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Detecting Incapacity

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so this is a gram Smithies curiously appropriately-named talk is of detecting incapacity I think we found it beautiful show you and there we go OK so that OK so I want to thank the organizers for for firstly incoming it give this talk and death and like blood said what I wanna talk about is is detecting incapacity this is this is joint work with the with the John small and we're still we're still got I I got a back going on about whether and actually to be allowed to to put that in the title and some real Journal in capacity and through the referees loud gonna get a ball 1 token all right so the problem that I wanna talk about today is of the more about communications than than not doing logic but I think it's it's still a really nice problem so here's here's the idea so classically so if you have some you know you have this whole theory of of information found by Shannon in 1948 so you have some access to some noisy communication channel it's got some input acts it gets mapped noisily to up but why and you have these parameters that describe the noise processes probability of getting out of Y given that you chose input X and you'd like to transmit with with low probability of error as many messages as you possibly can given access to many independent copies of this channel and the beautiful thing is that this has a simple and optimize over all these these arbitrarily complicated communication schemes and it turns out that this was just a simple formula for what's the best rate you can communicate at do given the channels so as a function of these noise parameters you just have to maximize over input distributions this correlation between the input and the output and the correlation is measured in terms of something called mutual information you don't have to worry too much about that and this is a sort of a reasonably complete and and and really beautiful theory and it's the sort of thing that we like to mimic or were developed in in the realm of transmitting quantum information and dealing with quantum noise OK and with C so specifically 1 nice
thing about about this theory is that it tells us which channels are useless so this is a channel that I I created in my lab at IBM and you know it's got a sender here receiver somewhere out that way and there is no correlation between what the senders tries to transmit and what the receiver get here so this shells totally useless and and actually in some sense this is the only useless OK so any channel that has some correlation between sender and receiver can actually be used for the error correction to transmit noiselessly classical information and what I wanna talk about today is trying to understand the similar question for transmitting quantum information it turns out it's a lot more complicated than this is sort of well basically because quantum information is a lot more delicate there are a lot more things that can go wrong where there are a lot more delicate fit with a lot more subtle things that can happen to you your quantum state that will make it impossible to transmit quantum information than simply having no correlation between input and output
OK so this is my little picture of the 0 quantum capacity channels that we really want to understand how so you know if this was the classical capacity channels then you just have a single point it's it's the uncorrelated channels but here the something what more complicated going on so so they're 2 different sets of 0 Kwan capacity channels they're both convex this 1 it's called symmetric channels were or anti degradable channels and these are just the channels that this server reason that they have no 1 capacity the reason is that that you know channels have environments and the environment of these channels is symmetric ones actually can simulate the output of the channel so if there was any capacity for these symmetric channels and you'd be able to generate a T able to clone the input stay right so so these guys have no no qualms capacities capacity because of the no cloning theorem end sort of the the main point of this talk least the starting point when we start thinking about these ideas with this this other class of channels the call PPT for positive partial transposed and I wanted to find some way to explain in in some operational in some operational language just something that didn't involve writing down a bunch of matrices and an appealing too complicated mathematics what these PPT channels and why they have 0 Kwan capacity OK so the point of this slide is basically 0 Kwan plastic shells very complicated and that set we have these 2 2 known classes as a upon capacity channels but there could be other ones outside the sets but we don't know and we know that actually these guys can interact and very very unusual ways so there you can take 1 of these PPT jails which is 0 comma decimal capacity you can mix it together with 1 of the symmetric channels that is 0 point capacity and the joint channel actually can develop quite a large amount of quantum capacity OK it is the object that were going to try to understand OK here's here's our going go about
doing it FIL OK so the 1st all just sort of review these 2 different classes of 0 capacity any degradable ones and these positive partial transpose channels and then I wanna show you a new proof that these PPT gels have no capacity that sort of appeals to more physical reasoning then than the usual map more mathematical thing like that I don't I'm not gonna mention anything called LOC C so you know I don't know some people don't like the elasticities way worse group theory OK anyway but then that then undertake this kind of simple proof that we these PPT channels have no qualm capacity and then try to generalize it to to search for a general way to detect whether channel has any capacity and they're the nice thing is that eventually will make it so general that in fact it includes any degree of these any readable jails and these PPT so we have this nice kind of framework for thinking about tests for for quantum capacity OK and then the motivation is is unifier thinking about incapacity and and try to appeal to some physical principles another motivation when when we're starting this work was this the they're sort of non quantum theories that you'd like to ask questions about information theoretic questions like like these generalized probabilistic theories you might have heard of I always something by Schumacher and Westmoreland called a moral quantum theory which kind of a nice generalization of were nice toy model of quantum mechanics where you have a vector space over a finite field rather than complex numbers and and it wasn't so clear how we could how we could do ask questions about this PPT criterion in in these settings but but in in this more so of operational way of thinking about pose a partial transfer so it becomes a little bit clear how to think about these things another motivation is that you know I like you know I like to be able to give sort of very simple explanations for why PPG were were for why things happen so so who going to get a nice simple explanation for why these PPT channels have no pun capacity any questions at this point that all right so this is just a reminder of of what the quantum capacity and the idea is to have access to large data center over here and a receiver over here she's trying to transmit some cubits by encoding them into the inputs of some some noisy channel this is a bunch of independent users of a noisy channel and I'd like to understand as a function of this channel what's the best rate at which I can transmit reliably cubits from left to right and because they said it's it's much harder to live classically but I'm willing to accept much less than actually 1 understand I just wanna be able to answer well can I send any cubits or not but I want understand whether have positive quanta passages can this is still actually quite a difficult question OK so here's what we know about the quantum capacity at this point but there's something called a coherent information which gives a lower bound for the quantum capacity it's just this sort of funny looking maximum of the difference between the entropy of the channel's output and the channels environment I should I should remind you I guess that I can't think of any noise process that maps side to his some input side to another would be as a unitary interaction with some some and inaccessible environment that's that starts off in some of the pure state 0 that and this turned out to be useful way to think about these these quantum channels when we're thinking about transmitting quick OK so this coherent information which is kind of the analog of the mutual information from from the the Shannon formula for the capacity of a classical channel is just this maximum difference in entropy of b and maximize the overall input states to the channel OK the reason I say it's the analog is that if you if you talk about random coding for quantum channel in fact this is the rate at which you can communicate reliably using randomly selected codes so it's a lower bound for the quanta capacity in and in fact if you do this thing called regularization if you take the this form this go here information and evaluate it for many users of the channel and normalized by how many channels as you got that in fact that gives you a formal expression for the 1 capacity I mean unfortunately this is a this is a disaster if you try to calculate we know nothing about how this limit convergence we know that it does converge and it turns out that you but actually this Q 1 this Cook here information is I can be strictly smaller than the 1 capacity in fact it can be a 0 when the conc pet capacity is quite large so it can be a very bad approximation at the point the point of this this slide is basically we know some things about the quantum capacity we have some bounds on it but there are a lot of questions we don't have an answer OK so here's a nice example of 0 comma decimal capacity channel that called symmetric a symmetric channel so what's a symmetric channel I have isometric extension of my channel so I have some input stateside it interacts with this environment that starts out in this state 0 it's a unitary and there's some output B and some environment and I call the channel symmetric if this for any society would in this state on these symmetric so I might as well just swap right so if if swapping the environment and the output of the channel leaves everything alone that I call the channel symmetric and the the power and the example to have your head is but a 50 per cent attenuation channel case so I have some fiber it's got input mode the environment it interacts with this vacuum as K and ii ring these things off a beam splitter and it's 50 50 so half the signal goes down to the environment you have the signal goes to the output right and and this is more or less OK so now I wanna show you nice old argument for for why symmetric shells have 0 quantum capacity OK so let's suppose that that I was wrong and that they did have some common capacity well the 1st the 1st thing to note is that quite capacity is is not a question about a single the channel to question about many users of the child you get asymptotically many and you have to do some encoding across many users and and you get to use the best encoding and decoding schemes so let's say I had an encoder that I could send through many users of the channel and I could take the upper the channel decoding gets I out OK so now transmitted some say Cubitt state from here over to the fine I wanna get a contradiction so remember this guys symmetric so who says I have to decode here this guy's I could just as easily to come over here I could just as easily encode on both sides and now what have I done i've taken my input stateside through some encoding centered through the channel that symmetric then coatings on both sides I get 2 copies of Siam this is a violation of KNO cloning this is actually not a linear operation it's completely and physical so my assumption that I could actually encode and decode reliably with this channel had to be false and again I really like this kind of art has it you can give the illusion of understanding to somebody who asks you Y is a symmetric channel have no capacity the reason is otherwise you'd be violating no cloning and everybody knows about cloning right then is obviously impossible OK so I want try do something similar to this but for this positive partial transpose criterion OK so what is the PPT so the basically remember what part of transposes poses you take a matrix and then you just sort of take all the stuff on the upper left and move it to the lower right and well you transpose no symmetrically OK so partial transpose is just well it's an operation defined on a bipartite system now on a b and basically what I do is I take the transpose the B system but none of these systems because a puddle transfers it's linear it takes I J K L 2 I j on a 4 K on the OK so this is well 1st of all this is not a physical map OK this is actually from it's positive but not completely positive kay so I it gives a nice test for whether whether a child whether an estate has any any sort of useful entanglement so specifically if I do partial transposed to a bipartite state AB and I find that it's not a it's not positive so the original really be is of course positive it's a physical state but when I do this non-physical operation of transposed to it it may not be positive anymore in that case the state is definitely entangled but if he is positive OK it may be entangled and made may not be a difference if a separable it's definitely positive but there do exist these entangled states that have caused a portal transport OK and the whether it's
intact what even if it is actually very very noisy and hired so noisy that that the entailment is bound which means that there is no you can't do any distillation protocol to make sort of perfect pure entanglement again out of the entitlement in this role in the OK in a PPT channel is just a the a channel that enforces PPT between its output and the purification of its input OK so another way it has try matrix that is PPT so if I take some state and put it into the channel the resulting state is also is always a PPT OK and and in fact because of this this bound entangled the property of PBTs states NHL that has a positive that that is PPT actually has 0 Kwan capacity OK so now what I wanna do is give you a short proof that that PPT channels have no pond capacity OK so 1st of all I'm just going give you some notation so let's let T of road just be the transpose of row and you can check from from Eitel before that any channel is PPT exactly when she composed with the channel is completely positive so it has to be physical alright so you should note 1st of all not every channels the PBT for example if if I to let the channel just be the identity decompose with identity is just t which is not physical it's not OK so this this identifies a certain subclass of channels that are are very noisy I wanna show have have no capacity for transmitting quantum information OK so let's let's suppose that we could transmit quantum information with with the channel that that is PPT OK so what does that mean that just means that there's some encoded I can feed my state into the I can take the output of the encoder feed into the channel I can take the alpha the channel field into a decoder and get the state back again get so that's just what it means to transmit quantum information so now what I wanna do is act on both sides of this this equation up here but with this transpose operation ends you so on the left I just get transpose of of the state and then on the right a get transpose compose decoder composed with channel composed with encode acting on the stage and transposed has this nice property that actually I can community through this this decoding operation so so I'll say a little more on the next slide but the ideas there's another physical map associated with this D actually it's just got the com it's got the same cross operators but you take the complex conjugate of and if I applied that chance if I apply transposed aside they the channel followed by the transpose I get the same thing as if I had applied transpose 1st and apply the this this D star for the complex conjugate of the channel 2nd OK so that that's the sort of nice property transfers that I wanna use but now at when business because actually on the left hand side would have I got I've got some encoded I've got this joint channel so it's a transpose composed with the channel that's a physical up that's a physical operation by by hypothesis as composed with this complex conjugate channel which is also a physical so on the right hand side would have got is some physical operation and on the left hand side would have got is the transpose of the state OK so this is the contradiction that we're looking for right so assuming that there was an encoding and decoding pair but actually allows us to derive a contradiction as long as the channel is P. T. and that and were done and if you want to talk about capacities actually need to talk about many users of the channel you have to worry about this being only approximately true but because transpose is is is continuous and actually many copies of a chalice PPT exactly when the channel is PPT but we're in business this is this is a good proof that the quarter capacity of a PPT channel was 0 Khot I want to have but that OK so now what I wanna do is it is try to generalize this idea this sort of simple per simple proof of of the 0 capacity of of PPT channels so couple of good good the spoil the surprise but anyway and basically I want to I want to talk about this I wanted to formalize this argument about and PPT channel so that I can include different kinds of tests for for quantum capacity and actually I want be able to take any and physical or almost any unphysical chance any and physical map on quantum states and turn that into some test for whether where some test for a channels having any quantum capacity OK so found I do with the picture and can anybody tell me the name of this picture did Johnson and with Johnson we give this talk before the telling yeah I think good this is this is the house diagram and I really like it and and here's the idea this is the kind of property that we want to have a 1 hour on physical channels and physical matter OK just focus on the bottom part of the house the foundation OK what I want is but I want a way to sort of commune my on physical channel my and physical map through any physical channels so here I've got are composed with the and I want to turn that into some other g composed with right so that's what that's at this diagram shows the show's happening I hear I have I I fly all are the missionaries OK here I can either applied and then apply or I can apply all our 1st and then apply some 2nd channel and that's related to decode the stock and get to the same place over here OK so the point is that about starting here you can either applied and then where you can apply R and D star it in any unphysical map that has this property actually all peak the peak commuting OK and the idea is that this on physical map can be commuted through any physical operation but with some additional modification on the on the operation left hand side OK CSO and and the little women that you can prove is that if is on physical and some some set of states S and are composed with your channel is physical but then the channel can be used to reliably transmit transmits dates from the set us OK and that the proof of the lemma involves both the foundation of the house and also the roof so what we're going to do is go through the roof OK so here's how it goes so this top part says well let's suppose I have my state node 1 and there's some way for me to encoded so that when I send it through the channel and then decode to the original state that I that I that I encoded OK so what I wanna do is use the fact that all that this entire diagram commutes to actually I show you based on that assumption based on the assumption that that this guy exists and also that and composed with this this bar is physical that in fact I can implement physically this on physical operation can so here's the idea we have the encoder the channel decoder but getting here the sign but actually what I'm interested in doing is going down the number 5 jumping from 1 to 5 that's on physical however if this roof exists then I can apply my encoder I can jump from 2 down before because this compose within his physical and then I can up the star of the encoder to get all the way to fight so what was the point is if this latest physical then these guys can exist otherwise I could implement this physical operations that and actually for any for any for any peak commuting unphysical operation like I I can never get a criterion now for for having 0 quantum capacity and freeze in this way I can now apply it to to to theories of of mobile quantum mechanics or whatever maybe it's not the right crowd for that but the in terms of foundations it may be interesting to about what of the capacities of of the noise noisy channels in in other non-quantum theories if look at the so I told
you this whole story about the key computation and and giving a general criterion for and for whether a move before in capacity and you might ask well is this more general because the only example you talked about so far is this transpose operation and it turns out that in fact after a while it apriori it's more general but but if you do a little work you can actually show that if you're talking about linear maps OK now of linear unphysical maps and that in fact transpose it and here's a little lamb other basically says that with a little bit more math from and the basic idea is from group theory and a concern so faithful representations of the projective unitary group basically there only 2 of them of the right dimension and 1 of 1 of them correspond identity or sort of the perfect channel which is physical and the other 1 corresponds to the transpose and that as a result if you're interested in using sort of peak commuting and physical maps that are linear you're kind of song that's kind of sad to me because there are lots of unphysical maps the linear and then I was hoping to use these in decomposable positive but not completely positive maps and it turns out that that this argument is not can work to show that 1 of those well will show that those guys give a criterion for about quantum communication capacity OK but so the answer is no it's not quite more general
but I can make it more general of twenties refine my definition of and the idea is because I have to get I am I have to get away from these linear and physical and unphysical maps it gets very difficult to make sure that things are p community which means in a volume of I wanted to I wanted to sort of the push my on physical map through my physical map in such a way that you know instead of applying the unphysical mapped to the output of the decoder I could apply the unphysical mapped to the input of the decoder and applies a modified decoding operation instead and so we can actually generalize this idea of the commutation the ideas right here basically I only have to well this is my and physical map parts may be nonlinear and I now have well I had the same equation here I can commute this arts of D through D. but now I've given myself a little bit more freedom basically but because I allow this arts-indeed to depend on on which decoded is that trying to commute through and now all of these guys have to be on physical grounds and that gives me now a new criterion for whether a channel to information but It's again of the same form if this on physical map it as composed with my physical map and is physical itself then and cats and quantum information and I should take some some time to introduce some some I think useful terminology again but when this happens we say that I end is so noisy that physical items are OK or you could say that all that to add is so physical very incapacitates this I think useful but but but and thank you thank you useful pieces of terminology the OK so now what I want to do with this more general idea of key connotation I wanna see whether I can get anything more specifically for nonlinear maps and the nice thing is in fact that I can what I get now I'll use any degradable a symmetric and the idea is well this again just as a kind of in math talk terms that OK 1st let me just remind you when any degradable cellist it's 1 of the symmetric things right it's like a channel and is any degradable if there is a sparse 2nd physical channel and 1 to be that has 2 outputs and when I traced out the 2nd output I get the original channel or factories at that 1st output I also get the original Châu OK so this is just slightly more math the way of saying this this a symmetric extension OK and now given 1 of these channels I can find a a man that's on physical that actually incapacitates the chair OK highlight partly speaking I take the inverse of the channel now this is not a physical operation and then I compose it with this extension of the channel this is a physical operations and it's a little calculation you can do but I this this map is not physical in the way you see it is that it can clone so it can take it in the input to this map and if you are if you trace out the 2 systems the set the right-hand system of this this output actually is just the identity but it's also the idea that if you trace out the yeah the left-hand system OK so this is a map a clones they might be a little worried about this but just I'll say 1 word which is pseudo inverse and this gets a little more complicated but basically you can make this work OK and then we can actually define this star operation that we needed because what we wanted you show that this this map which is is unphysical also has the sky community of property which is that if I apply all are to channel I can equally well apply our 1st and some 2nd channel a 2nd objective that that's the idea that that we want are to be able to commute through any any decoder and in this House diagram and four-unit unit Terry's this is easy you just let it be you tensor you I and in fact you end up with well this is the sort of argument this is the sort of you dependent on that we that we had for the more general notion of of the commutation and basically the shorter the short the short story is that this is how it works this
works so now we can we can draw this house picture and we can we can talk about the reason for is symmetric ready degradable tells having 0 Kwan capacity while the reason is that it otherwise it would cloning you talk about the reason for PPT channels having no capacity otherwise it would give a physical implementation of this transpose operation which is physical and this is nice a unified picture for for describing both of them now this is this is a thing this is I this is sort of at the challenge so so the point of this story was to try to better understand positive of partial transposed from and and why these PPT channels have no quantum capacity but here is I think in even better reason that they don't have any quantum capacity unfortunately I don't see any way to make it rigorous so let me let me tell you about it and most will see whether you like the idea OK so 1st this is a picture teleportation OK I have some states I wanna teleported up to here so what I do I take some some of entangled state the SI comes along backs with 1 half the MaxWay entangled states we do measurement I sense of classical information along to the guy who was the other half of the battle entangled the condition on a classical information but he does some rotation and out pops side it so that's how regular teleportation works OK and you know and in a way that that 1 1 could think about this is well I take this I I do this measurement and ice split-the the information about si up into 2 pieces 1 is classical information the gets transmitted up here and the other is quantum information it's sort of creeps along back in time down around this band and up over here and you put them back together again have pop side OK so now we can actually talk about about doing teleportation with with noisy states here and the rate at which you can transmit information from the left to the right is what you could present it is related to the the capacity of the channel that's associated with this noisy state and you can tell the same story the SI comes in do some measurement classical information goes that way quantum information sort of goes back in time it goes around the bend up pops out over here and outcomes I again OK but now suppose that that this state row AB is actually partial transfers variance OK so if I do a partial transpose on be let's suppose or a over here let's say that the state to the state pops out what the state is the resulting state is the same as before 1st of all this this state has to have no distillable entanglement so I actually can't do any it's teleportation through the state if it's if it's a portal transfers invariant because it means it's also PPT OK so why is is the story that I wanna tell you well so we have this information it's coming along and we do this measurement and the classical information goes along for just fine but this quantum information it's supposed to go back in time but because this state is actually a partial transpose invariant and transpose corresponds to time reversal in fact the information gets stuck in here gets confused it doesn't know which way's back in time which ways forward so it can't wrap around the bend and pop up out to here again so that's that's I think that even more intuitive way of explaining why why are PPT states have have no distillable entanglement or anti-PT channels have no a quorum capacity but I didn't really see a way to work to make that more rigorous so if you have any ideas that would be great and alright quick summary so the these channels was 0 comma decimal capacity but they're not trivial in the way the classical tells us recall capacity or it's actually difficult for I to that characterize which which noisy channels that transmit information which can these 2 known tests for incapacity symmetric extension PPT you can understand both of them as special cases of this house diagram that and the thing I like about this is that it is operational PPG channels having 0 comma decimal capacity but because otherwise if they had some form has to be good we can implement this and physical time reversal operation and and to go beyond PPGI we actually need a nonlinear and by doing that we can actually recovered this a symmetric extension gorgeously the questions and then you have some too if the I always admire someone who finds the most
intuitive explanation 1 that involves sending information that which in times of careers but any questions we have time for maybe 1 of the so a
You're using this teleportation thing is a stand in for your channel for your PPT channel is that you have the right idea that so what's the obstacle using with making it more generally rigorous well because it was the part where I said well at the status invariant under partial transposed the information gets confused about going back in time so that's the hand made it yet OK but I mean the picture here is very similar to these so so called post selected CTC setups that anyway I don't know that makes an intuitive difference but OK any any other questions we have to we have the next speaker come up let's think grounding in
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Metadaten

Formale Metadaten

Titel Detecting Incapacity
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Smith, Graeme
Lizenz CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nicht-kommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/35299
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract Using unreliable or noisy components for reliable communication requires error correction. But which noise processes can support information transmission, and which are too destructive? For classical systems any channel whose output depends on its input has the capacity for communication, but the situation is substantially more complicated in the quantum setting. We find a generic test for incapacity based on any suitable forbidden transformation---a protocol for communication with a channel passing our test would also allow us to implement the associated forbidden transformation. Our approach includes both known quantum incapacity tests -positive partial transposition (PPT) and antidegradability (no cloning)- as special cases, putting them both on the same footing. We also find a physical principle explaining the nondistillability of PPT states: Any protocol for distilling entanglement from such a state would also give a protocol for implementing the forbidden time-reversal operation.

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