a Bayesian say that she is more confident of one or two? Well, no, she can't. Well, I really shouldn't use sheep, but I have never made the female babies in this place. Um, so, uh, why not? Um, well, I said the, the whole, everything is, is, is, is about the decisions,

30:00 and if you, um, if you want to distinguish these situations, you have to look at repeat the trials, And then you have some probabilities to a bigger space. And then of course, this might be in high ID, and this certainly will not be in high ID in future repeated trials. So, single-repeated trials for a vertical basis, again, classically, there is no conceptual difference between single-repeated trials. But in a single trial you have a random variable taken from 0.1 and you have a probability which goes from the sample space to the interval, repeated trials. It's a bigger sample space. It's the same thing. If you haven't seen the case, it's a bigger space, so there's no conceptual difference here. Most importantly, a Bayesian cannot test that a priority assignment was correct, because it's not correct or wrong, it's just, it's just the assignment you base the decisions on and which you use to actually update, if you get new data, they're not used to test this assignment, they're used to update it and it will come from the material for fully approved assignments. So, for Bayesian, my initial form was development. The fact that I can't distinguish a quantum nanometer from a pseudo nanometer that is irrelevant in the sense that testing is irrelevant anyway. It's part of the score. So it seems that the Bayesian is a real problem in this issue of nanometer Bayesian. It seems that it's a real problem in understanding what's different from quantum nanometers. Okay, now, here's the way that Jeffrey introduces probability by a single theory. In this case, he considers logic worth one dollar if an event E happens to be true, not to be true happens. And there's a ticket price, Q dollars. And, um, the definition of probability in the simplest possible second, the introduction of the argument is that probability Q means regarding Q dollars

32:30 is the pair price for the ticket. So it's directly connected to the ticket. What I regard that their price determines when I buy, when I sell, if I buy, if I sell, and so that is a, um, is a, um, operational definition of, um, of, um, uh, of what we did in a decision zero in the context. Okay? So, the taskbook arguments, which I'm not going to do in the are not familiar with this, but, um, you can, uh, if you say that public assignments are inconsistent if they lead to a sure loss, I mean, in a single transaction, if your ticket valuations lead to a sure loss, well, you call them inconsistent, and it becomes consistent if they do not lead to a sure loss, and then, surprisingly, at least at first sight, consistency The efficiency alone implies that these probabilities, these thicker valuations, are going to be common ground axioms, except for counter-relativity. So it's just fine, excuse me, this is the definition. That's the definition. So that's an axiom, essentially. I don't want my thicker valuations to lead to sure loss in a single transaction. And if I, if I assume that, these follow. Sorry. Theorem. That's a theorem. Theorem. And, as I see, doubt. Theorem. Sorry. Theorem. Theorem. Theorem. So, and it's a symbol. It's a single, it's a single trial argument. So, usually, I mean, businesses think about frequencies that they come up for. A, that's not a great beat. But really, she will lose them longer, but that's what the argument says. It actually happens here now, in a single trial, she will lose it. And it doesn't have to be too close. So, now I come to my previous

35:00 Do you assume that there is a random device on which you are using, which follows the connection of the program? Oh, no, no, no, no. No, there's no random device here. It's just, it's just an event. But how do you do this? Ah, okay. I'll give you the dashboard argument later on. Ah, okay. But not now, because the connection is coming out very much. Um, so it's, uh, it's worth knowing. It's clear. Okay, now, um, the basins, it looks like, can assign different probabilities to, uh, two basins can assign different probabilities to the same event, and neither can be right or wrong. Or would be right or wrong. I mean, you can adjust different information if you want, but this works. So, I look at this case now that A and B assign different probabilities to the same event. Right? So, PA and PD, same way. So, that means that A regards PA dollars a fair price and B regards PB dollars a fair price and if they are different, well, they're certain that they're both happy to do. This is not a huge situation of a marketplace because their identity is such a player and other and low-organized players play a role. So this is a very simple situation in which two people can actually conduct transaction and everybody is happy to process it and they made money. Now, I'm not going to talk about inside information. This is a concept I will need for my punchline at the end. So, let's look at this situation here. A, Alice assigns multiple is the V event B, but Paul assigns zero volatility. So, what does it mean in terms of these tickets? Well, A regards certain amount, non-zero amount, as the fair price, whereas B believes that the event cannot occur. So, that means that from B's perspective, he will certainly make money. From his perspective, Alice has a schwulos. That shouldn't trouble Alice necessarily, because she thinks this is a disability, but from B's perspective, there's a schwulos. And this is a definition of inside information. So if, from B's perspective, he has a sure window of analysis, and his finalists mean that he has inside information about the event relevant to the event.

37:30 And there's a flip side to this, it can be, if A is probably less than one, but B is equal to one, Well, then again, Bob can make money with certainty. So again, he has been said information. And even more radically, if A is not over to zero and B is not over to one, well, then both that inter-termination with respect to the other, so I call this contradictory inter-termination. This is a, this is a, uh, uh, uh, uh, uh, a pattern about pathological case, whereas the other case is actually quite, quite normal. I mean, one person can, has just more information than information. This happens all the time. So, it's nothing, it's nothing pathological at all. Okay. So, um, I don't define what beliefs are compatible because these beliefs here are really incompatible. And they're always trivial, classically, but interesting, so I'll come to that in the next slide. So these two are incompatible in a sense which is defined. So if I have any party deciding whether it's the same event, then I say that I have compatible beliefs. If there exists a state of belief that some probability assignment could probably be that no party at any certain convention would like to do it. So there is one probability assignment at least that, well, no party could possibly exploit in this way. So nobody has this information, whether it is dead. Then I say that these are compatible. And it's trivial to say that we should have finite, sharpest-based atomic events. Then, I'm happy to believe it's the same thing I was saying that the support of the probability assignment overlaps. So it's non-empty. I don't believe, because this means, this doesn't mean, there is this one, if you just assign political reasons to the overlap, to the intersection, then I get just such a set of beliefs, which is time that nobody gets any information. And then I can come to maximum beliefs. I mean, I might want a set of beliefs such that

40:00 there is no compatible belief within teleformation. This is actually all I could talk about here. I want set probabilities such that nobody else that is compatible with me can actually explore it. Nobody else should have inside information. So I call this a maximum belief, but the idea is nobody else, from my perspective, nobody else that has incompatibility can have inside information about me. And then I still pay attention to the maximum. And classically, trivially, it just corresponds to certainty of a particular outcome. It's the only case. If there's certainty about a particular outcome, then it will be just maximum, and only then it will be maximum. Okay. Now we come to the final case, where, what happens? We have the loop map here, and... What is particular atom? What I mean by an atom. I'm thinking about a sample state with atomic events. It's a finite, finite sample state and the element of atomic events are called the atoms, atomic events. So, the problem case, I'm talking with a single system, as it's called a single element, I have any parties, I can state assignments, I have no A, no B, no C, etc. And, um, so, party, either the science table A, or the science table B, etc. And in order to come up with the decision theory, I need the measurements to affect the most general measurement. So, let's take a look, it's not really a name, sorry. You call this semi-spectral measures PODM, just generalized measurement. So it's a set of operators, positive state, positive state definitely, and that's up to one. And the probability is then, which stand by the alpha to the outcome K,

42:30 just given by the state's population. So that's the general situation. And now I define inside information as a constant case. Now I have to define this for all possible measurements because inside information was motivated by being able to make, to the other party, so if there is any measurement for which this is true, then I will always start with inter-information. So I say here is inter-information about system S versus system A, if there exists a measurement, such that Alice's probability for its existing measurement and an outcome K, such that Alice's probability is positive, but Bohr is equal to zero for the other one. So this is the definition of information, and this means that Bohr has a sure win over Alice, from his perspective. Okay. Now, what relates to the recording mechanics, I couldn't trust here, it's a small theory, but it solves this question to my satisfaction, so therefore I think it's nice. So the null space of the support of a density operator is the span of the eigenvectors that correspond amongst the eigenvalues. And so here is my definition of compatibility. So just slightly the same thing as classically, any party that compatible believes, if there exists the state of belief, grow, so because it's a nest operator, such that no part of any information will be developed. So this will be stated that none of the parties will be deployed in this immature, immature . And so here's the theorem. And part of the beliefs, if and only yet, the intersection of the supports of these operators is material, which means it contains just more than this, it's not the stage containing

45:00 I think it's zero-victor, but at least one will be cool, but not a major in some space. What does it mean to be a matter of? Well, this is a criterion that, um, Rund, Finkelstein, Mermin, um, I should try not to mention here because, um, And they actually derive this compatibility criterion here from a completely different So this actually, this way to characterize the compatibility is equivalent to that. Okay, so, finally, remember normally, instead of with these, again I'm talking about maximality, So, if it's maximal, no other party would be a compatible set of beliefs, as I mean that information. So this is just what we want. We want a state assignment, we want to assign a state to our quantum and number generator, such that no other party with this compatible beliefs or someone who we think is the same, basically, right, has any inside information on the issue. Excuse me, I don't understand. You have defined that compatibility already is the fact that nobody has... Sorry, sorry, sorry. Excuse me, you have already defined that compatibility as the fact that nobody has inside information that I can do wrong. No, no, no, not nobody. Compatibility of, um, of, of such, no fiber. Yes, any inside information that you've got. No, so that means that these end parties are... It's not circular. No, no, no, no. It's not circular. Because... Yeah. No, no, it's... Because, just because, I mean, in this, in this definition here, This is my penultimate slide, and yes, I have 44 minutes, so thanks because I can clarify this again in a couple of sentences. And this, this, no one says to me that's compatible, just refers to this two stages, this, this, this other one, so this, um,

47:30 Okay, I can't, I'm, let's cut it, it's not circular, okay. It's definitely not circular, and this, I guess, definition here of maximality is equivalent to, to, to go to the, to the cure. So, instead of a leader's maximum, it's an only if that is pure. So, if I can use that assignment, then I know that no other party that's compatible with me can have inside information. And that's precisely what the same could probably be what I want. And so my conclusion is that nobody can have internet information about my computer numbers and that that's my idea because that of course is my, or the experimenters, or the trusted salesperson's assignment to this device. And so if it is, if I style a pure state to my problem, I'm not going to be right when I have this zero and this characterization of randomness from a purely subjective basis perspective. Thanks for your presentation. one of a time, so, but that's ok, but I don't know. Excuse me, I have lost the last passage, can you repeat, can you derive again the principle of pureness of this? I've got to write it. Can you explain again this one?

50:00 So, you can derive this from the previous thing. So, let's... I can prove this to you. So, um, let's say, I mean, what is the rest of me wrong? Roe is pure. For my definition, why is it? No, Roe is pure. It's a consequence of what you said. You're like, why is it? You're going to believe that Roe is the right thing. Why is it? Okay. So, um, Um, no, where's my, my, sorry, I use the theorem not here, okay, that, um, that if I have compatible beliefs, then, uh, this is called, I suppose that this has to do with the fact that you think this is a state of maximum information, or not? I mean, this is using the concept of maximization that you would have to... Okay, I think so, but by... It's probably the same thing. So, the pure state... So... Let's see... So what I need to prove is that if there isn't, Well, can I suggest that a little bit of a change of terminology will make, never mind the deep nuts and bolts, it is natural in English to stumble a little bit on these ideas because of the word inside information.

52:30 It sounds like it must be true. And what's really happening in your lovely ideas is the idea of somebody being more opinionated than somebody else. And maximality is complete opinionatedness. And to have incompatible beliefs is, of course, the idea of some consistent, common, superior self. So I think if we eliminate these type of information and talk about more opinionating than... So if you have a mixed state, you have at least a two-dimensional support. That's a good idea. And so then somebody else could have a pure state assignment in that support, which would correspond to inside information. I mean, this is precisely the idea of inside information. So, therefore, if the state was purer, then, sorry, was mixed, then the set of relief could not be maximum because there would be another particle of inside information. Yeah, so that's exactly the truth. Thank you. Sorry for that. or that was your Miklos. Miklos. Oh, I'm sorry. Maybe this will be unclear because you said that you didn't have any intention to defend Bayesian's connection to the quantum state, but I think it's in need of some kind of support or defense, because one has the intuition, say a physicist would have the intuition like that, if you're asked to assign the wave function, say, to the ground state of the hydrogen atom, you're quite bound as to what kind of a state that it's supposed to be. So how can you be Bayesian, then, given this thing? If you can be... What this kind of answer is that the ground state of the hydrogen atom is the ground state of other Hamiltonians, so it's a mathematical type. So what I'm basing about is whether to decide the ground state to particular idea or not. So the question where it's a matter of basic experimentation and of prior information is that the idea of the human idea or the matter is find this particular state and the ground state too.

55:00 I'm trying to decide how to do it. You have to examine external data, etc. And then, of course, you can use it. I definitely wish to remain silent. Okay, I'm not convinced. You have a starting point. You should specify the algebraic core of events. We have events. In the practical case, events are subspecies. Is it correct? Because my events are semi-spectral metals, to give it a big name, here is what defines my events, the outcomes of these measurements. So, it may be defined by the PM. PM is a number of outcomes, and the outcomes are . So, it's not clear that you mean. In the classical things, you derive a one-shot derivation of the Kolmogorovian axioms, up to the same infinity of the Galitini, starting from one axiom saying that a pattern has to be consistent with no sure loss. then you define, then you obtain it as a theorem, the usual, and what happens here? If you start from, you say that this interpretation can be applied also to the quantum piece, the subject interpretation, is it possible to derive from the spiritual sense of all the The same hypothesis, the hypothesis to say that backing is consistent, not sure loss, is it possible also in this case to derive the axioms, the probability axioms? Of course, in this case you have to replace, but we have to be careful because in this case you have to replace the usual notion of compatibility because in the Murnian case, compatibility means that the intersection is zero, but not here because the intersection zero does not mean autoconality.

57:30 So I would like to know exactly this point. Yeah, this is really independent from this because this is a classical event space. Is it classical? Yeah, it's classical. It's a classical event space, an ordinary event space. These are ordinary classical orbitaries. Are you saying that events are possible with event dimensions? No, the events are the outcomes of the event. What you said before is that there are positive operations, there's a measure. Okay, but I meant the outcome was, I meant, when... When we did a sign, what happened to the outcome of the... So this means that the class, the actual instruction, that's... Yes, it's a class. So there's no quantum logic in this at all? Yes, because otherwise, I can see how it can structure... Just a brief clarification on the plasticity problem, namely, in the single trial case, It's typically assumed that a person is supposed to take any however complicated combination of this from a book and the combination typically is by classical connectives. So it's still the same here, classical connectives in, okay. So I would only, in this I would only condition on actual measurement outcomes, so I mean this is only, you say arbitrarily complicated, all I need is, I mean the base rule is a little, this is a little complicated, otherwise the only thing is I need a simple, simple compound that, yeah, this is, this is complicated to get, I want to get the, the, the, the rule here. I have a different philosophical question. I don't think that this Bionism is very hostile to the objective chances of the single case of objective probabilities, because there is this limitation of convergence theory in Bionism, that if there is an objective chance distribution, then in the long run,

1:00:00 And the probability of subjective probabilities will converge to the objective, provided there is some match with probabilities. Provided, one makes exchangeability assumptions. And exchangeability is a very strong restriction on the prior. So it is the kind of prior assumption I'm talking about. I mean, I just glossed over that, but the different theories are important, not because extendability is something you could, if it was given to you, by facts or by the experiments, but because it's an assumption, well, it comes closer to the usual experimentless assumption that they have repeated trial. So therefore, but that remains an assumption which is not So therefore this conversion is conditioned on a prior assumption and therefore I can firmly remain lazy in this type. Although I agree, I mean you can mix objective chances and you can certainly talk about having a subjective degree of belief about what the objective chance is. It's not contradictory, it's just not very economical. Thank you. Your last slide, I think it said, nobody can have inside information, it's relevant to mine, is what France is. So I have a little idea what the punchline is. It sounds as if this means that if I have a maximal belief state, so I'm assigning a pure row, then I've got quantum random numbers. Therefore, if I have the belief state concerning this sequence of qubits that they are in, as it might be, spin Z1, tensor product, spin Z1, tensor product, spin Z1, all the way, so that in fact if you measure it, you're going to get 1, 1, 1, 1, 1, so that, you know, a string drawn at random from this probability distribution my row is just

1:02:30 uniformly one, a lot of us would say that, you know, that's not random. I know there's a product-process distinction, random one versus random two. No, no, no, it's a process. Actually, it's rarely called spin-up ones, you know, spin-up, spin-up, spin-up, spin-up. maximum beliefs are the response, certainly, to probability 1, or particularly at probability 1, 0, and everything else. So you're just now recovering from the situation. Of course, I wouldn't call it, I mean, I say it's maximum belief, I wouldn't call it random. So of course, now that I have random, I need, I need probabilities. I mean, I don't have anything different on the test, and it will be that I'm not used as a zero-like, but I can have. Oh, you see, so your idea is that a quantum pseudo-ragic output temperature is indeed producing such numbers, provide, if the agent assigns a row, where row is the... Since I'm a radical base woman... Excuse me. Can I just wait for a second? I thought, I thought I had to finish this for the discussion, but I would never say that the device really gives, so it's all about, can I have, is there, is there, what is it, about belief that is different from the belief about a coin or about a student anomaly generator.

1:05:00 And I think that this adds this extreme maximum. So, what's different about the belief about the quantum anomaly generator, which I can obtain by experimentation, by examining everything, it would be that there can be no information. Of course, from my perspective, this is my belief. Obviously, this is why it's in mind, because it's not a fact about the device. This is the title, Problem of Information. Is it readable? It's not a very good tool in any way. Yeah, because it's incompatible. Back in Turkey, I mean, read those. Thank you. I believe this is the best problem. I have all some experiences, but I have seen more that it is not the best. First I wish to thank all of the Regulation Committee and the Special Olympics for ending I will deal with the interpretation of issue. I am only a modest scholar who tries to understand a little bit of these difficult matters, and I think that quantum information can be interesting to have an understanding of the problem of information in general. So, I will discuss Because we are so strictly related problems, the nature of quantum state and the nature of quantum information. I must be showing an abstract scholar because there are both abstract items, issues, as I understood this morning. But in my opinion, the reason why they are related is that according to John Miller, quantum systems are themselves information.

1:07:30 So I will try to follow this line of questions. First, let us consider why information is so fundamental. First, informational and entropic considerations apply to any physical systems, from black holes up to classical systems, I believe there is nothing in nature that could not be the object of informational considerations. Second, information is in my opinion a more basic quantity or a more basic feature than energy and then any other non-physical quality. This is due to a theorem, or due to another impairment, according to which classical reversible computation is possible when no selection of information is done. So, any irreversibility in any quantum processing comes around the stems from a rotation selection. This means that the quantum system, as it is well known, because quantum systems are also used in quantum contradiction as reversible systems in principle, quantum systems can be considered as reversible in rotation processing devices. Since we have no energy expenditure, evidently information is somehow more basic than energy. In my opinion, more basic than any other physical properties. Moreover, recently, in a recent paper of Oledek and others, I will give at the end the most but we are here to switch up and down, so we can see the end of these types. A recent paper published at Inesio, two, three weeks ago, by Orodecki and co-workers, in this paper is shown that information, one can communicate information for free, without possible. So, exact information must be somehow something that is perhaps interesting to discuss.

1:10:00 So, my first point about the nature of quantum state is that the quantum state is not known. In my opinion, when we have a very easy experiment, such as an interterrometry experiment, and these are full neurons, when we have a very easy experiment, we know that we can choose in that moment to postpone or to anticipate the benefits. And we choose in this way to measure either the path observable, that is to measure the path that is chosen, in the United States of interference observable or visibility observable if you wish. Obviously, obviously, obviously this is possible because we have here no event at all. This is clear, we have no event otherwise this would be not possible. But we have somehow something because I have an input forward and an output forward. It's very strange that the bottom disappears in nothingness and comes out from nothingness. So, it must be something in between, and this starting must be in a certain stage somehow. So, this is my first supposition. I need to sharply distinguish between two things. One problem is that nothing can come out from now. This is, I think, obvious. Another thing is the fact that any system, any quantum system and any quantum state represents to a certain extent the necessary but not sufficient condition of an outcome, of a measurement outcome. I will come back to this problem later.

1:12:30 The fact that it is only an essential but not sufficient condition of any measurement outcasts. So, the problem is that we cannot do a direct experience of this state, or the state of the photon, if you believe. It is well known that it is impossible to measure the state of a single system. This is the no-cloning theorem, this could also represent a violation of the unitary, of the Voluscio, this is the Dalyan and the UN theory, so it is simply impossible. And this is obviously a problem, because we have something that, in my opinion, is not narcissistic, but on the other hand, we cannot have a direct test in us of this option. So, what is this? This is obviously the central problem of my decision. So, my answer is that quantum systems, before being detected, represent a form of potential reality. I know very well that it sounds strange to physicists to speak of potential reality, but let me explain why I think so. And first let me begin with the consideration of the information approach. The connection between quantum information and the nation of the quantum state is the following. The information a quantum system represents is also potential. I already proposed this in a paper of mine one year ago, and the lecture paper of Rodecki shows exactly this. Why? It is well known that in quantum mechanics the conditional entropy can assume negative values. This is a well known fact. And this seems a little bit strange because the conditional entropy expressed in general, information that can be communicated to someone. So it must be possible in general. And here is negative, can be negative. Orodecki explains this point in a very simple and genial way in the context of an idea that EPR expected. According to Orodecki, it is, I quote the words of Orodecki,

1:15:00 It is the information, it is exactly the information, the potential information that can be later in a certain moment transferred by bond without a cost. This is the point. So we have, in my opinion, a strong reason for speaking of potential information. I will stress a fact. This negative conditional entropy, and this circumstance due to entanglement, because entanglement provides this condition, is something that is culturally impossible. So it is specifically a quantum fusion. So it is a potential to transfer additional quantum information in the future at no proper cost. This is the exact information of biological. So, that is, what I propose is the following. I propose classically, in my opinion, information is identified with the information that can be transmitted, excuse me, with the information that is actually transmitted. In my opinion, we should sharply distinguish between information and information transmission. In my opinion, part of information is information that can be transmitted, or a bonus, that can be transmitted, and in this sense it is okay, shared information. So, I propose to dissociate the concept of information from information evolution. As I would say, in a classical context, it is not so. Okay. As it is well known, any two-level pattern system can emit in the

1:17:30 the normal basis 0-1 in this world, and which can also be written according to equation 4, and as we know, this is called the qubit, the qubit. or now the basis 0,1 can be considered a code. And my first thesis is that the state 3 codes an infinite amount of potential recognition. That is, if you take the, if you take the the quarkly sphere, excuse me, I will now also play with this computer. If you take, this is also a node, if you take the parameter 5 as a phase parameter, that is the state according to a shift of the space can be anywhere in this scale of the equator, but on any parallel, and the delta parameter has the veridian parameter. the state of a two-level system can be a point of distribution of us everywhere. So, to this extent, it represents, it incorporates, in my opinion, an infinite amount of information. It is true, it is true that classically, when we measure an observable whose area states are exactly 0 and 1, I have zero or one. That is a classical thing. This is known. But the problem is that quantum mechanics, general conditions of information, do not determine signal effects. This is the central problem. And as we shall see, this is a general problem, as we shall see there. So, the question is how we can understand the information of changes given that a quantum state is already, already represents an infinite amount of information, or given that quantum

1:20:00 systems are information. How can we understand informational changes? To my opinion, I did a guess in the paper of mine, there are only two ways to modify the information quantum system represents, either by measuring or by time. Why I say so? This is, in my opinion, a corollary of the Landauer-Bernett theory, because the Landauer-Bernett theory says that we have reversible quantum processing or reversible computation, reversible transmission of information when there is no selection. When we have selection, when we have selection, we have a reversible process. So, in my opinion, since quantum systems already represent a reversible information processing, The only way to modify this information is to select somehow information. And this is only possible to measure it. Or you can share this information by entanglement. Because, in my opinion, entanglement is not only my opinion, there are a lot of works already done in this direction by Zurek, by Bernhardt and others. entanglement is, in my opinion, nothing else than mutual information, information sharing. The fact that several systems share a common amount of information. This is the reason, in my opinion, because in teleportation, we have no transmission at all. Nothing is transmitted apart the classic transmission of information. It is only a problem of local operations and information sharing nothing is transmitted in teleportation i think here is a big is a big misunderstanding of our transportation functions also it is it is only a problem of information sharing so we have only we have only selection of information from a pre-existing pool or sharing of information. That is, in my opinion, information cannot be created. Information cannot be created. Information can be only selected or shared, but not created.

1:22:30 If you will, this is another population of the second principle of thermodynamics. We cannot create information. Information can be only selected from a persisting form. My assumption is that quantum mechanical systems can be considered as the sources of any information in our world, of any information. My assumption is that the quantum system represents the reservoir of all information that we find everywhere, by selecting it in different ways. I think we should sharply distinguish between the concept of source and the concept of cause. Quantum systems are sources of information, not causes. Sources of information. That is, quantum systems cannot provide information by themselves. It is impossible. You need some factor conditions that do not depend on the quantum system itself. These factor conditions are normally provided by emission and environmental powers, for example, by the environment. additional conditions. This is the reason why I said before that a state of a quantum system or if you will the information continues in a quantum state is potential. For me potential means two things. One, it represents only a a sufficient, a necessary condition of a certain outcome and not a sufficient one. First, second, for these reasons it depends on external conditions. Without a measurement apparatus, without an environment, or without other open systems, if you will, it is impossible to extract information from a quantum system. In my opinion, this is a complete physical mechanism, In the sense that you have a potential pool of information and you extract some classical bits from this pool. And the mechanism is very, very easy. The mechanism is explained, in my opinion, by decoherence.

1:25:00 you only transfer locally, locally, some part of information from the system to the environment. And you have a local growing gap of the environment, according to this gap. But globally the environment is conserved because of the information. So, a source of information as a quantum system is only something that under a suitable operation that is a selection will provide information. But, as I have said, this does not mean that it will provide this information by itself. We need additional requirements, additional requirements. So, you will have here a qubit. A qubit can be raised to a bit, for instance zero, to regionally or by timing to a bit. So, these are, in my opinion, the only two ways to modify the quantities. So, when we measure the evolution of the world system, comparative object system, plus measurement apparatus, plus environment is secured by counter-mechanical laws, and is therefore unique. However, this global evolution does not in any way determine measurement results. because quantum mechanical laws only rule from ability to obtain a given result in a given environmental test. This shows that local and individual events are...