Reinhard Werner: "Systems and subsystems"

okay so everybody and thanks to the authors for this invitation I will be doing various little things today and partly talking about my favorite category to do in one mechanics which is not exactly the same as maybe everybody else’s going also to your infinite systems and things like that so I’m nothing the computer scientists are not bound and find that I mentioned something so you’ll see some of these things okay so first the basic approach that I would propose of course as always you want to have the connection between them a side of the theory and and the physics and the mathematical side here is supposed to be somewhere category theory is what some some some complete category that is supposed to map the things that we want to do in the corporate world and of course it’s very natural to having us to try to make the connection in such a way that the objects of the category some physical objects that we’re dealing with now that’s actually not what the what that doesn’t really make a lot of sense in the end so it’s rather than type of object that you want to talk about so with things like hella fun that’s one type of systems and so electrons or let’s say the polarization degree of freedom of photon things like that things like that would be would correspond to objects and scat now the morphisms would be then operations we wanted to form on these systems there are many things that would want to do for example one possible thing is leave the system alone for 30 seconds that would be an instance at the time evolution operation and of course their various other things that we want to do also changing the nature of the system also changing well may be obtained in some classical information on the way making a measurement that is and what we would call all these possible states of knowledge that that developed during that a big experiment systems or types of messages if you want us so so one object in different the category would become some kind of message and they get transformed and operated on his way so one one aspect on the classical on the bottom side is probabilities can you see down there I have to apologize I didn’t know that it wasn’t really usable so I hope that’s okay so so certainly as you know in quantum mechanics every experiment is a statistical ground at the end you always looking at certain things happening and that would be related to to linear structure that also appears in the category so there’s one special object that you would always have which is having no object or no system or whatever and that plays a role in transformations that are in making the typical process of preparation of the quantum system into a transformation between having nothing and having something so on the other on the other side you would have measurements and they take whatever your sister is and in afterwards you have nothing and then there is a probability evaluation so that if you first prepare something and then follow that by a measurement you get you get a probability to get a number so the the set of morphisms from this no object to no object will be identified with the unit interval that set of possible values for probability for all these operations there is I mean you it’s a bit strange to say that after a measurement II have nothing at least yet obtained some kind of information and this is a corporate by saying that all these operations may fail and they tell you when they fail there is a flag successful operation so for a yes/no measurement the successful outcome would be what you’re looking so then you get a ratio of success and that is this not the overall process so so for every object there is one

special measurement which just checks whether the system is still around and that is that I note that good at number one as it will be the case in the typical system that we look at that’s the identity operator and the state would then just be a particular preparation that always succeeds I’m soda and there could be a normalization factor that makes it allowing states that suck normalized so they can be normalized to something less than 1 the normalization factor would be just this evaluation the composition of with is one office so that gives you the probabilities they we don’t really care what what the particular how the particular operation is described we are only interested in those aspects that actually have something to do with getting probabilities in the end so we will actually identify operations that always give the same probabilities or ever we do so so here is the equivalence relation if there’s some transformation between ace a type systems and B type systems and for every preparation of an a type system and a measurement after subsequent measure after the transformation on the B type system if you always get the same probabilities then we say these two operations that we identify these two morphisms so so that’s a typical operation also known from ordinary pollen Kaiser a state is really equivalence class of preparation not a single one so we just that’s the same kind of identification that we’re doing so this means that these morphism sets can be embedded into vector spaces and so so one way to construct the vector spaces to take all formal linear combinations of the offices that you’re given somehow and identify linear combination of defined linear combinations to be 0 that have zero probability there’s a linearity on the side of the probabilities numbers there so that first prepare something then transform something that measures something that I always just get it done so this is an equation between numbers the same sense I can make this identification and okay so so we can actually add and scalar multiply these more feelings but of course that doesn’t usually doesn’t have to be amazing and the composition are to offer operation automatically becomes bilinear then become we will often will often look at the current generated value with actual physical morphisms those are the positive elements in this vector spaces so we naturally have ordered vector spaces and and multiplying so so this space more physics from nothing to something those are the preparations from something for nothing of emotions and just composition of morphism gives you the probability of devaluation which is then the canonical a canonical pairing between this preparation space and the measurements mess and then you have all the mathematics that goes around this like having week two apologies either of the two spaces by saying that something converges when all the probabilities converge with the fixed other element you get usual trappings for all of vector spaces like baseball or you they’re not going to go to go into that too much okay so here’s one operation that is very common and that is very important if you want to also handle like classical information he generated on the side is what I call the or operation so what type of system or message would be that you can either have a message of type a or a message of type B and it’s of course necessary to make sense of that operationally to know what you have so this is there’s a piece of classical information actually that tells you what kind of system you have and conditional on that you may operate on the system so so here’s what how you can express that in the language you actually have something like a byproduct

so there is an operation that tells you that so okay from from a planet collection of objects you can make this direct some object and so the type of information that this describes is well first you are giving this value X so you know in what component you are and then you know what kind of how to operate on that if you know how to operate on ax so there is this injection map that tells you that if you know that you have an exercise object you can this is a special instance of having some one of those kinds and of course we didn’t have that with certainty and on the other hand you can check what object you have declare failure if you don’t have an X and otherwise you take the system that’s the projection operation and then from that exportation these these equations would be obvious you postulate that which means that on the media level you have something like a bike are so the imbecile i just gave you ok so so actually the category that we starting on which is the category of physical operations so the arrows there will be something that you can do not those linearly extended things there you only have a co-product and not this my product because there is there’s some missing information there so it’ll cook this this refers to the situation when you go from a single system to a direct some of things that is not automatically well defined because you don’t know with probably with with which probability you end up in which of the branches that’s an operation that you could define and that will let later be automatically coming out using a tensor products but at this level it doesn’t visit it doesn’t there’s there’s insufficient information there and that that is the one way this shows up is that this normalization condition is usually not service so that you would do it if you just you can formally build this this direct sum morphism in that situation as well for a product but that might turn off given probability is not in the one and that this is the reason for that is that some other is some missing information in this branching process you have provided and in the other direction is okay and also operating in parallel on different summons with the same labels is also an operation that we postulate on the hang on the morphism themselves okay so rather than having time a system or the type pieces you might have both and then it’s also clear that your operations will be how to operate on something like that you can do things in parallel and the of course they the notation for that would be a tensor product and would have a reliable category you would is from the front of interpretation it’s clear that you would have some that would have distributivity in the usual way and this is no system thing acts as an identity for the for the tensor product then you could use this you need this tensor product information to to define forgetting of information and but if you have that you can easily show that all these more for some sets will be calm we could have pasta dated at an earlier stage so it comes out ok so then from this kind of general scheme oh sorry okay so so the it’s clear that the in the in the quantum model that we’re used to since we have a tensor product operation from the outset these woof isn’t automatically completely positive terms you can’t answer them in their sylvie morphisms and so but otherwise well there’s a lot missing here and get this the scheme is the master general so actually here is a giant axiomatically well I’m not going to fill but i understand that morale is going to do that to some extent at least okay so so there’s a lot that can be done at that level that so that because jump two to

one possible realization of this which is the standard category that most of quantum information happens in so here we have two kinds basic types of information which is what on the classic let’s say and an irreducible quantum system will given by will be given by a finite dimensional matrix out since we have tensor products of those will be at the same time tensor product matrix algebra solar gain matrix algebra ii just modify the conventions and direct sums are in the game anyway so there’s always this possibility of hybrid information between quantum of classical so we must have the direct sums of all of a finite collection of full matrix ultras and that is exactly a description of what is finite dimensional see style so this sort of the minimal model if you don’t like operator algebras actually at this at this stage Cecil trees are just piece of fancy language because it always boils down to just a classical combination of pure quantum systems this these vector spaces would be the Commission part of the algebra and the dual space actually the real part of the Commission part of dual space that’s where the state’s live so that’s all pretty much what you used to and then the more fit of the linear space of all visible Peter there’s the span of the completely positive maps and they actually you allow arbitrary completely positive maps with a sub normalization property there should be a preparation on the right-hand side it forces you let me get numbers too okay so I is I call this planetary I’m not sure that’s as good English word but this this is the finite dimension approach to quantum information and now there are many situations that might prompt you to go beyond that and here are some subs in the in the quantum information business many people look at continuous variables that always requires an infant dimensional space and actually if you’re a physicist you find planet dimensional systems a bit hard because every natural system like the particle always has infinite dimensional divot space I mean there’s no way around that so even one part of your definitely needs it space and very often you would like to look at well I at least like to look at infinite systems that are seriously affilliate in the sensor there are infinitely many particles so like field theory or statistical mechanics or things like that infinite lattice systems then then you definitely get go beyond this is finite dimensional approach or I mean you can you can you can say well essentially everything should be expressible in this finite dimensional language but then this makes you this is very often because you have keep making excuses and so and so uh for proxy measures that are not really what you want to say at the end so it’s kind of you can it’s very artificial if you insist you keep saying stupid things all the time just in order to preserve the Dark Matter finite dimension and physicists are certainly not related to that place so I’m so if you mentioned that means that you have to there’s one investment that you have to make which is a functional analysis force so so you have to you have to look at you have to tame these things by looking at topology is late and in this case actually to look at manageable classes of maps that are widely singular operator relations in the and the toolbox of function analysis that we don’t really important ok so so we certainly go way beyond your spaces and I think you’re also going beyond the dagger compact lowest setting where which is somehow I mean you can you can visit with this doubling construction you pretty much get this planetary you may be also like discrete Hilbert spaces type 1 systems phenomena out of a classification going beyond that is I don’t know how to do so how to take the square root of a phenomena after I don’t know and I would know for be avenged for the operation Hilbert space you can work instead with

the vectors and that that you can do do a lot in that setting but with higher type phenomena dress I just don’t know how to do that phenomenon that fascinating my boss says things like continuous geometry ok so here is a one way to enlarge that so the cisa out for us at the at the level of the finite systems were pretty much just what just unnecessary fancy language so no but now let’s take that series so the observables will be faces observers that will naturally be now that we take as a seaside algebra which means that we also complex if I a little bit that’s the sea there and the preparations will be also enough space preferably separable and such that they lose again an operator si staff now these objects are called W star algebra or when they are represented on a difference basically from online feels so so there are actually two algebras around one is the dual of the state space that’s a phenomenal run and there’s a seesaw sub the seas are off crutches will be subbed out about that thing and it should be it should be dead sweetie dance so so what you could say these two spaces that I feel their one for the states one for the for the operation cooking a sister algebra and the schedule now the the morphisms will be completely positive maps of course but else could there be and they have to be such that their map these sub arrows into each other which you can reef reef raised by duality are locally convex topological vector spaces which you can rephrase as the continuity assumptions on the on these offices so so that they actually they have two Raptors well this is you can phrases in both ways but you solve well you have completely positive maps between these siesta add referrers but they also have to extend by continuity to the phenomenons that’s one way foot now the tensor products our Father exams are obvious actually in this second you should not stick with fine a direct sounds but allow continuous direct sums and things like this integration that’s kind of more technical than I want you to be be right now for the tensor products you have to it’s actually relatively easy because well okay if you just take the potential problem sister afros is in general not well defined so there are different typical for to give in C starters there can be different for the phone and algebras that’s much better so there is a canonical tensor product so we take that and then the embedded system the product of the embedded c start occurs which means that we automatically go for the minimal so-called minimal see start answer so so actually if you if you think of these objects as pairs of a an outgrowth nice observables added out and a set of nice states then the tensor product is actually unique operation oh there’s not much to choose so let’s look at the class is in the classical case of course you have something like phase space and the objects there would be a topological space that give you on sort of compact space actually and there would be a measure on that space with full support otherwise going you can restrict to the support which is again in compact subset we don’t care about things that never happen so we would throw that away anyway so we say is possible and then these these various structures that i mentioned would be this space in which the state’s lid would be the l1 space for this measure and the observers would be continuous functions so actually the states i always is related to measure theory l1 spaces the duel is al infiniti that’s the big phenomenal deidre that contains the continuous functions and and the observable side would be related to topology continuous functions and the dual of that would be the measures all Burrell measures on the space X and that

contains singular measures like Delta functions and so sorry because every every a beauty sister algebra is isomorphic to the algebra of functions on a compact space so if you take if so if you take something like the bounded continuous functions on on r RN there is also a siesta algebra and it’s the algebra of functions on a compact space which will leave the stone shape compactification so it’s something that is well anyway so it as long as if you talk about the billion see styles because there’s always a context-based project elephant’s back okay probability from Luet evaluation will go like that so so somehow when we when we when we only have a topological side we get pretty wild combinations of measures we choose a nice class by fixing a reference there and from the other point of view well okay I guess if you know so you haven’t had a nice class of both sides and that the combined structure is something that so I come to my second part which is talking about subsystems and how to say how Alice’s bar are embedded into such a situation here is the first example that that actually directly fits to this structure supposed Allison Bob share infinitely many singles countably many more so that’s that’s the sea star algebraic view so if you have to construct it the observable algebra for such a system that has infinitely many copies there is a canonical way of doing that you can take the tensor product for every finite collection and then there is an inductive lipid construction that give you this leave the algebra for all so actually the classical case this is exactly taking the top of the compact topological space and taking its product with a product apology and then party can offer you get you again again get a compact spaces so this is chris is the algebraic version of of this product of complex spaces and if it works in the noncommutative case in exactly the same way so then you have to choose something like the reference measure and now i already said should be infinitely many singles so you take this product of all these signals as a reference state that you do something called the gns construction and that gives you a double space and in that little space you build operate and phenomena reproduce and clothes in that the week to apology in these spaces and that gives you did the phenomenon for us now Allison Bob will be embedded into that but we know at for every finite number of singles for a refinery collection of subsystems we know what the telus out of brown to Bob algebra is so that that’s that’s assume and now we can go to the limit then these outras become larger and larger and they and if you take this from online closure of that algebra the way I described you get what is called a type 2 1 factor in 11 and of course it’s clear that the outer probe Alice will commute with the other football nothing changes of that that is still true so so you have the situation that Allison Bob I love our flu built into this structure by specifying the phenomenons how about a drop and well okay so these two these two towers commute that’s all you can say at this level and this looks like pretty much the ordinary thing so if you’re on this finite category you would immediately conclude from that yet you can make a tensor product between now this is no longer the case so actually there is no there is no product states at all between else which is what you would want to clearly have in any tents of product situation to just test products to preparations is that it’s a legal preparation but here actually did you don’t have any product state in this for no man in this in the set of normal states this in this set of nice states so so another way of putting that is that if maybe if you want to think in terms of measure theory there is no you can formally of course define these chronic states but they will not live in that here with space there will be singular with respect to your reference

measure which which came from taking infinitely many signals so only finite deviations from that are contained in this servo gns civil space so and the products they would be something like an infant occasion and that would be singular that is we live in a faraway Hill into the one that which is constructed far away just meeting a thermal but okay so so you can have these commuting savvy articles that are not tensor products and this actually happens in physics in various places for example you can take one field theory now in quantum field theory to every space-time region you associate such an algebra let’s say phenomena from so a 0 would be the observables measurable in that space-time region all and Alice Bob situation here would be one in which they live in space like separated space-time regions they set up the lab very quickly actually destroyed afterwards so it’s a space-time it said and if there if there is no light signal that go from one set to the other they called space like separated and that means that they can it’s one of the axioms of you that in that case these out was commuted now I’m interested in in this situation and I i will look at the particular case when they when these two space-time reaches a tangent so they’re closed so there’s so that is you take let’s say to take two disjoint spatial regions that however touch and then take the cause of shadows of those service that’s all the mission drawer space so this is about the bar and then we include always this little icicles that that is the situation I’m talking about my surface are separated by marshal and they watch their local back vacuum fluctuations so there is a canonical state in this theory which is like having nothing but there is no way you can really have nothing because there’s no counter that will always give zero probability and measure something option in the factor that that will respond to the vacuum with zero always and still measure something else in the launch of your way and be localized so so there’s always vacuum fluctuations around the unity field and they watch that and then there’s an ancient fear and by myself in summers saying that this actually gives you exactly the situation that I talked about before so just in the vacuum fluctuations within such a situations you find infinitely many singers meaning that we have these to contain this time to one high-profile a factor of those sides in Eclipse so across these others will commute and the restriction is this state the restriction of vacuum to this subalgebra will be exactly this infantry many singles example okay so so well I hesitate to say that this is the this is a piece of real world one of your theory it’s a bit more realistic though than abstract to it okay so here is a more information type scenario where this questions of how to how to identify Allison Bob in a in a bigger system comes up that’s device-independent ography of course they will have devices but they don’t know so much about the devices in that sense let’s device-independent I mean the scenario here is that the adversary actually sells them their equipment so I also God get their devices from from the from the adversary meaning that’s it well the NSA would be in any possible and and they then the APIs are also distributes the basic quantum states that ever this will be a entangled state somehow so they each get some quantum systems and now

comes a critical moment in this scenario they lock their labs so it’s true that ‘if distributes devices that one of the things that she’s not allowed to do is mount the camera analysis lab watching the screen so that clearly if you allow that amount of cameron have a signal to the outside if you allow that there is no way to even talk about seizures so you have to assume that somehow even though the device is built bad is bad i eat and she could have some devious stuff set up there that when they close the lab we assured that Alice owns this all can be sort of has complete control well no but if has no control of that or whatever is in that laughs and this end so this this locking of the labs is a crucial step in any scenario that it was it now then they make some measurements on their farm systems using that equipment they check for certain statistics which is a violation of the Clausura my hot inequalities actually here nearly maximal 10 it’s play metaphor that works and then they do the usual stuff some classical rounds of reconciliation error correction privacy application and in the end they get some secret key that that’s that’s it so the thing I want to point out is this this thing in the red box here they need to lock their labs and you have to say what controlling a subsystem it now if you accidentally minded having control subsystem means that you can make all kinds of measurements on the nurses with some class of measurement make and in this in this situation between Alice varmint also e one way of saying that is that that that you might want to try is that actually everything that Allison measures can also be measured jointly with company that ball Sam so there is a joint measurement for any to any of these any of their choices Allison vodka juice a measurement there will be some and that they don’t interfere with each other so they can choose freely from their set measurements and that people need that it will give combined only the Brende but they have to keep data can communicate but that will be the public line and a public classical language so um so just assuming that all of alice’s operational choices must be allowed together with Barb’s is one thing that you might postulate but that means you have lots of drawings measurements and it turns out this is much too weak so if you only want to postulate about controlling the subsystem then you cannot do this device intended to that the independent security and it’s really a very unclear situation actually they could get any know any Mon signaling correlations that’s really my boys managed with controlling the subsystem so if you just add a little bit to that you have this thing about the operational choices that’s it just measurement simple and you assume also that the full set of Alice’s observance even if she doesn’t use it in this particular protocol the fourth set of Alice’s observables spans a suburb then you can show that actually these tools are and that’s the same Bob then these Alice’s about South African us to be pointless and you know from quantum mechanics that if observables commute you can you can make a joint measurement there is a partial converse of that namely if two observables POV ms have a joint extension so they can be measured together and one of them is projection value then you can show that they actually must commute and the product is the drawing measure that’s the only way to do so you need to have these projection valued measurements in the game in order to conclude something about commutation this this if you don’t have this insert if the other halfs algebraic structure around somehow you cannot say this and this I find this a little bit mysterious why would have to go to the algebraic structure which is somewhat he’ll interpret it if you have that you’re in business for also this

device independent security so you have to make an assumption is not just your assumption of Alice’s controls her lab means she owns a sub algebra or that said intensive factor in a big divots base decomposition would be the same thing from this party ok so the sabbat subsystems here would be subalgebra and with a kind of separation between ours and bob i had in this infinitely many simple examples would be just fine for that now of course you cannot also have this other version i started with a degrees and tens of products and that that’s a much that gives you much stronger view of what a subsistence you would end in that context that I started with bureaucracy subsystems just intensified and so these 22 views are what I want to prepare so so here’s how device-independent cryptography works with the lemma I did in that this lemma i did in 88 not thinking cryptography actually we were doing this quantum field theory stuff so so we just take this commuting out of a scenario and we assume that we have a maximal violation of the CH sh inequalities then the content of the lemma is that then ascend well apart from things that with that happy with probability 0 for some sense these these these operators that use for the violation actually have to be the ones from the canonical single example so these span latuda two-dimensional matrix algebra or generated to damage and restrict to that the state is exactly the singlet state so so it’s quite amazing that just from the value from this equation we saw for the witches I mean an arbitrary state has less than or equal to twice a square to this is 40 songs inequality and from from the case of equality in the in that inequality is extremely specific there’s only one way to do that and okay so the corollary of that is this device dependent independent security in the 0 error case mainly if they can check that they have this kind of Croatian there is rejection to this 2 cubed subsystem one cubit for Alice one cubic foot bob is a pure state which means that it factor Rises with respect to everything else including Eve so they so so if they only check these these correlations which they can do without looking inside their boxes they got got for me then they can still be sure that Eve can’t listen it that’s a kind of security that is totally impossible in the classical so this is this is actually I think this is the most interesting type of lot of cryptography although it’s totally irrelevant from the practical point of view because the entrance ticket to this kind of cryptography is the loophole free Bell experiment and that we are still waiting for it may happen in the near future but it’s still don’t have that so from a practical point of view forget it for the moment but it’s it’s certainly conceptually the most interesting kind of cryptography quantum theory last okay so so one thing that I’ve been interested in for some time is this is extreme specific specificity what is it bad that’s all you have so this this the fact that this maximal violation happens and only this this unique situation from the truth if we use very strongly the algebraic structure of this expression rewriting in terms of as a real part of some operator expression something like that and that looks very they’re very much Taylor to this particular inequality to the CHS age and now not so sure anymore at something like that wouldn’t happen for other inequalities similarly correlation inequalities basic build this is the same you would have a finite set of observables at each end with finitely many outcomes and combine them make a linear combination of the probabilities somehow and you look you asked for the largest value this

expression can take in any quantity so if you look at such correlation expressions there is there is this upper and lower bounds to them to this maximum to the quantum accent yes but we are not even fixing the observables so you still have one one one more quantified error so so it’s the maximum over all these observables once those are fixed it’s clear that that you’re wearing of the convex set of states and the maximum will be attained at some pure state but it’s not clear that that there is only used for example and it’s even less clear if you have allow these I mean very different observed you don’t specify the examples so it’s just the structure of the expression that is given and you’re not saying what the A’s B’s are that’s part of the of the maximization ok so the lower bounds are easy to get that you make examples you build think of observables and then you look at the maximal violation that you get for those very of the state’s coming up now for the upper bound there is a program champion by Tony RC from Barcelona and where you can find upper bounds by fire the hierarchy of semi definite programs in some sense you take the given observables you invent all kinds of products are then you invent probabilities and correlation functions for those and check feasibility of a semi definite program built for that what this does asymptotically well ok asymptotically it’s clear that if you the example that you make will typically be tensor product examples so so you filling it from approaching it from below and you a syntactically would get the best balance for tensor product separation now in these in these and what this hierarchy does for you is if you take it to the to the to its logical conclusion infinity but beautiful you take the full hierarchy then you are characterizing exactly states in this commuting algebra sonography building arbitrary commuting algebras in which these correlations are realized and this the semi definite program tech that what fun is actually checked is the positivity of the gns representation and that and the only thing that you put in is a commutation there is no way you can put in the tensor product structure at this level so so actually there might be a gap between these and the question whether or not there is a gap was hosted by Cyrus on Boris Arizona our problem page it’s the last one there should be more lazy editing and well this problem is really over series on conjecture that it’s like it might be related to the thing to to this farm and algebra class so if you have type 1 algebras only if these two commuting outputs are type 1 which means that they’re isomorphic to pound operates on tibbett space or another way to put it you have minimal projections which which correspond to raise in a double spaced them then if you have that you automatically are in this in this tent across commuting outras then are separate as attention times now type 2 is this is for long runs classification actually it’s related to the structure of projections in such an algebra type 2 and type 2 every projection can be subdivided into in some sense into two equal parts so there are no minimal protections this is why a phenomenal is continuous geometry immediately the geometry of subspaces of the double space is an instance of projective geometry and this would be some kind of geometry without points there are no minimal protections that’s why it got of continuous challenge type 30 drivers are a bit wilder than that they’re the eight can say that the the there’s in each of these if you have a

basic building block or a factor then here there is a trace functional which takes integer values here it takes continuous values here and here it takes only two values may be 0 and infinity so these these when mama did that type 3 algebras word of the untreatable monsters now one of the things that turn up in one few theory is that all the local articles are tax-free actually they’re types 31 I don’t don’t want to bore you with the details of it specification so type 3 is actually quite ordinary and so so so terrace on says that this possibility of having commuting some algebra which cannot be represented as a tensor clearly must be related to this to this guy and the modest contribution that we did the other day together with a PhD student mindfully shots so say no actually this is not the crucial but it’s part of the story but it’s not it’s not really the crucial point because in the examples that I gave you both this infinitely many signals case and they and the quantum field theory case there is no gap without refers like that you cannot I mean you can show that the maximal violation that you can get for any correlation inequality it’s the same as you can get in the terms of pharmacy now the reason being that there is a strong sense of finite approximation these operas so the terminology here is that in the sea star case cisa acquisition ice from the if you finite dimensional approximation about nuclear they have the property as you can find by the property that for them the minimal at the maximal 10 subscribers always coincide and but basically the good definition is nice approximation by finite dimensional phase and the same is there other on the phenomenological level there is called hyper finance and our property property injectivity they are equivalent although they I mean there’s a non-trivial theorem that these two things are equivalent and the algebra so quantum field theory due to an axiom that bounds the number of local degrees of freedom at finite energy will always be hyper finite so actually this is what people have thought of that and made an axiom of this madness in some sense of about the possibility of fine an approximation and for the for the case of infinitely many singles I actually describe this finite dimensional approximation to you so this will be a type of fun at algebra always actually the hyperfine at 21 factories unique object that’s defined up to four modes okay so the the gap if there is one is related to a breakdown of this pineapple profitability okay here’s my conclusions and there’s some things I didn’t talk about for example if if you build axiomatically a category with tensor products and you put two electrons together how do you take it out of the of the of the Fermi symmetry reputation symmetry that’s that’s something that has been puzzling me also but the quantum field theories have a way out to say there’s a local region and these community those observables commute with each other they get the Fermi statistical and also this would be some in some sense an answer why you would be confident what it means to close a lap well this is a space time reach and what a few theory tells me that should be an alpha associate they don’t have a good axiomatic reason for pasta veiling alphas either but anyway the two things that excite okay anyway the first I think everybody thank you very much for you as you go to infinitely many degrees of freedom and therefore potentially to larger quantum systems are there any new tests that emerged as to whether your linear quantum mechanics is breaking down for larger systems but the larger systems are only one thing that you can do nicely the small sisters

really measure all zones and I think we agree very quickly that this is not a good option for large systems and even even a block chases eight cubed state so they do complete tomography that’s a mess whether we’ll do that he couldn’t do it for larger because he did you have an entanglement witness that he could have deserted with a small number of measure of it so complete to complete tomography is out there will be there will be entanglement witnesses also fill our systems but we should expect that they become more tricky and in the end we believe artistical mechanics so if you really go to the internet limit there for certain observables we’re not going to see any quantum effect but that’s but here I mean these I I gave you examples of there was a witness so the one of your theory system is one with infinitely many degrees of freedom I gave with this we can write down these operators the battery is in China yes so in that sense there are tests ccr roles is a ton to one factor do you see any possible relations that the CCR is as a trade CS but but but usually you don’t take that completion on so the CCR you mean is the algebra of canonical commutation relations with let’s say family video into diminutive is free okay so that so the you have a trace on that the trace function is singular however it’s discontinuous and this precisely not the kind of state that you take a few teeth this would be the temperatures in poor or is it just a coincidence a little time to own practice where there is only one type to one factor that is also if you have okay so another interesting state is the original epr state the original upar state is continuous matters and it the Delta function that they write is nonsense what they say about it is completely correct however this is why the balm spin example is so much better because you don’t have to make I mean implicitly invoke some some non mathematics F so it’s all explicitly as important but but what they say is that that you can have a state which is sharp for for they well let’s say the difference of positions and the sum of the momentum because these two things community so they can be both simultaneously sharp but these things have continuous spectrum so you can build a singular state does that and you can defy that on the CCR algebra and it was not the property that restricted to Alice you will get a trace so it will be a maximally entangled state the sense that it will be a pure state on the ccr drop the combined system which is a restricts to a trace on the suburbs but you can’t have a V of a church so there is no plan bana extensional retains that kind of property full set of operators there saw that for this particular example I think there is a closeness and draw operators with our operators I didn’t hear a copper Dirac I’m not sure I see the collection why I’m staying expected objects just extending instructions I think we talked about and i’m not sure i get i get your context yeah you may imagine not about the tank equipment can dvds a day and the dublin consideration rested blood last year to say you don’t actually don’t need contact us for that so you can actually do this for even imagine local space to be sure to mention this is this these type two phenomena fees are way worse yeah the vigilance so sure i think so that the result Jamie keeps on telling me the nails dry quantum field theory there is Academy image shows up which has a particle systems and particle interactions as morphisms and which turns out to be all my clothes I don’t

really understand how this happens but yes you have to talk to you i prophesy yes but for that you have to you have to throw away momentum and effing all these sort of non-continuous aspects of the the gauge group so actually you do have there is a very category laden part of the computer there were you thinking local operations that might take you from one child sex with your mother yes they reconstruct the gates to the page the more distorted question I go from one node but i do want factors that he wanted them because of having it out an appropriate notion tres leche because ready so in our program we also perhaps a commercial space which is usually invalidism around which extended angling as abdolmaleki this stage which windings after why so was right back to walk thank you so much and do all the human race it’s more like question gentle rolling so you’re so actually I the theory the trace is very closely connected to this to the notion of equivalence of protections and this is just a very convenient way to see the structure of possible projections and maybe thinking about equivalence class would be more interested actually thinking about races like to make one marker for 10 years ago rammstein do tonight invented a kind of category value to your ideal system utilizes an opening charity ok button what you are network oh yeah sure