Information-Theoretic Derivation of Quantum Theory

0:00 We relaxed one hour for each talk and we decided to schedule three talks for the session, three talks for the office session, and each talk is supposed to consist of a 40-minute lecture and 20-minute discussion so that we have more time for discussion that makes it more lively and more relaxed. which is very interesting and which you cannot see on the program which is available here and which you have. Auletta's talk is replaced with, or exchanged rather, with Roberto Juntyni's talk. So the president is going to speak at the second speaker this morning, and our letter's part is moved to the president's part. Please note these changes, and then, uh, . Good morning to everybody. Uh, this session, uh, this morning's session, is devoted to two main topics. of quantum logic and quantum information. And that leads to an author of the speaker. It's a . And this topic is information, theoretic derivation of quantum theory via quantum logic. Thank you. Well, first of all, thanks to the organizers for having me here. Let me start right away by telling you what this talk is going to be about. By the way, it has a reference on the vision. So here's something I'm going to argue for. In several stages, I'm going to show what the products are good, that quantum theory must be viewed as a general theory having a general character, constrained by several important information-theoretic principles. So we poster the principles and from a very general view of the theory of information we get something resembling quantum theory.

2:30 Now, views of quantum theory can be formally derived from the corresponding information-theoretic axiomatic system. So in order to show you how this works, I'll try to take these principles, information theoretic principles, and to try to show you a derivation of the formalism of quantum mechanics. Well, let me start by a few remarks about the historical context, which is too cold On one hand, this word inserts into the context of the information theoretic approach to include quantum information, but more precisely information theoretic approach in quantum theory as such, starting with, well, it can go back to quantum logicians of 1930s and 1950s, but recently started with Wheeler and other people in Wooders around 1980, and there were a few names, but the list is certainly not complete. And on the other hand, this work inserts into a tradition of axiomatic approaches to physical theories, which again, not only are limited to quantological approaches, but in a more I'll send all those practitioners and friends to play it for you, can be characterized these axiomatic approaches as follows. In more than physical theory, more than physical theory manifests a certain tendency to look for an axiomatic representation based on the model of axiomatic systems in mathematics, which means here, by the way, that axioms are not viewed as ultimate postulates about nature or reality, but just as in mathematics, axioms are viewed as postulates for using mathematical theory. Now, the axiomatic ideal taken from geometry means that one finds all the initial objects of the theory only by relations and in a way by substantial qualities. Let me do the following, I'll start, in an unusual way, certain philosophical remarks, and then once we clear the ground for philosophy, and I'll explain why we need it, I'll move

5:00 to formal results and show you how one can derive the elements of the formulas of quantum theory from information theoretic assumptions. So, concerning philosophy, I want to start with it, there are always internal questions that we hear every year now with one space about information and the questions are, information about what? What is the physical support of information? How is information stored? How does one operate with information? All these questions are certainly a bit erroneous, but there is a certain approach that I'm taking, and let me make clear here the core points, the core philosophical points, which in my view give a certain answer to these questions, or at least show how one can a little bit of an information-theorated approach and not worry too much about these physical questions, to know where they're placed. So my first point, my first philosophical point is that I'm taking an epistemological attitude. So one is concerned with theories, and remain deliberately agnostic as for any ontological content of these theories. So whether physical systems or information are ontologically prized on this couple of these. show you a theory based on an information-theoretic approach. So science is viewed as the construction of theories taken as an objective description of certain phenomena, while selection criteria for phenomena and the understanding of objectivity of description can vary depending on the particular theory question. So concerned with these theories, and without taking Let me move to this second point, universality. I will play in the following, well, in the main third and fourth point, but there is no reputable trace of any phenomenon, so nothing, basically, that escapes from a theoretical description. So we're speaking about theories, but anything, well, any, almost any reputable trace,

7:30 and you're described by a theory. By a theory. of universality seems very possible, given our version of thinking about quantum theory or other physical theories as universally applicable. Now, here's the first of the two main points, third and fourth. So, depict various theories or classify various theories, axiomatic theories I mean here, in the loop for it. So if one classifies theories, but what they assume and what they explain, then the theories can be sort of connected one to another. Something which is assumed in one theory can become an object of scientific inquiry in another theory. In this way, I view the set of theories as a sort of loop, you know, where here we can have certain things assumed in one theory, but another theory from here will actually lead to an explanation to an account of these notions of things assumed. And now, construction of a particular theory, whereas this loop depicts the whole enterprise if you inquired for me, construction of one particular theory requires a loop cut. What does it mean? The loop cut separates explanation from action on loop in this particular theory. Now, here, take loop cut. Something which, let's say, abstractly is in this part of the loop, will be assumed in this particular theory, while something which falls into this part of the loop will be derived. is unique for this theory, but of course, some other theory may replace the loop code. And that way, something which was previously derived now will be assumed, and something which was previously assumed will be derived. So theories interconnect in this way. And in this tool, I'm going to look at this loop picture concerning physics and information. And there are, of course, two possible loop

10:00 in which physics is viewed as basic and informational notions are derived, and that there is another look-up in which information is viewed as a primitive notion, as a fundamental primitive notion within this theory, within this theory. And then physical theory, or physical theories derived. So in this talk, I'm going to be concerned with this look-up and will not be concerned with this look-up, whereas I fully acknowledge the importance of the necessity of looping and the other loop cuts. So the questions of what is the physical support of information and what information is about or such and such, fall for me into these other loop cuts and will not be analyzed in the circular-based tool, whereas I'm going to look at information, but I'm going to give you a certain primitive notion of information that will be among them, and the physical in this case, will be derived from this system of basic notions and certain postulates that I'm going to give. OK, so with this philosophical part clear, let me now move to the main part of the talk. I'm going to, as I said, select certain information theoretic principles. With the help of these principles and the formalism of quantum logic derive the elements of the formalism of quantum theory as we know it. Now to start with, in order to formulate the axioms of the principles, I need to show you the notions of the language which will will be used in these axioms. And here's this first step. Here are three fundamental notions that I take for primitive in this particular derivation in this series, other notions of system, information, and fact. Fact is seen here as an aspect of bringing of information, or actualizing, giving birth if you want, to information. Well, in formal representation, because we're quantum logic all over the gold hole during this talk, so there will be no on the court at least. These fundamental notions will be translated into a form of representation of those systems, again, physical systems.

12:30 Information will be seen as yes, no, or binary questions. And facts, or pretty much information, will be treated as answers to particularly yes and no questions at a certain time, or time will be until later. Now, in this system, in the quantological formulas, we have a certain set of gifts and questions that we're going to study and put a certain structure in it, and we have answers to these questions that give us facts or facts and brain-boughts information. Now, let me adjust with these terms, without yet going into any formulas, formulate the two main axioms that will play a crucial role in the following. And these are the information-theoretic axioms largely responsible for this theory becoming quantum theory. Here is the first axiom. There is a maximum amount of relevant information that can be extracted from a system. There is a maximum amount of relevant information that can be extracted from a system. And then the second axiom, it is always possible to acquire new information about the system. Now, certain things have to be said about these axioms. Well, first of all, not by any originality here, these axioms appear in the work of different people. Well, as such, they were formulated in, for example, in the paper by the Bradley that we've spoken about here today. The first axiom also very, very much resembles an axiom used by Brubner and Salander in their word. So, the idea of having a limit on information and then requiring some ongoing renewal of information is not new. Now, why is there no contradiction? It seems that the first axiom imposes a limit and the second axiom tells you you can always get new information. There is no contradiction, but there is this word relevant. And of course, I will need it probably to give you a formal sense of what I mean by relevant information, to give a definition and show how this definition formally works for the derivation of the formulas. So this term, relevant, is not viewed among the fundamental notions and yet needs to be defined. We'll do it in a few moments.

15:00 Now, what am I going to do with these axioms? I'm going to reconstruct the elements of quantum theory, and these elements are the Hilbert space, and I've clearly both proved that in this Hilbert spatial it's a quantum theory and not a classical theory. Now, I've taken the Bourne rule with the state space and the unitary dynamics in the form of the future evolution or I'm sure I've been related to life with certain supplementary assumptions. Now, the main part of this talk it seems, will be about the Hilbert's case, and all the other things will receive an information theory interpretation of their patient. Now, at this time, let me take a few moments, I'll leave this for a second, and tell you this. it seems that in the enterprise of interpreting quantum theory that people have been engaged into since late 1920s or 1930s, there have been a relatively new paradigm or approach of reconstruction, reconstruction which removes the interpretational task, not completely of course, but at least if one succeeds to reconstruct the formalism, then the question of mystery of the formalism, you know, this famous Hurling the Cat question, does not arise anymore. So the question, once one reconstructs the physical theory from certain physical assertions or assumptions, there is no mystery in the theory as such, there is only question Where do these assertions or assumptions come from? And this work, at least its first part, the Hilbert space, I think inserts into this relatively new in modern foundations of physics. Well, of course, people have been doing reconstruction back into the 1950s. But it seems that the idea of reconstructing physical formulas from certain simple assumptions can actually add the whole enterprise of interpreting quantum-mechanical formulas and quantum-mechanical here. So, where I'm going to achieve a result in reconstructing something is the devoted Hilbert's case, and the

17:30 rest of this will be interpreted and not reconstructed in the information theoretic language. Let's go directly to the Hilbert's case. Well, there are seven sketches of reconstruction And on these stages, we will look at the following one. So the first two stages basically just define a certain structure, and I'll explain which on this set of yes and no questions. Now there is a third stage, the definition of relevance and proof of modularity. This will be a crucial result for the following. Now with this result in hand, we will move to discuss how from this one gets the Hilbert what one needs to add to a normal point of a lattice to get a Hilgert's case. The first two stages before I move to relevance. So relevance of the third. The first two stages, basically imposed by definition, that a motivation here that otherwise it seems to make a lot of sense to speak, but it's no question if there is no structure, impose a lattice structure on yes-no questions and define a complementation, so the notion of negation in a way between questions. And these are generally defined as corresponding to our common sense notions of being able to ask a joint of two questions, you know, is A equal to one and B equal to zero, or being able to say that to a question A are responsive questions on A. Here's the third stage, railroads. Once we have a structure of a lattice on guess many questions, let's think for a minute of that. What can be characterized reminds us as a relevant question. To consider a sequence of two questions, A and B. And consider B such that at the level of ordinary understanding of what negation means, B entails a negation of A. So I'm trying to motivate the plural definition of a public evenism. B entails negation of A.

20:00 Now, if the observer asks A and obtains an answer to A, but then the same observer asks a genuine question B, so what does it mean ordinarily for us? It seems that if the observer asks a genuine question B, then he expects an answer which can be positive or negative to B. Now, a positive or an negative answer, both, are only possible if information previously obtained in A is no more relevant. Indeed, otherwise, the observer would be bound to always obtain the negative answer to B because he already has an answer to A. So if you have an answer to A and you still ask genuine new question B, then it means that you sort of forget, or render irrelevant, or leave aside your previous information that you've got, that one got in asking A. So, at the level of motivating the following definition, we will say that by asking B of this kind, almost, the observer renders A irrelevant. But of course, this notation here doesn't have any particular sense yet, so we need to write something instead of this in order to use this motivation to give a formal definition of prevalence. Here is a definition. In a lattice, question B is called irrelevant with respect to question A, if B means that the complement of a is not equal to zero. Let me illustrate this on this picture. What does it mean? So in the lattice, there are zero and block elements. Then there are some questions A and B, not necessarily, well, this picture is illicit to be less or equal than needed, not necessarily connected directly in the letters by one step. Some A and Bs. Let's have a orthogonal complement for A here, and this definition states that

22:30 if there is a question C here, not equal to zero, such that C is at the same time less equal than B and less or equal than the complement of A, the A curve, and then, and since C is not equal to zero, then B is called irrelevant with respect to A. Well, what's an intuitive sense of this definition? Intuitive sense of the definition is as follows. This means basically that there is a quote-unquote component, or we aren't speaking about any components in a strict sense, but there is something called what we can understand as a component common to B and to A curve, which is non-zero. So then by asking B, one sort of evokes some part of a curve, and thus renders a irrelevant. This is, of course, a very wrong intuitive explanation, and the forical definition remains fixed, shortly and strictly. Now, here's something which one needs to say straight away. This definition is trivial in Hilbert's latinx. The one is dealing with close-up spaces of Hilbert's space. This definition is trivial because it just means that we take the notion of self-theoretic inclusion and give it a unique. In Hilbert Gladys, question x, if it is less or equal than question y, is always relevant with respect to y, and all other questions are irrelevant with respect to y. Now, so in Hilbert-Lattice, this definition of relevance enhanced to no more than a reformulation of set theoretic inclusion. If we look at closed subspaces of the Hilbert space, this is true. Now, what is interesting about this definition is that it is non-trivial used to derive what in Hilbert-Lattice is assumed, because we're going to derive the structure of the Hilbert space. And here's a very simple example. Look at this primitive lattice of six elements. In this lattice, question B can be larger or equal than A. And at the same time, B, just

25:00 by simply changing the definition, B is relevant with respect to A, because B means A curve So here is a very simple kind of lattices where this definition is non-triggeral, unlike Hilbert lattices. And from these elements, one can, of course, build other lattices from these primitive things, construct longer and longer chains, build other lattices where the definition is non-triggeral. Now, with this definition, we need to say two more things which appear in axiom one. that there is a maximum amount of relevant information that one can extract from the system. Now, we now know what relevant means. We need to assume something about the phrase amount of information. What does that mean? And I will limit myself to two very seems intuitive and plausible assumptions that will suffice for derivation of both modularity. assumption about the amount of information. If relevance is not lost in a sequence of questions, so if one asks questions, A1, A2, A3, et cetera, one after another, if relevance is not lost at all, both of these are relevant with respect to other ones, then the amount of information grows. So if we don't, in other words, if we don't forget things, then the of information grows. That's all we need to assume. And the second assumption is more abstract, is that the lattice contains possible yes or no questions. So there are sufficiently many questions by assumption. Sufficiently many questions as to bring about any a priori allowed amounts of information. So if the theory tells you by asking, you can get such and such amounts of information, let's say, and this, then there will be a certain structure, a certain question in the lattice that, given your current information, will allow you

27:30 to get this allowed, if I override allowed of information. Now, let me show you a proof of quantum modularity. I'll sketch the proof without going into all formal details. A proof of quantum modularity of this lattice uses XM1, and as follows. By XM1, there exists a finite upper bound of relevant information called M. question A, consider a question A tilde, which brings about n bits of information in most of the sequence. Then, the first result, which one needs to prove, I'm not showing the details of this group, that A per mid A tilde equals zero. Now, second, one proves the dilemma that in all the complemented letters is often modular, if and only if from these two assumptions follows that A equals B. So that's enough technical results. Now, we're going to use this lemma, so we're going to assume these two things and show that A implies that this implies that A equals B. And if you look at this, by definition, this means that question B is relevant with respect to A. and also one shows that question A tilter from here is relevant with respect to B Finally, if one looks at a sequence of questions A, B, L, tilter then one shows that relevance is preserved here so nothing is forgotten as I said, and the amount of information grows so the amount of information is strictly more than any standard from this that A equals B. With this result, from accent one, which says that there is a maximum of relevant information, we've derived the structure of orthomodular lettuce from an orthocomplementic lettuce. Now, having an orthomodular lettuce, we need to make a step further and move to the Gilbert's base, not just Hilbert's case. In order to move to the Hilbert's case, I will use

30:00 Combox, the term Combox theorem, which is just one of many correlations of results obtained by Tron years ago, which allows one with certain assumptions that one needs to to justify, to take all the modular lattice here, assume something else about it, and then obtain, on a certain space, meaning a product which renders this space Hilbert's space. Now, let's just enlist the additional assumptions that I'm making. First, and importantly, there is an assumption about being lattice of space. So what I'm doing here is I'm saying, here's our long-mortible lattice of yes-no questions that we've built. Now, assume that this lattice is isomorphic to the lattice of some spaces, close-up spaces, of some vector space, vector space. And now on this vector space, we'll build the Hilbert space structure. The assumption that the research vector's case, of which the lattice is isomorphic to all the lattice, hope he has no questions, is a strong assumption. And there is no formal justification of it for me. This is about being willing to do something like continuous descriptions in physics. Once we want to go So from discrete mathematics of lattices to the description of body type spaces, we assume that this lattice isomorphic lattice of subspaces of a certain space. And on this space, we impose a Hilbert space structure with the help of this important assumption. Now let me take one minute to discuss this assumption. This theorem, well, if you agree with the formulation, a vector space in this theorem, in this particular formulation, is assumed to be interdimensional. There is an analog of the theorem, finite dimensions. So the problem is not with the dimension, the problem is with the numeric field on which the vector space is based. Not only we assume that there is a vector space of which the lattice is isomorphic to

32:30 yes to our letters, or multimodal letters, if you have no questions, but to also assume that this vector space is based on real complex numbers or quaternion. This assumption, while here, is taken as an axiom, called it axiom 7. And there are many alternatives to this, of course. Well, the main alternative is the Solaire's theorem. There are also reformulations by people who've been working that says Olin or Lawrence on, all these are formulations, except the layers, so all these last three reformulations, instead of postulating that the numeric field is such and such, postulated that a certain function from an algebraic structure is continuous into, let's say, one sphere of 0, 1 interval. So it's about to reformulate this assumption in an assumption of continuity of some function, some . Unlike these three assumptions of continuity of some artificially overplaying the structured function, Solaire shows that one can get this structure of the numeric field from a purely algebraic assumption of the existence of an infinite sequence of orthogonal and normal vectors. Now, this is an alternative correlation to just reading as an axiom that the field is real on the examples of fraternions. I'm not taking this hilarious way, because there is, it seems, no motivation of why a sequence, an infinite sequence of all the normal vectors would exist. And furthermore, importantly, this doesn't work in finite dimension. It only works in infinite dimension. And I would like to keep all this open to finite dimensional reconstruction. So that's why, in this approach, we simply assume that the numeric field is such. So with all these results in time, we move to the last step of this reconstruction, showing

35:00 that Hilbert's case can be reconstructed based on the assumption of, based on nexium 1, so the assumption of nexium 1 of relevant information, and certain mathematical assumptions with axiom 7, about a numerical figure. OK, so this is the main reconstruction result of this talk. Now, let me quickly go through other elements of the formalism of quantum theory and show you not how one can reconstruct them, because I don't know, but how one can interpret, still in this information-theoretic sense, how one can interpret the assumptions needed for getting the two other elements. Well, there is still quantumness of the Hilbert's case of the theory as quantum character. Well, in this work, I'm assuming that criteria that they say that the lattice must be non-discriminate and non-bullient. Altschep, for example, showed in his work that one is a bit more than that. This is certainly a question open to discussion. And from AXM, too, about the possibility to always get new information, one can show quite directly, but it is non-ruly. The question of whether one needs more structure for the theory to be quantum theory is open. and I'm certainly open to critique about this. Now, here's another element of quantum theory, which is not derived, but interpreted in this approach. It's about the appearance of state-space and vulnerable, where do these come from? And the approach uses Glissant's theorem, except that the condition of Glissant's theorem is formulated in an information observed, or information through a language, if you wish. One postulates, or takes an accent, that if information is obtained by an observer, then it is obtained independently of how the measurement was eventually conducted, or independent of the measurement context. So what does this mean? Speaking simply, this means that there is no meta-information in the system. So if the observer obtained information i about the system, then there is no further information j about how information i has

37:30 been brought about. So there is no meta information about the active measurement. The observer doesn't have anything else that he knows out of that information i that he obtained. And And this i can be any set of yes or no questions or any yes or no question. Would this assumption sort of pre-formulate the assumption of Gleason's VRAP in the language of information theoretic and Kavich? And this, of course, is open to discussion and I'd like very much to have questions about it if you think that it's interesting. Now, here's the last step of this, how one gets time dynamics. Again, time dynamics is all derived. It uses all the old theorems, Wigner's and Stone's theorem. But the initial assumption gets a very nice, it seems, reformulation in terms of information theoretic language. Because we only need to assume, to use Wigner's theorem, that time evolution, which is introduced as indexing of sets of yes-no questions by a same T variable. Now, if we assume that this T evolution commutes with relevance or with orthogonal complementation, we want to show that it's all the same. If time evolution commutes with relevance, so if question A is relevant with respect to B, then some time passes, then the image of A at this new time will soon be relevant with respect to the image of B at this new time. That reverse theorem is applicable, and with an assumption of continuity, some additional mathematical assumption, one can get even Jordan-Bear evolution with B.S. So, finally, in the few minutes, I think, which I just have a minute and spend a lot of my external amount, don't mean more about that. Let me also tell you about this approach, this information theoretic approach, it seems, allows one to construct a natural framework in which QVM description of measurements arises.

40:00 This is again an interpretation, but look at it, look at the observer, the observer, because there was this claim of universality, if you remember, the observer is a physical system as any. I certainly do not claim that the observer cannot be described as a physical system. Yes, the observer is a physical system as any. At the same time, the observer is a sort of informational agent, because these answers to yes, no questions, it's the observer who has them. So we can speak about a sort of two-fold role of the observer, at the same time, a physical system, and what I call an observer, or informational agent. So the one who gets this information, these answers to questions. Now, P-Observer, just as a physical system, must be described by physical theory. There is no reason to get an exception to that. While I-Observer, this informational adjunct is not described by physical. At least it's not described in this loop kind. So what we want to achieve, if you have here the observer, which is seen as I and P, and here you have the system that we're describing, S, what we want to achieve is just a description of s and information that the observer has about s. So p-observer is something to be factored out of the theory. One needs to take into account that there is a physical counterpart of the observer, that the observer is a physical system, but in the information theory approach, that's not interesting. We're interested in the information about s, not in some physical interaction. Now, if one does this, then by using Yasser-Paris's theorem about ancillars, one can treat P. This, of course, needs more time for it to be shown in the detail. One can treat this physical part of the observer, P

42:30 observer, as an ancillary system. And if the basic interaction, so yes, no questions, corresponds to projector measures, to projectors in the Hilbert space, yes, no questions. Then, by vectoring out P, and treating it as an ancilla, one can obtain a description just of the interaction between R and S without P, and this will be described by P of the N. So it seems that this idea, an information-theoretic approach, also allows one to sort of naturally accommodate the ATO POVM measurements in quantum theory, with something which one, of course, needs very much for information theory, for the theory of quantum information. So to finish this, let me again tell you that there were information-theoretic axioms and supplementary assumptions about the mathematical structure of the lattice and about the numeric field. And the information theoretic axioms are about the maximum amount of relevant information, the possibility to always require new information, and then the known meta-information axiom, which allows one to use the license theory. There are open questions still as for the meaning of these supplementary mathematical assumptions as for a numeric field axiom, function F, probability function, is not information-creatic interpreted in this approach. Then there is a question which arises in almost all approaches, information-creatic approaches, of where the dimension of the Hilbert space comes from. And finally, there is a question of super selection rules, because this builds the Hilbert space without and does not believe how to motivate them information to your right employee. Well, thank you. Two questions, please. Yes, you could emphasize for me once more, what is, so to say, new aspects of the information I mean, somehow, one could learn this from all the books

45:00 and one of them started from . So somehow, these structures are completely well known. So what is the new event here? So, yes, quantological formulism as such is not something that all the things except for the definition of prevalence of this particular result, but all the big things in quantologic are the same as before years ago. What is new, what is new, is that this formulas, quantological formulas, is just a means to And the idea is to take information-theoretic axioms, derive the formulas from simple assertions, simple constraints which one puts on the availability of information, the kind of information one may have. Now, these two axioms, as such, don't have anything but a lot of logic in them. These are just two axioms, one, two principles, two constraints, one puts on the kind of information one may have. Now, one uses quantological formulas in order to translate these into formal terms and show how the quantological structure of quantum theory arises. One can equally well, let's say, use algebraic formalism, take these two axioms, give them an algebraic formal representation and show how algebraic structure of quantum theory arises. So the formalism itself is a means to achieve an end, and the end is to get the elements of the formalism of quantum theory from the assumptions about information? Well, somehow it seems to me that you exercise the formal aspect of quantum theory. I mean, there are also concepts really detailed. But for instance, the staff of ,,

47:30 where he investigates the operation of . Let us, somehow, the so-called sequence of quantum logic of Stahel seems to be very close from the general philosophy point of view than what is here. So I cannot see what is the new element that brings something more. Well, again, the new element it seems is that this starts with information I certainly want and do not want to emphasize quantum logic here for two reasons. First, I do not want to emphasize quantum logic because it's not the main thing in this talk. The main thing is that one has these axioms, one obtains formalism quantum theory. Now, why I want to emphasize quantum logic is that this is a formal derivation. And that's the most important problem. This is a talk about reconstructing quantum theory, not about interpreting quantum theory. So, the idea of formal reconstruction has to be taken to, you know, to the formal end. That's why I want to emphasize that there is a mathematical stage there, it's not about interpretation only. That's why, yeah, of course I acknowledge that there is a body of origin in quantum logic, but that is within quantum logic. logic to interpret my information to write the axioms. My worry was vaguely similar, just what exact work the word information is doing in this approach. And also related in the statement of your first axiom, right, so you said that information was understood as yes-no questions. I don't know what it means to extract a yes-no I might know what it means to extract an answer to a yes or no question from the system. But then your action should not read there's a maximum amount of relevant information. It should read there's a maximum amount of fact in your terminology that can be extracted from the system. And also the word extracted is undefined. Well, yeah, these words extracted or acquired are the same as what I hope bring you about

50:00 in a notion of fact. So is the right way to read an axiom one roughly that there are a certain number of propositions that can consistently receive truth values at the same time? Is that the basic idea? Yeah. The idea that there is maximum... Oops. Here is this notion. Well, one of the fundamental notions was about bringing about information. So, of course, this means that there is maximum amount of relevant information that can be wrote about, or acquired or extracted is this fundamental thing about asking questions. This, if you remember, was translated as answers to questions. But there's a sort of suggestiveness about by the extracted that seems to bring in the role of the observer of your informational agent as an active agent. Whereas if you just say, there are a certain number of propositions that can consistently receive true values at the same time. That's just a formal fact about the system you're interested in. And, you know, arguably that what's actually going on in your axiom is expressed in the latter way more clearly without using the word information at all. Well, I'm not against, and, well, I wouldn't say active, but irrelevant, but what is important in all this, sorry, is this idea of relevance, of sequence of questions. between assets. And there is something which I would call, let's say, memory. You know, the observer asks the first question and gets an answer to this, then another question. So this answer to the first question remains relevant over time, there's time over there. And then the new question may translate it irrelevant or not translate it irrelevant. So, you know, this is a sort of consequential process of, you know, being on customs from GLADIS and saying, here's the question that I'm asking, now I have this and this part of information available, some of that holds on, some of that is irrelevant now, here's a new answer that I'm guessing. And the question of what information means in this, what's the meaning of the word information, is something that refers, for me, to this overlook, because information taken as a fundamental

52:30 Fundamental notions, of course, cannot be defined formally, it's a fundamental notion. It can be intuitively motivated. Yes, here I'm intuitively taking information, the sort of knowledge available to this observer who's asking questions. It's not quantum information of pubis or something else. But in order to be able to give a definition of what is information, not to take it as a fundamental notion, it seems that one needs to go to another look and explain information. If you hear information, one would be explained, and one would be motivated as, you know, why would we take this as a fundamental notion? But it's a primitive. But when you define it, when you say it's an answer to a question, yes to a question, so that doesn't do it. Yeah, but you know, if I, instead of quantum logic, I took algebraic formulas, that information would be a state over an algebra. So that's a formal translation, that's not really what defines it. I don't want to laugh. such a question. This is a first-order language in which you are created this axiomatic, or second-order language, or what it was to follow. Probably the two-third or nine percent of mathematics you are using here doesn't exist in first-order language. But, my serious question is that your axiomatic theory is formulated in a simple language without using these verbs, these intuitive, finding these verbs, having different meanings and so on.

55:00 to say digital system, yes, no experiment, information of the class, I mean, it's very disturbing that in this formalization of the COE you are using these kind of terms and then, yeah, Which is okay if you are just telling now what is going on in this program, but what is the primary program? Let me add, okay, so that was my original question. Actually at the end of your talk something shocked me. I mean, it seems that the whole program brought to the edge to the table, because you claim that the probability measure appearing in a recent theorem has nothing to do with information directly, so what's the reason? to follow this program. OK, so there are two questions. Let me start with the second one for the answer, I think, for this very quick. The only question about probabilities. There are two questions, it seems to me. There is a question of where does the fundamental structure of quantum theory come from in the foundations. And there is a question of where do the probabilities, which we build on this algebraic or logical structure come from. So where does the Gilbert space come from? Where do states come from? Now this answer is, this gives an answer to the question, this derivation gives an answer to the question of where does the structure come from? So where does the Gilbert space come from? I'm not claiming that it is impossible to give an answer to the question of probabilities. I'm saying at the moment how to interpret that.

57:30 is not treated in this work. I'm not saying it's impossible at all. Other people may have very good ideas. Now, something about axioms. What's the difference between these formulation in linguistic terms and formulation in formal terms? This whole exercise starts with an idea that quantum theory should be derived from a set of simple physical assertions in the same way discussed yesterday as Einstein said, there is a speed of one, which is maximum speed of all. From this, it derives special relativity. Now in the same way, there are simple physical assertions, in order to derive something, we And these simple physical assertions are translated into mathematical language, so to me, the idea that there is a fundamental physical principle or information theoretic, because we are speaking about it as information theoretic, there is a fundamental principle which is not horrible, which is almost at a phenomenal level, which restricts our phenomenal literature, this is a fundamental idea. And the fact that it translates into a formal language for us to be able to formally derive something is important. But it's only a translation of this into one I'm sorry, that was another sub-programming that wasn't about yours. Initial circle or rationality. Yeah, I'm building a theory. I'm building a theory. Yeah, I'm building a theory and saying we want to see how physics or how physical theory arises. It arises from a set of simple physical assertions. Well, here are two assertions, all the theory is about just formally developing these two assertions. But where do the assertions come from? Well, if you want to motivate the assertions, if there is a maximum amount of information, then take information as a derived notion, take something else as a primitive, and make another theory.

1:00:00 So... Good question. Well, it's very late, I'd like to talk a couple of facts, but let me just add one specific Can you show me that the transparency which is obtained in the theorem, the main theorem, I didn't quite understand the hypothesis of that theorem. The hypothesis which contains that the WP is ison-working to the soft spaces or the vector Yeah. Yeah. Now, what soft cases? Closed linear soft cases? Or not necessarily closed ones? What's the... Which one? Yeah. Closed. Closed. It's actually... We should assume that they are... Yeah. It's fine. Yeah. It's more than vector. Well, there... I said there was this step called step four here. step forward, which if you reduce it, is state-based structure. So this is the weakest part of this closed iteration I did. So this is a non-modular lattice. Now, this is assumed to be adiomorphic to the lattice of I'm saying there is a vector which is not a Hilbert space. We need to assume something about it. Well, if you want to look into the details of the paper, there is a discussion of what What is the minimal set of assumptions that we need in order to be able to get Hilbert's space? So there is an assumption that there is some space, sort of, as general as possible, of which the lattice is isomorphic. And this is the step four, which, of course, is responsible for the appearance of the Hilbert's space, as opposed to just a discrete structure. You're just one minor . What if you drop the assumption that WP is high school or UQB instead of all ? Yeah, that's what I'm doing, yeah. I'm saying take something smaller and then see.

1:02:30 But the theorem, well, the theorem I'm using works if it's about all substances. Now, there is axiom 2, which motivates how you can pass from the assumption that it is isomorphic to some, to the result that it is isomorphic to all, with the help of axiom But that's a separate story . Jeremy? I mean, I want to ask you something along the spirit of technology's question to try and get you to say that all of our information is really going to add to a kind of quantological traditional story. And there were two particular places where you could have said more about how the proof goes. One is in number three of the class theory of the proof of orthomodularity. You had B and A tilde. And at the third part, you said, roughly, in this context, relevance is transitive. You proved that from A and A tilde relevant. Right, so you prove that, A, children is relevant with respect to B. Now, it's not obvious how you could do that, and maybe you could say a bit about how you think her axioms help. The other example is through the consciousness, because they smell the reactions, like they're not compatible with classical physics. That's what everyone says. So more detail about how you prevent distributivity would be helpful. You see, you say now that I can prove it's not distributable, but I mean, it's great to see the landscape of how you're going to give information lots of the traditional quantum logical moves, but I want to be convinced that we get good mathematical level of theory by your notion of relevance or axiom one, seven, three, on distribution. Well, as for this part, well, I would certainly acknowledge that I skipped certain parts of the tool to have this work done. Here, this is a crucial thing which motivates that we can ask the sequence and get to a

1:05:00 large number. At this step, well, we're using this landline. So there are certain things which are granted to us by the conditions of the landline. Among these things is this. This means literally that B is relevant to respect A, because we did some work by showing that this limit is the same as the definition of post-mortularity. So the number of conditions granted to us actually render this significant. Yes, I agree. Yes. Now, A tilde is relative with respect to B, requires two things. We're taking this, but again, the vendor's text, which I don't have any transparency, but things we're using in this, this, and the idea that A tilde is such that it brings to that, again, maximum. So, from these two things, the definition of a tilde, this result, and this follows that there is, I can show you, a part of the article which covers this topic. And about quantumness, very quickly, yes, if I had time, I must have showed how So, the logism is non-distributive, but why the logism is non-distributive? But that proceeds by reasoning from the opposite use. You take it to be distributive, and you show that axiom 2 is not verified. You cannot get new information. The research sets up questions, but you can never get new information after that. So, yes, what you've said is very, very precise. The idea that these two axioms are non-classical is sort of in the air when you notice the axioms. And to show that it is non-classical, one needs to assume that it is classical for a moment, then show that the axioms don't stand. I think we are running out of time, so we should finish it here. and to be here again at 10.15. Thank you again.

1:07:30 Thank you. So, the subject in my talk is the logist, not the plural, arising from quantum computation. Along the spirit, I mean, along the philosophy, the Birkhoff philosophy of extracting, reading of algebraic structures from structures arising from the concrete framework of Hilbert's matrix. So this is the still. Of course, this does not mean that the logics arising from this, from quantum computation coincides or are comparable to vehicle from normal logic. I mean, I will call it orthodox quantum logic. By orthodox quantum logic, I will mean automodular quantum logic. Logic semantically characterized by the class, the variety of all automodular languages. That's the definition. So, quantum computation has suggested new forms of quantum logic, we call quantum computational logic. What is the basic semantic idea? Unlike the Buick-Oppon-Narmann's approach, where the meanings of the sentences are identified with the class of all possible closed-up spaces of a human space. This is the basic framework of the tradition approach. Here, meanings of sentences are here identified with quantum information quantities represented by spatial states of quantum systems. So this is the basic semantic idea. And what

1:10:00 is quantum information quantity? It may be a qubit system which corresponds to a pure space to the universe space. In other words, from an informal point of view, a maximum information of the observed about the system under investigation. Or a quantum information quantity, maybe a Q-mix is another word to denotate mixed state or statistical operatives or density of the reference. This represents, of course, a non-maximal information. It's a mixture, okay? So, let us start from qubit. What is a qubit? Let us find ourselves to the two-dimensional liberal space C2 with the canonical, the so-called computational basis, the autonormal basis, cat0, cat1, where cat0 and cat1 are defined as usual. I will call the canonical autonormal basis or computational basis. A kibit now is any unit vector of C2. So, is the superposition of the cat0 and cat1 So any vector satisfying is normalization as each. What does cap 0 and cap 1 represent? From an intuitive point of view, the basis element of cap 0 and cap 1 represent the classical bits, 0 and 1. So a qubit is a superposition of classical bits. I use the annotation X, the variable X rays over the set 0, 1, and cat X rays over the cat, cat 0, cat 1. As is well known from an intuitive point of view, a qubit sign can be regarded as an assertive piece of information about the physical system under investigation. Well, the answer no has a probability of the square modulus of red, C0, and the answer yes has a probability of the square modulus C1. This is the usual, the normal interpretation.

1:12:30 Now, the second fundamental ingredient in quantum mechanics to mold the compound physical system are tensor products. So, let us fix our bi-dimensional hyperspace C2. Then I will write this way to indicate the M-fold tensor product of C2 N times. So, I fix here H with index 1 is C2. H with index N is an M-fold tensor product of C2. Now we can define the notion of n-cubic system or register as it happens in classical information theory. In the case, what is a qubit system is just a quantum realization of a classical bit sequence of length n, of a classical register. In other words, an encubit system is n vector in the n-fold tensor product, in this n-fold tensor product. Instead of x1 at x1 times xn, we will write simply the sequence at x1 xn. Please note that xi raised over 0 and 1. 0 and 1. So, now, QNICS, the generalization, what is a QNICS is a density of the filter space of the M4 tensor product filter space. I will denote fixed and M4 tensor product of this kind. I will indicate by D, M4 tensor product C2, the set of all possible density operators of that space, of a tensor product, and by this simply, the union of all possible density operators for any possible space which has the form of a tensor product. So that's a sense of all possible. Of course, any q-register psi, any qubit, any q-register, can be represented by a mixture,

1:15:00 by a unidimensional projection, the projector P psi, the projection onto the one-dimensional close spanned by the vector psi, this is usual. So, we should write sign instead of the projection, which is associated with sign, so there is no difference. So now, logic is coming. What is a true register, a false register? A true register is a register, a vector, ending with one, whose last element is one. Dually, a false register is a register ending with zero. This is a total notion. In any space, this is the angle tensor problem of C2, two special projectors can be defined which represent the falsity, the false property, and the true property. What is the falsity property? The projection P0, whose range is the closed-up space spanned by the set of all false registers. Meaning, the set of all registers ending with zero. And the truth property is the projection whose range is the closed-up space spanned by the set of all true registers. Meaning, the register ending with one. Now, we can define what is called the probability, probability maybe is not the right term, fix a density operator's role by the means of the statistical algorithm of quantum mechanics. What is the probability of role of a statistical operator? It's just a trace of the composition of role with the truth property, with a projection which project onto the set of rules possible to register. From an intuitive point of view, P-Row represents the probability that this is a possible interpretation, I don't know, represents the probability that information stopped by the cubic's row, because the row stops in an information, non-maximal information,

1:17:30 is true because it's composed with a true projection. Of course, in the particular case where rho is a pure state, it's a qubit, or a q-register, in this case a qubit, what is the probability of the qubit? It's just the square modulus of C1, because this is the only register ending with 1. So far, how quantum information is stopped, now how quantum information evolves. So, quantum logical gates. What is a quantum logical gate, shortly gate, I will call gate, is a unitary operator that transform Q-register into Q-register. Of course, being unitary, gates represent characteristic reversible transformation from this tensile product into this tensile product. In this way, it's sufficient to define the action of our operator on the basis of the autonomal basis of the space. I concentrate on two different kinds of gates, which are interesting from a logical point of view. What I call semi-classical gates. What are semi-classical gates? Semi-classical gates are gates that transform classical registers, I mean sequence of 0,1, into sequence of 0,1 of classical gates. So the quantum behavior of these semi-classical gates emerge only when they apply, they apply to genuine superpositions. Otherwise, their behavior, if you restrict classical gates to the classical register may behave as purely classical connectives of operators. For example, what is the NOT? The NOT operators of C2, x is 0 to 1. It's just the k to 1 minus x. This is the usual definition. Of course, the NOT operators can

1:20:00 can be generalized to any tensor product. I recall n depends on the dimension of the hyperspace. This dimension of the hyperspace is 2 to n. This is just the action of this operation, just reverse the opposite of the last element. So not x1, xn is equal to the We had x1, xn, minus 1, and 1, minus xn, just the opposite of the last experiment. And now, an important, very important, gate, the so-called patent-toffoling. What is the patent-toffoling? We divide only for c2, times c2, times c2, okay? So, Toffoli applies to the case, x, y, and z are 0, 1, x, y, and z is just x, y, and the sum modulo 2 of the product of the last two, x, y, and z. Well, this is to be understood as the sum modulo 2, x, y, and z. I use these symbols because then I need the O plus symbol for other reasons. So, and of course, it can be defined also for hyperspace of dimension 2 to n2 to n1. It's very easy. So, what is the action of the Toffoli gate? The Toffoli gate is in the case of the silver space. It leaves unchanged the 64 spaces, you see? And then it acts on the last bit if the first two bits are one. It's a control, control, control, not, control, control, not. The first two elements are the control bit, and the last element is the target bit, which carries the result. So, on the basis of the Toffoli, a Toffoli gate, as a classical logic, we can define the connective, a reversible end, because as is well known, the classical end is not reversible because it's not invertible.

1:22:30 Okay? So, what is the end of the two vectors? The end of the two vectors is just the top of the gate of the tensor product of the two elements which represent the control bit, the input, and the third element affixed to cat zero. And cat zero carries over the result of the application of the end to the first two elements. So this is a reversible definition, a meta-linguistic definition. And all can be defined, of course, in terms of the modern logic. It's not so important. Now, genuine quantum gates. What is a genuine quantum gate? A genuine quantum gate is a gate that transforms classical registers into proper genuine superpositions. So, the behavior of this gaze is quantum in the sense that even if you apply to classical registers, it transforms classical registers into real quantum objects from the superposition. One of the most important for us logicians, but also for physicists, because it has nice, very nice physical implementations, for example, in the interferometry, in Mach-Zehndel, for example, Mach-Zehndel in Tecronica, it is a model, a very direct model, is the so-called square root of knots, the square root of the negation. What is the square root of the negation? Let us consider just the action, the square root in C2. It transforms the cat zero into the following superposition. Maximally uncertain because the probability of one is one-half and the probability of zero is one-half. So it's the maximum uncertain superposition. And dueling for the one. What is the, of course, the main, the basic property, the main property of the square root of knot. If applying twice the square root of knot you obtain knot. It justifies the name of the square root of knot. So, the square root of the square root of a vector is just knot, the knot.

1:25:00 So, what does it mean from a logical point of view? The square root of knot can be regarded as a tentative partial negation. You try to regain something. you don't succeed, you try again, you succeed. So, can be regarded as a tentative partial negation, a kind of hot negation, that transforms precise pieces of information, cap zero, classical bits, into maximally uncertain ones. And what is important, that this square root of knot can be realized physically, for example, by a Mach-Zender interferometer with two beam splitters and phase shift. So it has a real implementation in nature. So it's a connective, a new connective, coming from physics, from the interferometer in particular. What is the point? Why is so interesting the square root of nought? Because it has no classical no Boolean analog, but it has no fuzzy analog. In other words, what does it mean? There is no continuous function from the real interval into the real interval 0,1, such that, applying to y as a function f, you obtain the negation. This means that the square root of nought is typically quantum. So, even if you enlarge the set of opposite truth values to the 0,1, as in Ukasiewicz logic or in fuzzy logic, you cannot obtain a square root analog of of a classical fuzzy square root anode of this function. This is quite interesting. So no fuzzy countable, purely quantum. These are gates. Of course, you can define how many gates you want. There are unountably many, but we can find to some connectives, some gates which have some logical interest. Now, these gates, which are defined on qubits on Q registers, can be also defined in a quite trivial, natural way,

1:27:30 also on the mixtures, the qubits. So, I write not this way, the red way, instead of the blue way, to denote the gates when they are applied to density, to statistical curves, distinguish the two different. So, how can you define what is the knot of a mixture? It's just the composition of the knot previously defined, rho knot. And you have, this is, it transforms Q-mix, statistical operators, into statistical operators. Okay? And the square root, you have to be careful here, because the square root of knot is both complex composition of the square root of nought rho and the adjoint of the square root of nought. So, quantum gates defined on qubits and queregisters can be naturally extended in this way to gates whose range is the set of all possible cubics, statistical operators, density operators. So, now some problems, for example, which is the probability of the negation of a mixture, as expected, is 1 minus the probability of a mixture. Again, the square root of not, applied twice the square root of not, of rho, you obtain the negation of rho. This is, uh, okay. In other words, it's quite a mysterious property. I am not able to understand, but it's essential. The probability of the square root of nought, of a conjunction, is always fixed to one half. And this is, uh, I am not able to realize from a physical point of view, for example, the meaning of this, of this, but it's essential. Now, Q-mix is, you have Q-mix, fix a real number, lambda, in the interval 0,1, then you can define rho-lambda, right at the end, what is rho-lambda?

1:30:00 It's the mixture of the truth and the combination of the truth and the false property, see, and d1 by means of lambda and 1 minus lambda. So it's a complex combination of the two, of the false and the true properties. For example, if lambda is one half, is one half, you obtain the so-called semi-transparent effect, one half of the identity, of the completely unpolarized state. So, now, this part, which is quite new, concerns irreversible operations. So far, we consider only reversible operations, that is, unitary operators or generalizations of unitary operators to the case of density operators. Now, to find ourselves to density operators, we can define also irreversible operations on the set of all statistical operators. For example, which is interesting from a logical point of view, This is all called Lukasiewicz disjunction. In the case, you know, in the case of the classical case, Lukasiewicz disjunction of x and y, where x and y belongs to the interval 0, 1, is just the minimum between, this is called the truncated sum, because it truncates the sum to 1. So it is the usual sum of the two numbers if the sum is less or equal to 1, if x plus 1 is less or equal to 1, is 1 otherwise. So it's called the truncated sum, is a Lukashevich. So we can define the Lukashevich thumb, take two, take two density operators, tau and rho. What is the Lukashevich thumb of these two density operators? Is the density operators indexed by the Lukashevich thumb of the two probabilities of the element. As I mentioned, given any lambda, any real number in 0,1, you can define the convex combination based on lambda.

1:32:30 So, we have an Okashevich sum here, which is clearly reversible. And here we have some properties, but it's not so important. So, now we, step by step, we add structure to the density of corridors. So, we have the class of all density of corridors. We can define a binary regulation taking into account the statistical behavior of two statistical operators with respect also to the square root of naught. In other words, rho is less or equal than tau, if and only if, as expected, the probability of rho is less or equal than the probability of tau. And then you have a sort of...