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Polar codes for classical, private, and quantum communication

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they might well all codes OK so sorry about that thanks red made this conference and in light of the panel discussion last night I look in the palace one-stop throwing tomatoes at me but my talk is about codes for quantum communication and the silly fault tolerance all we're trying to find code so they can achieve the capacity of a quantum channel for communication and the fact M so interested in this you can partially but blame on top brand you never make you take an oath to preserve protect and defend the quantum tutor and so in the classical world then the pet since about 2008 there's been some work utterance called polar codes which have pretty remarkable properties and with be going to this information theory conferences for the past few years and so we decided that we should be a good idea to see ahead of make these codes work in the quantum domain in parallel or not surrendering his group in Zurich it turns out that a rental lives in the same apartment complex as the chief editor electrically Transactions on Information Theory and they have these birthday parties for their kids in of this information the people averted parties and rat serenity s the the chief editor what he thought the biggest advance in coding theory in the past 10 years was and the chief editor said well without a doubt it would be polar codes so their group started and about the same time we had papers appear in the archive so we we did this classic alkaline I did these 2 papers here and I recently 1 of co-authors on the paper as we put our heads together and we came up with that of polar codes that can achieve this of exotic superactivation the graphs with was talking about where 2 0 capacity channels the together have nonzero capacity it said this is a the explicit code construction for doing that OK so I'm
just a little motivation from quantum Shan theory quantum information theory we have a pretty good idea of capacities of channels for sending classical data a private classical data for quantum data only only for certain channels we have this idea but there's still some open problems and and so there's been a fair amount of work on quantum Turbo codes of by the people on others in an attempt to achieve the capacity of 1 channel but also Kwan LDPC codes and then there is explicit constructions when you can actually these codes the capacity achieving and the actually very little work on codes for sending classical or private data even though this is an important problem we can actually demonstrate a quantum supremacy as was said last night and turns out for a bosonic channel if you have your detector perform a joint collective quantum measurement there can be a significant increase the amount of classical then you can send over this channel when there's a low photon number OK in and so as the saying this these polar codes are promising in the classical world and from what I hear the being incorporated into standards for communication so we thought why not explore the Kwan generalizations and result we found is that the scheme that's near explicit there's still some work that needs to be done in this direction so we can prove that the capacity achieving for certain channels
OK so also by describing this channel polarization phenomenon and it will do it for the Kwong case in particular I realize this is not a a CDC conference but I'm gonna start with the classical case the build up to the Qantas eventually all get a curiously near the in the talk we focus so will begin with a channel that is a classical Kwan channel CQ channel because w there's of classical binary input X and conditional on that some states in row X is prepared out but it doesn't matter that much of the state a case of sudden 0 conjuror 0 when 1 outcomes row 1 an important channel parameter is something I'll call the information of W of in more detail will call the symmetrical Lavoe information it's just the Mutual Formation the correlations you can establish between the input and the output when you select the input uniformly at random it's that's why it's called symmetric 11 information if the form a work out of this and then you know I can evaluate this information quantity with respect to this classical state which corresponds to and some like hair using this chip channel at the output OK and just to give you some idea that this parameter is equal to 1 of the channels perfect and it's 0 if the channel totally useful so the use of cases when the cable car with graphs with was in this talk OK so this is an important channel parameter and so the inventor of polar codes as a profession and R. E. Kahn and his idea was the take 2 copies of this channel and just do a classical see not care at the input and what you can observe this this information preservation conditions OK so if I could if I computed twice the information of the original channel FIL this quantity when you work it out and this is an information evaluate rhythm with respect to these 2 and inputs and since the state is reversible but it the information here is equal to the information of these but when these 2 inputs input but what we can use this something that's well-known Information Theory of the chain rule the Mutual Formation to break this information term into 2 parts OK and that this is very important for the polar coding idea OK and this chain rule it's
suggesting new that we can actually think about 2 different channels is a channel from the bit you 1 to the 2 outputs and a different channel from that that you to to the 2 operates with the new 1 side information OK so to give a picture of this this 1st channel is a bit channel from you want to the 2 outputs where this but is acting as noise not known to the receiver OK I'll call that W minus maybe think could be a worse channel than the other 1 where you want is available as side information rights and taking this I mean this channel here is a bit channel from you to to this these outputs you 1 and the 2 quantum outputs and what's happened is that this new 1 is available at the side and from OK so when you look at these pictures maybe it's already hinted at how the decoder could work in decoding these bits and we came up with is an extension of our Kuhn's ideas we call a quantum successive cancelation decoding it the name comes from information theory but the idea is that at the decoder would 1st decode you 1 assuming this is noise and other receiver would do this by doing a simple quantum hypothesis tests for a conditional on you 1 there's some statements prepared the output of acknowledge the source said 1 and the receiver would just do the what color Hellstrom measurement project onto the positive eigen space the difference of these 2 operators the prepare the output OK and then assuming that the decoder corrected you 1 that that that the decoded decoded you 1 correctly it's that available as side information to do a 2nd quantum hypothesis tests to determine you to say that's the year behind the decoder
but what you do is you continue this construction workers a cake is really many are copies of the channel to to make some interesting statements OK so what I'm doing here is I'm taking I had this
channel this channel w to write this block here and I take copies of it in this recursive construction right someplace in them here and here this operation which that just places all the odd indices you want and new 3 1st from here and it's they can be even in a season but in that so in this case it just amounts to a swappable the middle 2 bits and then I do a see not on every pair of bits community case of the idea is just you know but to see not swap things around the knots walking around and happens her which builds it up but OK and you can use the same idea of the chain rule this information preserving condition of 4 times the original information of the channel using the chain rule will distribute in this way such that I can think of 4 different channels now the channel where the receivers trying to get you 1 1st conditional on that then you 2 and so on and so forth so picture that again just the point
home of the decoder is trying to get you 1 of these bits a random assuming gets this correctly he should too and these are random so on so for the continued threat to this OK so our you
continue this work but construction recursively many times and so the chain rule a recursively will be this just were you trying to get hit high condition on the previous 5 bits for the Prius I minus 1 bit and the this is the important statement of channel polarization OK so were thinking about these induced channels channels from bit parts of the channel upwards right and the statement of channel polarization is that a fraction of the channels will be completely perfect OK you can prove this and the complementary fraction will be totally useless OK and the fraction of channels that are perfect is equal to the capacity of the original channel in the fraction channels that are complementary number OK see to prove this result using that model 3 that's right conjured and reuse Kwan generalizations of Connes inequality for this result that so I'll go into that in a little detail to give you some idea of how it works there's a different parameter we can think about that will call the fidelity of the 2 output states now to mine work were not considering Kwan communication here were considering classical Communications of the interpretation of fidelity is a little different we interpret as a distinguishability measure it's a how well can distinguish rose 0 from row 1 quietly we know it's this whole Richman normally these 2 states and so the fidelity actually is 0 if the states orthogonal if they're perfectly distinguishable by some measurements and it's equal to 1 if the states completely indistinguishable OK so it's it's since of classical transmission it's it's a different protection OK and this generalizes the the classical fidelity which in the classical world of the court by a charter press OK and certain operational interpretation of this quantity and this is a parameter of the channel CQ channel if I'm trying to do a quantum hypothesis test thing was that from each other then that the fidelity serves as an upper bound on this error probability and we use this in in the proof to show that the scheme works well OK so you'd expect these parameters to be related in particular if the channels geared for sending classical data you expect the states to be distinguishable rates of the of the channel if the information is 1 of the channels perfect and that the state should be indistinguishable and if the information is 0 if the channel is useless you'd expect the state state of the completely distinguish OK and you can make this precise with these bonds here right so if the fidelity is 0 so this will this will this term here will lower bound the information which the shows 1 and if the fidelity is 1 of the states the not distinguishable and then the information so we can prove this this bound using far results of Lavoe and some recent results of program and others OK and what's nice about this is you can prove things about holidays in little inclined similar things about the information quantity and by these bands through 10 so recall this recursive channel construction we had before we do this many times and and they're called the channel induced from the i-th that's a say I too would be thinking about you too in the nth recursion level so here the recursion levels to this sum bit channel there the only and what you can show in this recursive construction is that the information of the i-th bit there and the next iteration of the person will spread away from the middle OK so there will be 2 of places in the circuit that will serve spring forth from the i-th bit channel and what happens is they they move away from the center of this construction OK so this is actually how the polarization effect takes hold of their the this information choirs are moving away from the middle of Central between 0 and 1 they're being pushed Towards an extreme 2 words your 1 OK so that's that's what this is showing and at a complementary sequence a set of inequalities holds for the fidelity perhaps but then the some other inequalities and you need to show this but this is the basic idea OK and we can prove that all stations of items ideas OK so then were thinking of this the splitting process the way that the charter created under this recursive encoding and you can actually model the chill splitting process as a random birth profits OK so I take the i-th bit in ants recursion level and I'd represented as a binary sequence in this tells you where I'm going on the street right so all this is a random process and what you can show is the fidelity of the i-th channel at where I is a random parameters right it's this it's this binary sequence this process is a Martin and you can use of martingale convergence results and ii the inequalities in the previous slides to show that this random variable will converge to a 0 one valued random variables that's the polarization but and what you can show is that the probability that this side channel is 0 the fraction channels good is equal to the capacity so that's the basic idea that we were able to show the point is so what you had this immediately leads to your code instead right minutes pretty
obvious and there and if you know which channels of the good ones so this is sort of the existence parts and we're users a probabilistic argument to say that this effect will take hold so if you know which channels of the good ones you send your information that's for those and if you know the good ones the bad 1 so you just send a pre agree upon values for themselves to those other channels but and then some the decoder as this quantum successive cancelation because she performed the sequence of conditional hypothesis tests 2 1st to figure out the bit and so on the key tool that we use to analyze the airport shows that the scheme is in fact a pathogen is a recent bout of pre perhaps and found in the context of quantum information theory and I call this bound the noncommutative union bound as I've said a different conference I think this bound is really important and can be used in many different contexts and it generalizes the union bound from probability theory and this is used all the time in many a probabilistic arguments and so the reason this is like a union bound there may actually go to war the so Europe which is the show generalized the so the union bound from probability theory to remind you have less if I'm interested in computing a correct the probability for a correct sequence of events they can think about and of the and the I want the intersection of a princess correct sequence of events right so I can instead analyzed the complements of that but so this 1st term here corresponds to right so I do so I applied to Morgan's law right panel yet the union of the complements of events then using in the union bound this bounded by the sound of the complements of images of individual events so that's that's what this is doing right except with trajectories so this sequence of projectors course monster correct sequence of events and 1 minus that is the complement in its upper bounded by the complements of each individual problems these projectors with the exception of there's a 2 square in front and that comes from the knocking at the if the project is to meet you can remove the 2 square root getting about but the doubts in general you need this to square root and find counterexamples were we demonstrate that this is needed OK you don't think so the idea is we can balance each of these events the events for the individual channels by an exponentially decreasing terms and there's a linear number of them so there's an exponential find a linear term in the fall of the that's the basic idea behind the scheme and Bernhard she's the classical OK so now I realize that this is not the PC conference that of the conference for private part indications that we need to finally get to the Jewish again so we have this is in relation to what they Printers talking about you relations between and private data and to that and so as saying that we have a quantum wiretap channel there's a class a lot at child there's some classical binary input it leads to the art with shared between me right so we can think about the channel to bob just by iteration of relief and we can think about the channel to you just penetration of about nickel tells what w want you don't you start and what's known is that if the channel is degradable meaning that Bob can stimulate new by points and channel is open to other sciences it is output is not as noisy as these then you can prove that the private capacity is equal to this information to the ideas of science you know about private information consent is just the difference between what bop units and what is that and so what happens in this setting we have 2 different channels ready to polarize in their own way this can be channels there would provide about about the ones that are good for an even better the so overall the channel will polarize in 4 different ways OK will be those that are good for bob and good for you as well and we don't want you to get any information so we will be sending information that those 2 the good we want to randomize which is dead and so we send random that's the but and this is this all is important because it's going to make coherent version with slides OK ones that are good for bobbin battery well and that's where we can put our private data because you will you get the information there about well and so we just send the information that's into those and finally the there that will not find it there once they're bad for bobbing good for you and we can't guarantee that Bob will be able to decode the in there on but leave might be able to do so we need to randomize the so we need something that Bob will have access to in those faithfully we also need to randomize the so the best thing we can put in there and has a secret key I don't 3rd point tomatoes yet and we can show that the amount needed here the consumption reciprocating goes to 0 for certain and and then and then finally there channels that are bad for bargain bad for you and since you can get those will just put the insole that's there the Bob notices such that it can help from the information that sent throughout OK and what we can so is that the channel is degradable such that the map to the environment is classical so there's some density operator for ro 01 and if it's diagonal and some basis those and this is what I call a channel degradable classic environment then we can show that the scheme provably achieves the wiretap capacity using this successive cancelation now they're actually many sales that have this property that amplitude damping channels the phasing channels cloning channels rated channels them and so this this is a useful scheme and what we can also shows that this reciprocated require gives the Zero Net
OK so now finally where at the the basic idea is to run the wiretap codes into it these are ideas that were used by then attacked approval but the coherent information is an achievable rate for quantum communication and so on and so what we're doing instead is using a coherent version of the same encoded the encoder just gonna to be seen NOT gates and swapped right and when you consider the channel will be producing any channel will have an extension to the environment as well as part of the overall channel unit area and so this this induces a wire attached so then there will be channels that polarized in 4 different ways just like we have the wiretaps it's a little bit channels that area good for bob and good for remember from before we put random that's in the system and I mean there were just put the coherent version of a random bit which cluster into those channels their channels the good for Boston bad for information units and the but there were those channels that were bad for bob and good for so remember the bad for Bob so we need something there they can help and decode Wall what a randomized the with Pacific Jean but the before version secret that isn't even entangled the bell states but remember for certain channels we can show this entanglement consumption rate goes to 0 and then finally there were these channels that are bad about that for you hands-on science a better if we can put something that's agreed upon for Bob and Selten decay and so that so of the
decoder in this case the scalar consists of 2 steps but remember we had a measurement this quantum successive cancelation because the that worked well for getting the private information and so we're gonna run a coherent version of that any measurement you can do a sort of as a controlled operation sold in doing that and so that's the 1st then find the 2nd step in it is a controlled the coupling unitary it's guaranteed by all means give to this part is not efficient The 1st part does not be there but it's explicit OK there's a linear number 1 hypothesis tests and for poly channels Renner and others were able to show that it is that it is efficient but then and so in analyzing the proof of this scheme can possibly we just use the fact that the the decoding machine with the decoding measurement is reliable and also that it's secure OK to guarantee the quantitative in the cold war a case indicating circuit will look for who knows this this is the encoder consisting of all these scene arts and permutations and Alcom something for evolving something for you this part right here without this is just the measurement that was this quantity of cancelation because a can writer coherently and taking the outcomes and putting them in the Z registers coherently and finally this is the speech control to companies a case this was the the 1st thing we have in this paper then I
recently within the past 2 weeks working with children as and we can use some of his ideas from his scheme to come up with a slightly different spin but there are relations between the sky OK so during this is written all these papers about how sending quantum data is like sending classical information continental based and so were using this idea except in this explicit context where we have this whole occurred constructions OK so the way he thinks about quantum information I've had to learn recently in some if we look on channel we can think about a classical 1 channel where were sending some bit into the amplitude basis which is used to basis of the linear but and then if were able to decode that successfully coherently we can think of the next channel in line as this phase encoded channel where a bell state is available at the this this comes about from the decoding process through creates these bell states have phase information can not about so this is this phase encoding channel and his coding scheme is well so channels that are good so channels that are both good for amplitude and good for phase the ascension which will be able to decode the shells because decoding Kwan data is like the code in a classical data well in continental basins OK and the channels that are good for Apple to bad for phase will lose some phase that's there to help body code of the phage child because the bad in this OK and ones that are bad for both PCs you will need halves of the that's there because in a sense of the that analysis of measurement locally is the item basis but will be able to predict the outcome of that measured and the same happens in the phase because of a phase measurement if the should does a measurement of the X operator locally and bomb We'll to predict the outcome of that measurement and related to that is so why we need the bits here because of bits will help decode the channels the bad about and then finally the to channels the bad for the purpose of basis but good for the phase basis and bobble need information and the the the amplitude basis to help them to curve right so we really want this archive number here so it's really symmetrical you know we wait for a while to get this nice pretty numbers here you for this does not to OK so the way that this works and we had this attitude channels that details and aperture basis and we know from the polarization results that this number of channels will be good for the application that and then from the phase results this system here is the half that's available to Bob for helping including the phase information so you know this number will begin and then what you can show is that of the net rate we just considered the sizes of the set in the basic set theoretic operations of the sizes of sets though that the net rate of quantum communication will be the sum of these 2 minus 1 and and so were subtracting off the bits that were used to communicate when you can show which I was done in previous papers of his favorite heroes the this rate will be equal to the coherent formation and so this is a picture I lifted from from 1 just papers and this is how the decoder would work 1st a decoding amplitude information the coherently copy to this end so register and then this this process is what induces that phase channel previous 1 with about it was available phase information coming on to get in a coherent copy of the amplitude information in this register OK so then the phase measurement run coherently acts on these 2 systems and copies of this register and finally doing hole in a number of control the 58 you can be a couple of the environment of this as a dynamical coupling with operation and so use linear linear number those you can to couple the environment gets Kwan data that house trying to transmit about what's nice about this is that this is explicit this is exploited and this is exposed this is a linear number of coherent 1 hypothesis 1 in succession this is the same type of thing so this this work reduces the problem of the decoder for general channels to figure out how to make these 2 things efficient again this is not so this is an improvement over the previous game but now what what sort of the the
cool statement we can make is that this scheme will achieve superactivation for the other channels that transmit John are you can also look at this from and so the 2 channels of that they gave word this 4 dimensional and put it in PPT channel positive portion that has 0 capacity the other channel by GM the argument was an erasure channel and it has a 50 % irrational and that has 0 Kwan path the case but put together all of these have nonzero capacity the result is a 16 dimensional channel so we can factor the 16 dimensions as a tensor product before inputs basins and so we can use this that of polar coding schemes to achieve the coherent formation and the way this works is you coherently decode this amplitude and phase variables in the Saudis' so 1st you go the amplitude variable for the 1st few the channel Interfax and then using that other side information in this coherent encoded because the 2nd amplitude variable for the the 2nd channel in the for full time at Nietzsche doing so after 4 rounds of get all the amplitude variables you can use those inside information coherently Kwan side information to decode the phase variables this scheme will achieve their core information 1 thing we can't show then for these channels how to make the entanglement assumption rate goes here think before we needed and this condition channels were degradable but this doesn't hold for these examples so what they happened I should say why we consider this a superactivation is if the quantum capacity is 0 then also the catalytic 1 capacities Europe that many not obvious and the Catala quanta passengers your Langerman meant to be consumed but then the net rates of communication is many Q which you can get out minus entangler consumption a case so when you can show is that if the quark capacity is 0 the catalytic quantum pattern is also 0 because that's why what I consider the superactive you can subtract catalytic this that's all I have for now the polar coding scheme is nice what's known as the encoder the decoder problem thinkers efficient you can do it with all of em long and operations butterfly-like Fourier transform FFT tied operation can decompose it that way and the decoder is a linear number of quantum hypothesis tests which the working and there has been progress agile and rattle and Frederick if we had this paper on the archive with they show the the decoder made if the child have this this would be useful in addressing some of then Radisson's questions from and so this is the most important problem to make this because the efficient for general channels and then the other 1 of course is figured out which channels of the good ones and this would make the scheme of the so there's been progress in the classical information theory community this problem using approximation of the to compute which channels are the good ones and then we we may want to extend this to other scenarios like compression were lost due to thank you very much at the questions up in the classical case to achieve good bullet ation you need very long codes like to the 20 it is the case in quantum cases well as ours that's another and that's going to this very long your skin your village was in the U. B. or half of the bits than boys tends not backed but it didn't show up in your diagram so goes the other half to answer those 2 servants was the figures but there there that you could draw different diagrams so if you get out I will send there's shared between Alston beforehand and the years and you know the indices of the channels there good or bad for him in that it is is the outer half is already a very available long yeah that that's that's show it works but there there was some work in the classical world on these schemes for a prize the classical wiretap channel and when evaluated in practice how many secret key bits were needed the of the show in some cases that there were not me also for the Joe radius for power channels and they were official American they were not for a certain culture 1 last question he described a recursive construction of these induce channel's always wondering his apostasy beforehand which 1 of these 2 to the n induced tells you gonna get 4 for a fixed term to begin with the release of a the fixed channel the construction is a possible see which ones are going to be the good ones which ones are going to be about because we don't know evolution channel value actually in this and which ones the but to in general and there was work on these approximation of of OK could rest whether we intend to paper really a commission is hopeful Committee to the fact that she kicks in it relates to Alex's question is that right understood from the like that it's understood all I think it's the trail that that understood and that's because the block have long molecules and to the but such take my again and it could
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Metadaten

Formale Metadaten

Titel Polar codes for classical, private, and quantum communication
Serientitel Second International Conference on Quantum Error Correction (QEC11)
Autor Wilde, Mark
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 10.5446/35316
Herausgeber University of Southern California (USC)
Erscheinungsjahr 2011
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Informatik, Mathematik, Physik
Abstract Channel polarization is a phenomenon in which a particular recursive encoding induces a set of synthesized channels from many instances of a memoryless channel, such that a fraction of the synthesized channels become near perfect for data transmission and the other fraction become near useless for this task. The channel polarization effect then leads to a simple scheme for data transmission: send the information bits through the perfect channels and "frozen" bits through the useless ones. One contribution of the present work is to leverage several known results from the quantum information literature to demonstrate that the channel polarization effect takes hold for channels with classical inputs and quantum outputs. We construct linear polar codes based on this effect, and we also demonstrate that a quantum successive cancellation decoder works well, by exploiting Sen's recent "non-commutative union bound" that holds for a sequence of projectors applied to a quantum state. In addition, consider that Mahdavifar and Vardy have recently exploited the channel polarization phenomenon to construct codes that achieve the symmetric private capacity for private data transmission over a degraded wiretap channel. We build on their work and demonstrate how to construct quantum wiretap polar codes that achieve the symmetric private capacity of a degraded quantum wiretap channel with a classical eavesdropper. Due to the Schumacher-Westmoreland correspondence between quantum privacy and quantum coherence, we can construct quantum polar codes by operating these quantum wiretap polar codes in superposition. Our scheme achieves the symmetric coherent information rate for quantum channels that are degradable with a classical environment. This condition on the environment may seem restrictive, but we show that many quantum channels satisfy this criterion, including amplitude damping channels, photon-detected jump channels, dephasing channels, erasure channels, and cloning channels. Our quantum polar coding scheme has the desirable properties of being channel-adapted and symmetric capacity-achieving.

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