Information-Processing, Quantum & Classical Theory / Relational QM and EPR

0:00 Michael Bittbog and the board of Lettree for making all this possible. It's a really calming and serene place to think and I think it improves the quality of your thinking. And also Anne-Marie Gruner-Schlenberger for having created this whole place. have this meeting. Please feel free to ask questions, especially questions of clarification during the presentation, because I'm going to try to present the formalism at a level that I hope everyone will be able to understand reasonable mathematical background, but not necessarily highly sophisticated background. So I am going to talk about this project that Jeff was also talking about, not necessarily committed so strongly to the idea of a principle theory or that we will find some relativized a priori principles or something which will become the foundation for information-based quantum mechanics, although maybe that will happen. But partly just sort of in a playful sense of, can we find some principles in terms of information processing what is possible and what's not possible within the theory? that will allow us to pick quantum mechanics out of a wider space of possible theories. And I'd really like to work in a space of theories that's quite wide and unrestricted, although maybe not the broadest you could think of, and that's what this convex operational formalism is. So, let's see. The main results I'm going to present are, I'm going to identify some features that are generic to all non-classical theories this ROD framework. Those are going to include no cloning and no broadcasting. I guess no broadcasting being viewed as the proper generalization of cloning to mixed states. The existence of information disturbance trade-offs, the inability to extract all the information about a set of states that are not classical, and also some results on non-generic features that might be a step towards characterizing quantum mechanics from among this whole space of non-classical theories.

2:30 And in particular, we know that teleportation exists only in theories whose state sets are isomorphic to their outcome sets. They're dual to them. Well, the outcome set is the dual in this framework, but the two are isomorphic, which is a fairly restrictive condition. I think in the abstract, I actually announced on some results on bit commitment to the effect that exponentially secure bit commitment is possible in any of these theories where you don't have entanglement. So I guess the time I wrote that abstract, I had just seen how this was going to go, and a problem arose with that approach. So that's still absolutely an open question, although I think that is probably true. Don't have a proof yet. So I probably won't discuss that too much. And the motivations for such a program, well, I should say that people have been talking about axiomatic attempts to derive quantum mechanics axiomatically in a broader framework for many, many years, probably starting with perhaps Mackey in the 50s, certainly Ludwig in the 60s, in a framework very similar to the convex framework I'm going to be working in. we're looking at this people in the quantum logic tradition once they've got to structures like orthomodular lattices and you know some substantial progress has been made in fact you could almost say there is a lattice based characterization of quantum mechanics Alexei has worked on getting a more operational more conceptual understanding of some of the assumptions that one makes that seem rather formal. And so I don't want to claim that what I'm working towards is going to be the ultimate or the only axiomatic derivation, or even the only conceptually illuminating or information based axiomatic derivation. I mean, there may just wind up being many alternative theorems that are characterized upon mechanics. It'll be illuminating for us to know all of these theorems, all of these different ways. some simple set of principles are going to be agreed upon as the sort of foundational ones, the real conceptual essence, I don't know in the end. So I should mention also Lucien has a paper, which

5:00 I urge you all to read, or several papers, where he derives quantum mechanics in a framework that is very similar to this operational framework I'm talking about. I mean, you're sort of naturally led to work in something that's closely related to this, when you try to think broadly about theories. So I urge people to look at Lucien's derivation. And I guess I'm just saying that alternative derivations are worth working toward, not necessarily to the exclusion of the quantum logic approach or of different ways of understanding quantum mechanics in terms of principles. So from within the quantum computation, community. The advent of quantum computation information seems to have revived interest in this program of axiomatic characterization or characterization by our principles. Some people are working in sort of an ad hoc framework or just make a toy theory and don't necessarily initially situate it in an overarching formalism for these theories. Other people do create an overarching formalism or work with some existing one. But quantum information, cryptography, computation, I would say, the researchers on that have just naturally become very interested in characterizing quantum mechanics in terms of its information processing power. Partly to understand the source of quantum information processing of quantum mechanics non-classical powers, like quantum key distribution, the speed-ups that seem to be possible from quantum computing for things like factoring large integers, Grover's quadratic speed-up of sort of black box search, also to understand the shared limitations of classical and quantum information processing. It's provable that you can't do bit commitment in either theory. Jeff, did you describe what bit commitment is in your talk? No, I missed the first bit. So what bit commitment is, is essentially somebody, let's call the two participants Alice and Bob. Alice wants to be able to effectively put a bit in a locked box and do something that assures Bob that she has indeed committed to a bit in this locked box, and now she's unable to

7:30 change this bit. Also, Bob should not be able to access this bit until Alice does something else which is like giving Bob the key to this box that allows Bob to open the box and see the bit. And it will have to be the bit that Alice put in. She was unable to change it during that time. So, of course, a locked box is not secure according to the laws of physics. You can get a big hammer and get the thing out. Something like quantum key distribution, on the other hand, eavesdropping on that is detectable according to the laws of physics. Again, you need certain background assumptions that Alice can do some things in her lab. I mean, no one sees her encoding the message because they just look over their shoulder and see the message. But subject to those background assumptions, the laws of physics assure that eavesdropping on the channel will be detected. Nothing like that for bit commitment. classically or quantum mechanically. And it's a very powerful cryptographic primitive that people do try to either implement that and then in a computational and secure way and use it for other protocols. And in fact, you can reduce a lot of things that people actually want to do. Secure computation, multi-party computation to bit commitment. Why can't we do this in either quantum or classical? Are there theories in which you can't? I think that in theories that don't have entanglement, you can. It's sort of an old idea. We'd like to understand this. No exponential speed-up of brute force, or so-called black-box search, which means that there's no sort of bone-headed way of doing NP-hard problems in polynomial time. If quantum computing can do NP-hard problems in polynomial time, which I personally doubt, It's going to have to be because there's something in common to the structure of all circuits that, well, of all NP-hard problems, it's particularly suited to exploitation by quantum mechanics. You can't just sort of take the device that verifies, that gives you a witness, let's say, to a solution,

10:00 which you can verify witness in polynomial time, you can't just take that as a black box and look at all possible solutions and just find it by running the verifier as a black box in some clever, superposed way. So, you know, there are even people like Scott Aronson who feel like, well, it just must be the case that reasonable physical theories don't allow us to do NP-hard problems in polynomial time because the world would just be so radically different. Thinking would be so radically different. We would be so much more mathematically powerful that in some sense, we shouldn't be able to do it. I don't know if that's Kantian or not, but people who do not necessarily read Kant think in these ways that might be considered a little bit Kantian. Okay. we'd like to understand some possibilities for new theory beyond the currently understood physical regimes that might be a motivation for some people and it's just fun to generalize things that's a big motivation to be honest okay, so here's the framework we've got a regular, and for me that means convex pointed and generating cone, we call it K It's in a vector space V, which is isomorphic to Rn, because I don't want to worry for now about infinite dimensional issues. And it has a distinguished base omega. And you can think of this base as distinguished by some linear functional U. It's the set of X in the cone on which U takes the value 1. So let's make a new page here. All right, so here's an example of a cone. Here's R2. And again, I don't know, does everybody, who here already knows what a regular cone is? So I think I'm doing okay to actually give an example. So here's a cone. A cone is just a set in Rn that's closed under multiplication by non-negative scalars and under addition.

12:30 That makes it convex, the addition. Some people don't do that in. Pointed just means, well, let's say if I made this R3 and I just ran this whole set down here so it looks like a wedge. That's a non-pointed cone because it contains a subspace. We don't want it to contain a subspace. It's just got this point. Oh, and here is the level set of u, u of x equals 1. And here is our base. And, of course, if we just take the set of all positive, all non-negative multiples of things in the base, we get the cone back. So the base is a compact, convex set. I should say one more thing about regular is that it has to be topologically closed. makes this base compact. Okay, so this is the base. And we think of the base as the set of normalized states of the theory. So if you think about Lucian's type of theory where, oh, we've got a whole list of all possible measurements we could make, let's say. And we empirically figure out what's the probability distribution for those possible measurements, and that's an enormous real vector space. But it may be the case that there's a lot of relations between these things. because several of these experiments are just different ways of measuring the same thing, or maybe they're coarse grainings of the outcomes of other ones effectively, and also measured in some different way. And if there are these relations between things, that means this set of probability distributions can span some subspace of possibly a much smaller subspace. In fact, in many cases, if you think of a continuum of possible experiments as in quantum mechanics, we imagine it would do. It'll be a finite dimensional subspace of an infinite dimensional space. But in any case, you project down. You can think of Lucian's compression, at least in this level one compression, perhaps, as projecting down to this subspace. And that's why we just want to work in this subspace and not worry about being in this much bigger space. And that's why another component of being a regular cone, is generating. The cone generates the vector space. Linearly generates the vector space. Ambient vector space. Okay. And then, of course, having this base is just like saying, well, we like to have all these unnormalized probability distributions in there. They're useful, but let's look at the normalized ones,

15:00 and those are going to be a compact common mix set. And if you like, you can have one vector space representation of this. for this space that's spanned by the states is to pick a fiducial measurement in Lucian's sense, which I think I may call an informationally complete measurement. Okay, what about, how do the measurements look in this formalism? Well, there's this important thing called the dual cone. So what's the dual cone? Well, it's the set of, it lives in the dual space, V star, V star is isomorphic to V it's just a set of functions from vectors to the reals or in general to the underlying field and since it's isomorphic to V you can actually use an isomorphism represent it in V and represent the valuation of these linear functionals as the Euclidean inner product so let's do that so the dual cone is the cone of functionals that are non-negative on every element of the primal cone so they're kind of a good candidate for representing the probabilities of, or representing the measurement outcomes, because when you combine a measurement outcome with a state that's supposed to tell you what the probability is you should get something positive so here's a right angle here's a right angle and everything in here is the dual cone in this case it's much wider if you like we could have represented the empirical theory that comes out of here can be represented by just making the dual cone and the primal cone actually lie on top of each other if you can do that the cone is called self-dual it's not just isomorphic to its dual it doesn't just have the same convex structure but there's actually a way of sticking them both in the same space right on top of each other by picking the right isomorphism between the dual space and the primal space

17:30 and so forth And this is just a two-dimensional, classical probability theory on two outcomes, in fact, is this theory. It's a simplex, as Jeff was saying a little earlier. Okay. All right. So, the set of things in this cone that represent allowable measurement outcomes is the so-called unit interval in the cone. 0u between this unit functional that gives unit probability it has unit probability in every state and effectively gives you no information and the 0 functional that's identical to 0 and when I say there's an interval cones are the same thing as ordered linear spaces essentially where we just define the order by saying well x is less than y if y minus x is in the cone and then this interval is just what you'd expect it's a set of things between this and this in that order. So in quantum mechanics, what's that? I don't think I even have a slide for it. Okay, well, I'll talk about quantum mechanics a little later, but let's do it now. Okay, in quantum mechanics, you know, the ambient vector space is the d-squared dimensional real vector space of Hermitian matrices, so the Hilbert space The cone is that of positive semi-definite hermitian matrices. K-star is the same thing. It's a self-dual cone. The interval of measurement outcomes is the set of positive semi-definite operators E that are less than the identity operator. Functional evaluation, well that's just A of omega. We use this inner product. A of omega is just trace A of omega. and then the normalized ones u is just the trace with the identity so the normalized ones are just the unit trace operating to the density matrices back to the general formalism so a measurement is a resolution of the unit into elements of this interval the sum of the measurement outcomes has to equal this unit and that guarantees that whatever measurement you make

20:00 Probability is going to be positive and add up to 1. So I won't go through why that is. It's extremely simple to verify. You know, I'll also, I'll evaluate measurement outcomes on states or states on measurement outcomes. It's the same thing since the double-dual, the space of functionals on functionals on the vector space can be canonically identified first vector space. So, you know, this is just like effectively omega of AI or AI of omega is just like an inner product and it doesn't matter the order because it's real, valued Okay, a measurement is informationally complete if its outcomes span the dual space and the probabilities for an informationally complete measurement determine the state omega completely I'm not sure if at least it's reducible or minimal they may be. A minimal informationally complete measurement is one for which AI are linearly independent, so they're a basis for the space of measurement outcomes. That's sort of the minimal number of outcomes you can have that will determine the state. Every system of our type contains a minimal informationally complete measurement. That's proved in this paper with John Barrett, Alex Wills, Matt Leifer Jeff referred to. And I should say the best way to find references on this, if you know the author's name, is go to www.arxiv.org go to the QuantPH archive and do a search for the author's name. Later I'll give some more precise references. Okay, dynamics in these general theories. Well, dynamics are going to be normalization, non-increasing positive mass, sometimes called operations in quantum theory. John Barrett likes to call them transformations, which is a better word, but his operations is pretty entrenched. They're linear. They're positive, so they take the cone to itself, look at a map from one cone to another one, they take the cone to the other cone.

22:30 They don't take things to function. They don't take a state to a functional that has negative values on some elements of your dual cone, because how would you interpret that as probabilities? It's normalization non-increasing, which means u of Lx is less than or equal to u of x. So that means we can't go to sub-normalized states. And the reason for doing that is it's useful to be able to think of these dynamics as conditional associated with some measurement outcome. And then the value of this unit functional on LX is associated with probability of the measurement outcome associated with L. But L also tells you what the state looks like afterwards. so it's not just an effect it's not just a element of dual cone but it's telling you what the conditional state is and then if you want to calculate conditional probabilities you have to normalize that state and then go from there right and so a sum of a bunch of these dynamics that adds up to a normalization preserving dynamics is sort of a procedure in quantum logic it's sometimes called an instrument it's kind of like a bunch of measurement outcomes complete measurement, and for each outcome, we say what happens to the state. So for any operation L, of course, the functional that takes x to u of Lx is an effect. It's an element of this unit interval in the dual code. Composite systems, how do we treat those? Well, we make something like a tensor product. our notion of composite system is going to say I can have all the product states any state of this system and this system I can prepare the independent state for which no matter what pair of measurements I make the outcome probabilities will just be the products of the probabilities on each side maybe that's too strong an assumption but that's what we have, at least for the moment. You can do other things. So the minimal tensor product is just the convex hull of these product states. So in quantum mechanics, that's what we call the untangled states, or the separable states, sometimes they're called. That's sort of the smallest thing that contains those products and is the kind of thing we want.

25:00 Why do we want these things to be convex for the state set and the outcome set? Well, it's because it's easy to think of experiments that you can do by flipping a coin to decide how to prepare this state or that state that will result in a preparation of the convex combination. And similarly, it's fairly easy to think of experiments where you flip a coin to do this experiment or that experiment, and some of the outcomes are naturally thought of as convex combinations of outcomes of the original experiments. So there's an operational justification for that. maybe this is a good point to sort of pause and say that this style of theory is it's operational so it's a theory for an agent who is looking at a system that's part of the world it's manifestly not a theory of how the whole world is going to look unless you just sort of imagine some agent outside the world who does experiments on it but if you want some, if you don't like everything that can do experiments on anything. You're going to have to have some way of combining a bunch of theories like this as perspectives within this structure, and you don't want to have some infinite regress of operational theories looking at... I mean, maybe that could be consistent, but probably in classical world it is, but probably in the quantum world that's not going to work. So these are sort of perspectival theories, This may not be the way to formulate your ultimate theory of the universe if you ever have one. Okay, to get back to composite systems, there are other things we can do. So the maximal tensor product, which is sort of the biggest convex set that could reasonably be considered a composite of these systems in the sense we have in mind, well, you take the duals, you do the minimal thing, and then you take the dual again. What does that mean? Well, this minimal tensor product, the duals, is just saying, let's look at the unentangled measurement outcomes, if you like, just the products of a measurement outcome here and a measurement outcome there and convex combinations of those. And let's say, we allow as a state any functional that's positive on all these product outcomes. And that's sort of the largest set of states that doesn't permit signaling. You just set up this formalism, and there'll be a well-defined marginal state, which is independent of what measurement is done on the other side, so there's no signaling.

27:30 And there's no real, so you might say well now we have these two candidates, we're going to define provisionally a tensor product of K1 and K2 to be any convex set that's between this minimal convex cone with all our nice properties, it's between this minimal one and this maximum one. and later one might want to add additional things like, well, if k1 and k2 are isomorphic when we swap them, we get the same thing or even if they're not isomorphic, we swap them we should effectively get the same structure up to some canonical transformation, so particularly if they are isomorphic you should get pretty much the same thing, except for relabeling no matter which way you do it. Okay, but for now, A tensor product is just anything, any convex set that's in, any cone that's in between these. Note that in this formalism, you automatically get that the span of the tensor product cone is the, it's a linear span, is the product vector space, the tensor product vector space. That means we have also determines the joint state, which is an important property that I think Lucien... I don't know if Lucien talked about it in his talk. He talked about it in the conference. So by local tomography, I just mean that if you do enough experiments locally that you find out and you have to do experiments simultaneously locally and then analyze the correlations between the outcomes, right? Alice and Bobbi, a bunch of experiments, they keep telling each other what what the measurement result was each time. They look at the probabilities of outcomes, the joint probabilities of the outcomes. They know the state. There's nothing more to be found out. So the idea is that composition of systems does not create any new, completely locally inaccessible degrees of freedom that appear. So, for instance, John Smolin of IBM came up with a so-called lockbox theory. to sort of show that there were physical theories where you could have bit commitment, and that he actually wanted to get other protocols in there as well. And if you look at Smolin's theory, it's sort of saying,

30:00 well, by fiat, we have totally secure lockboxes. And his lockboxes are essentially, you know, they're one-dimensional. I mean, he doesn't formulate it, Mr. Franklin, but you can think of them as one-dimensional cones that have got no information, except that when Alice sends her thing to Bob, And when you send, when Alice sends her thing to Bob, suddenly there is information. Okay. So that's how you lock up a bit. But there's an extra degree of freedom that's non-local in that theory. So our framework doesn't allow us to do this. I'll skip this category of business. Positive maps between systems. Right. Okay. One thing I should mention is that once you have the notion of composite systems, you need a notion not just of positivity, but complete positivity. And that means it can happen that a map that's positive on one system is not positive on the tensor product. if you naturally extend it to act on a subsystem, it can mess up the positivity of joint states, even though it never messes up the positivity of a marginal state. And so you need to require that it's positive on the joint state, and that's complete positivity. I think that'll just fall out of a category theoretic analysis that I don't have time to discuss. Simplicities, I think Jeff pretty much gave you the idea here. If you have R, N, a convex body is a simplex if it's the convex hull of K affinely independent points. So you could have at most N plus 1 of these points. It's a classical state space with K outcomes. That's the way to think of it. Simplices are the only compact convex sets in R, N, for which every point has a unique expression as a convex combination of extreme points. This fact that in quantum mechanics, there's not unique decomposability into extreme points is one thing that people have sometimes identified as one of the weird features of quantum mechanics. It's another of these weird features that's generic in non-classical theories. Okay, skip the proof, it's easy. Cloning. Right, so Jeff has pretty much defined cloning already.

32:30 I will skip the definition. And just say that we, so in 96 we approved a no-broadcasting theorem, so this is Chris Fuchs and Carl Kays, Richard Joseph, Ben Schumacher, and me, where we defined cloning of mixed states to be producing a tensor product state. So rho goes to rho tensor rho. Well, with that definition, even classical probability distributions cannot be cloned unless they have disjoint supports. So we thought, well, this is not the right generalization to mixed states to capture classicality because classical density matrices are sort of a commuting set of density matrices. So to capture classicality, we defined broadcasting. Broadcasting just says you need to take row to some state whose marginals are both row. And I think this is, if you look at what John said the other day about Cartesian, non-Cartesian categories, I think broadcasting is very, if you have broadcasting, it is Cartesian, let's put it that way. I'm not totally sure about the converse yet. Okay. So we proved Let's see. Well, here's the reference. Quantum state represented by density operators row 1 and row 2 are jointly broadcastable by a completely positive map if and only if they commute. So this does characterize classicality. You said genuos, which means the opposite of genuos. Is that what you're saying? Oh, I thought this was funny. Goran Lindblad proved it in a C-star algebraic way, which is nicer than the original way we did it, which was just incredible, blasting through a bunch of algebra. And he said that we had done an ingenuous proof of this. I like to think he meant ingenious, but I thought it was funny that he meant ingenious because in a certain way it was. Didn't use C-star-Apple? Algebra is even binary-dimensional. So, I mean, it was ingenious, but it was also, you know, ingenious. So, okay, here's our generalization of this theorem to this broad, convex framework. Okay, and here's the reference. For all choices of tensor product, K, between the minimum and the maximum inclusive, a set of normalized states in the state set of this tensor product, no, sorry, in the state set omega of K, is broadcastable by a positive

35:00 map if and only if it's contained in a simplex, which I call delta, within the state set whose vertices are jointly distinguishable. So jointly distinguishable means there's one measurement I can make such that the measurement outcomes tell me with certainty what state I have. so these vertices are jointly distinguishable sounds kind of classical and indeed they span a simplex you can prove easily that there can't be more jointly distinguishable states than the dimension so in some sense in this very broad setting it's still the case that broadcasting is associated with classical subsets of states. Well, not only are they contained in a simplex, but if you look at any map L, and you say, what is the set of states that it broadcasts? Well, usually that's going to be empty. But in general, it is such a simplex. Although, as I say, it might be the empty one. So it's a slightly stronger result. Okay, so no broadcasting, also generically non-classical. Non-disturbance. I think I'll actually skip the statement of non-disturbance and just state the theorem let L be a map that's non-disturbing on a cone C then L is the sum of a bunch of maps that are proportional to the identity map and the cone decomposes into irreducible cones it has a unique decomposition into irreducible c sub i and the way to think about this for instance in quantum theory with super selection rules so all the density matrices are block diagonal in some block structure i labels the blocks it's saying I can divide up this space into subspaces such that the extremal points all lie in every extremal ray of the panel, let's say, lies in one of the subspaces, C sub i. And so figuring out what the value of i is, that's classical information. There's some intrinsically classical information in this theory. There's some intrinsically non-classical information. The intrinsically non-classical is about where you are in the irreducible component. The classical information is which irreducible component. Only classical information can be extracted without disturbance.

37:30 Again, a very generic result about something that had been thought of as uniquely quantum. Let's give at least one uniquely quantum result. Here we go. Teleportation. Teleportation in general. So quantum teleportation is actually, in my opinion, not peculiarly quantum, because classical teleportation is easy. What is teleportation? Alice is supposed to pass on an unknown state to Bob using only perhaps a fixed joint state and a classical channel. Now, you might add the requirement that classical information contained no information about the state because that is in fact the case in quantum mechanics. It has to be the case because if it contained information about the state, The state would have been disturbed, and you couldn't pass on the state undisturbed if you have to send information about it, classical information about it. So classical teleportation exists. You just either pass the state on on a classical channel, or if you require the channel not to tell you what state it is, you make a one-time pad. That's your perfectly correlated state. It plays the role of the bell state in teleportation. and you use the one-time pad to encode the identity of the state and send it on. Alice, I mean, Bob can then decode and know what the state is. Okay, so the surprising thing about quantum teleportation is that you can actually still do it, even though you have information disturbance trade-off. So a deterministic teleportation scheme in general is a state, omega, on systems B and C, and a measurement, A sub i, on the joint systems A and B and a set of normalization-preserving positive maps, tau sub i, from C to C. So Bob has C and Alice has AB. Sorry about having Alice have B, but Bob not have B. Okay, so we have this setup. And for all states, sigma a, it has to be the case that if Alice does the measurement, whose outcomes are a sub i, sends the value of the outcome, i, to Bob, Bob does his correction operation, tau i, c.

40:00 and we do this of course with our fixed joint state and any sigma a and let's say we need some way of saying did the state get across or not so that's an isomorphism eta from c to a that just says what does same state mean so we compare the states we get the original state out we do this isomorphism let's compare states we get the original state out So, in quantum mechanics, it's the Bell measurement. These are some poly operations that fix the thing up on the other side once the measurement outcome is communicated. Okay, so that's a deterministic one. For a conclusive probabilistic one, there just has to be one measurement outcome, such that this works with positive probability, gamma. And maybe gamma, probably I should let gamma depend on the state, although it will turn out in an irreducible cone, it can't depend on the state. So a conclusive probabilistic teleportation scheme just has to work conditional on one of those measurement outcomes with positive probability no matter what the state is. Okay, so the theorem is that if you make any tensor product, say A tensor B and then we tensor in C, Whatever tensor product of those ones that I showed you, you use, it supports a conclusive probabilistic teleportation scheme if and only if A tensor B star contains an effect whose corresponding map is an isomorphism. What's the map corresponding to a state, an effect? Well, a measurement outcome, a bipartite measurement outcome, allows you to stick in a state here, a state there, So if I can view it as a map from states here to effects here, because an effect here takes a state there to a probability. So do this thing. There's a similar thing you can do for states. States map from effects to states, which is sort of like a conditional state. So A tensor B star has to have an effect whose corresponding map is an isomorphism. state whose corresponding graph is an isomorphism. And A itself has to be isomorphic to its dual. And this is under the assumption that all these systems are isomorphic. It gets a little,

42:30 it's just slightly more complicated if you don't let them be isomorphic. So there is something non-trivial you can find out about the structure of these theories beyond that they're non-classical by some kind of conceptual thing that you impose about what protocols you can do. If you say you can do teleprication, the state has to be, the system has to be isomorphic its dual. It's a very strong condition, actually. So it's a step along the way of a complete principle-based characterization of quantum counts. Thanks. Thanks, Howard. Mr. Dermot? I just have a question. At the beginning, you just were talking about finite-dimensional vector spaces. Yes. Does this theory be extended to the infinite-dimensional case? basic framework has definitely been extended to the infinite dimensional states. We haven't proved anything in that context yet, although we're sort of working on it. Some of the things like dual or dual... Yeah, they won't be isomorphic. But you sort of... There are operational things you impose about, well, I would only like functionals. You want certain continuity in this functional evaluation. And I think... So I don't know the details, but I think it tends to pick So probably, it picks out the dual with additional topological conditions. That's sort of the nice thing for functional analysis anyway. So I think it works reasonably nicely. I don't understand the details. Chris. In the general definition of the teleportation, have you looked at whether these, in more general theories, have violation of the benefit qualities in a way, I don't know if you can understand what's going on in quantum teleportation. So John Barrett said, look, here's a way of thinking about what's happening in teleportation, where we can see quantum teleportation, which shows why they seem fairly important. Is there anything, I hope, anything analogous in the general? Well, so no, I haven't. And I wasn't actually very familiar with John's work there. I mean, you should be able to, you can violate it worse in these theories, right? This includes what Pesquibro-Lake boxes. Yeah, but we're teleporting via this, if that's the question. Yeah, I don't know if those things are equivalent. I would tend to doubt it, but I really don't. Let me generalize on this one. Well, no, you can teleport classically, so that's my answer.

45:00 Yeah, but teleporting classically, one of the differences between the classically and the quantum case is that constantly you are via this development, while also classically you are not. So this is, quantum teleportation is really important. If there is something, and to generalize this, if there is something you clearly lose, when you generalize the framework, which, you know, we're used to thinking as being reasonable. What do you mean by something you lose? Well, things can be generalized in the interpretation. Well, currently, things like, you know, there is a problem of seeing how these continue to go through the generalization. Are there some features of quantum theory that can clearly not be taken to this Gerald Frey work. Something which is their, you know, I don't know, usually dynamics. Well, I mean, even, you know, you can't do teleportation in all these theories for one thing, right? yeah, I mean, the set of reversible dynamics can be very restricted. To see what we gain and what we lose. Well, you don't lose anything because everything you already had is still inside you. So that's it. But it's, you know, if you make, if you go to a very arbitrary theory, So a nice property, for instance, is homogeneity of a cone. But it's also a pretty strong property. Quantum mechanics has it. It's classical probability has it. It says that the automorphism group of the cone, so the set of things that takes the cone onto itself and takes extremal rays to extremal rays, and it's invertible, reversible kind of dynamics. Well, there you're saying that acts transitively on the interior. It's homogeneous. That's a lot of symmetry. of these things can have a very restricted onomorphism group, and that's when you said unitary dynamics there's a lot less there are some things that are like unitary dynamics in that sense, but there's a lot less of them and you can do a lot less with it sorry it seems to me that this sort of program of deriving information-directed constraints um is one should ask this question so that there are kind of high level information constraints and low level so the high level information that might be the possibility of a low level one might be the um

47:30 for example the possibility of the uh of the state so it seems to me that um one should always of low-level constraints over high-level constraints? So I think, as I was sort of suggesting at the beginning, I mean, I think one should let a thousand flowers believe me. And I think it's interesting to understand both. Ultimately, I think the most satisfying thing would be to understand a whole bunch of ways of characterizing things in terms of what you're calling high-level, relatively specific protocols in terms of a few basic principles, like local tomography, which you can see local tomography is defeating, I think, Smolin's big commitment theories. So you've got this lead to that. Yeah, it's more satisfying to get from the low-level things to the higher-level things, but the low-level things are also more satisfying once you get high-level constraints from them. So I'm wondering about what other features are non-generic. comes to mind, I just wanted to verify that in quantum mechanics for any pure state, there is a measurement which affirms that you have that state with probability one and it doesn't do it with probability one for any other state. And I'm just wondering if the duality, if the fact that the output space should be dual to the state space, will that ensure that this feature of death nature? Absolutely not. What you just said is, I mean, it's been used in various characterizations or attempted characterizations for a long time. There's a paper by Rocky, Communications and Math Physics, that I think uses essentially that. Alphstein and Schultz have this book, Geometry of Operator Algebras, something, it may even be exactly that, but things like that or generalizations of that are used as So these things are called projective faces if there is some functional that identifies the fact that you're in this space of the cone. And it's even stronger to say that there's only one functional that identifies this space. And those are pretty powerful, very mathematically, geometrically natural. They have some operational content, but it would be nice to tease out more what they mean in terms of information processing protocols.

50:00 so I think they're important axioms and no, this is a much more general framework than that this book of Alves Saint and Schultz shows you in an infinite dimensional setting something that was proved earlier by Vindberg which is that all self-dual homogeneous cones are basically quantum mechanics over R, C the quaternions or they're cones whose state space is a ball which are somehow things in some spin algebra that I don't understand very well. And then there's one weird case that's like three-by-three Hermitian matrices over the other side. So that's like a stress base of Jordan algebras. Yeah, I mean, these things are states bases of Jordan algebras. That's the result. And the Aufstein-Schultz thing is in the infinite-dimensional case, they're states bases of JB algebras, or maybe JBW. I forget which. There's a theorem for both of those. I'll have a question. Yeah? I didn't quite understand your response to the idea that you could sort of swap. about self-dual, not the whole framework. Yeah, but self-dual. Oh, you were asking about self-dual. Yeah, so at the end it seemed to me you were saying that if you do have self-duality then you have this. If they're self-dual and homogeneous, you have that. What is homogeneity? Homogeneity means the automorphism group acts transitively on the interior. So it's sort of like they're kind of very smooth. I mean, you can take any unnormalized state that's sort of non-singular to any other that preserves the entire cone. And, you know, something like a unitarian. So the combination of those... So the answer is, I think, yes, if it's self-dual, there might be a relation. I don't know if it's strong enough for that axiom to hold. It might be that there exists a way of picking this out, a face, but it's not unique. Or a state, but it's not unique. but I believe that does hold in these homogeneous counts, I'd have to double check homogeneous self-dual thanks any more questions? I've got to write homogeneous self-dual homogeneity is a kind of feature that we would normally identify but continuous transformations between pure states well except that simplices have it too right Right, okay, so it's less general.

52:30 Simplices have it too. Okay, so it's less general. It will not allow us to distinguish a sphere and a ball of states from a symbol. Homogeneity, no. No, okay. Yeah, all right. Although there are no more questions, we have a cotton break. It's a pleasure to be here. I want to thank warmly because really, I didn't expect this at all. Maybe I should blame them because this being my first session eating, my standards will be far too high, I'm afraid. So let's go. After a very brief introduction, I'll present the relational interpretation of mechanics. And then, after some elaboration on it, I will confront it to the famous and paradigmatic EPR scenario. And then I will try and open some doors to the philosophical implications of this view. My introduction, I want to say just a few words. What I call physical objective. So within the context of scientific realism, the notion of an object has changed during the history of physics. So let me ask, what primary quality can I do to an object? This, in the history of physics, within those were the state of motion. But then Galileo came and said, no, the state of motion is not the relational property of an object, the body relative to a reference frame. What remained somehow was its weight. and said, no, weight is not a primary intrinsic quality, something which is in a binary relation between two bodies interacting through the gravitational force. Well, maybe spatial extension shape volume remains as intrinsic qualities of bodies. But then, no, Einstein kept up and explains through the special theory of relativity that even those are friend-dependent and relative. So I'm defending here twofold thesis on scientific realism. The first point I'm making is historical continuity. Objecthood, as I put it, has been continuously desubstantialized, relativized. So there's continuity in the history of physical objecthood. And the second point is this long-lasting

55:00 reluctance, which is, I think, typically human. The old picture remains for a long time. It's very hard to get used to the new picture. This is just a simple introduction. So let me go to the problem with quantum mechanics and listen to the way Chris Fuchs puts it. I like it, it's very true. Go to the end of the meeting and it's like, you will find all the revisions with all the priests including the entire world, the tokens, the consistent historians, the transactionalists, the spontaneous class leaders, the ideal selectionists, the contextual objectivists, the outright, the practice, and many more beyond that. They all declare to see the light, the ultimate light. Each tells us that if we will accept the solution as a failure, then we too will see the light. But there has been something wrong with this. There has to be something wrong with this. If any of these priests had truly shone the light, then it would not be this year after your conference. So that is being screwed up. If we don't care about car mechanics, Then he picks up the community to ask why these meetings are happening, and then try to find a way to accept it. So, I'll follow Ropelli on this and try and expand on his hints for a solution. He procloses that Viannese with quantum mechanics may derive from the use of a concept, which is inappropriate to describe the physical world at the quantum level. So he clearly takes his cues from special relativity where the problem was lying in the absolute time concept. Maybe something else is lying in quantum mechanics. We need to find out. Okay, let me go ahead and introduce Revelli's interpretation more specifically. So let's consider the following situation. He calls it the observed sound. A quantum system, S, is being measured by some observe of O. And both of them are observed by a third observer, a second observer of E. I think he drew this picture. Let's consider the most simple, the simplest physical system I can imagine, mainly equipped with two possible states. And let's consider an arbitrary quantum state of S. Let's say some alpha O and beta S. denotes the system which is considered.

57:30 So, there are two perspectives on this measurement process. From O's viewpoints, when measured, the state of S, this is tended for mechanics and not adding anything. Collapse is to one of the two possible outcomes and the probabilities of given by the formal rule. But from P's viewpoints, the S plus O system is isolated and can be described by the Schrodinger equation, and so the measurement process is nothing like a specific interaction between S and O. So by the linearity of the Schrodinger equation, the compound state of S and O, psi S and init, which denotes the initial state of O, of the pointer value of O, would approach to this superposition. So, from O's viewpoint, after the measurement, the state of S has collapsed and the value associated to this observable has got an actual value. But this is not the case from P's viewpoint. So let me ask the question, what is the actual state of S after the measurement? This is the certain problem. If you have any clue, it's time, so I go with it. Let me introduce two criticisms on this presentation of the observer's picture. An Einsteinian criticism would amount to say that if QN does not predicate the state of S, which seems to be the case, I have presented only standard quantum mechanics, then quantum mechanics is incomplete. and from this line of thought we go to the whole hidden variable program and we know where it is but on the contrary Rodelio assumes that quantum mechanics provides a complete and self-consistent scheme of description of the physical world appropriate to our present level of experimental observations I could justify this argument with the no-go theorems which you know or by the experimental success and the absence of new data contradicting with quantum mechanics. Anyway, let's take this as a hypothesis and see where it leads us. Another criticism would be more buoyant in style. Some hypothetical boards say, OK, the confusion comes from the application of quantum mechanics to the observer. Oh, we know that there's a cut between the system and the observer in quantum mechanics. This cut is sometimes called the Heisenberg cut.

1:00:00 It's canonical, namely, between the macroscopic environment, which includes the apparatus and the microscopic system, the cut is clearly well-defined. It's between the micro and the macro. This was the boring perspective on something. But this is hardly terrible nowadays because experimentalists are not able to do quantum measurements on meso-systems. And then it's clear that the boundary between what's micro and what's macro clearly vanishes in time. So let's make the following hypothesis. All systems are equivalent. This is a form of modeling. Nothing distinguishes a priori macrosystem from quantum systems. If the observer O can give a quantum description of the system S, then it's also legitimate for an observer P to give a quantum description of the system formed by O. So this amounts to reject the canonical distinction between the quantum world and the classical world. Of course, Bohr was very much aware of the fact that the so-called heisogicab, this boundary between what's quantum and what's classical, is freely movable, but he still kept this idea that it should be somehow fixed in a given experiment. we want to contradict. So, standard polymechanics suggest that different observers can give different descriptions of the same sequence of events, but following Reveille, we assume that this description is complete, there's nothing more to it, and there's no difference upper array between observers. So there's clearly a tension in these three statements unless you accept this tentative resolution, namely that any column description should be understood as relative to a particular observer. I can reformulate this by saying that any column state is always implicitly indexed by a particular system of observer claims. It's never the state of a system, it's the state of a system