Detecting Incapacity
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Smith, Graeme
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.
 So this is Graham Smith who's curiously appropriately named talk is detecting incapacity. I think we found it. Beautiful. There we go. Okay. Sorry about that. Okay. So I want to thank the organizers for letting me come and give this talk. And like Todd said, what I want to talk about is detecting incapacity. This is joint work with John Smollin. We're still got, I got a bet going on about whether I'm actually going to be allowed to put that in the title in some real journal incapacity. If you're the referee, if you allow it, it's going to get me a bottle of wine. Okay. All right. So the problem that I want to talk about today is more about communications than doing logic. But I think it's still a really nice problem. So here's the idea. So classically, if you have some, you know, you have this whole theory of information found by Shannon in 1948. So you have some access to some noisy communication channel. It's got some input X. It gets mapped noise-elitism and output Y. And you have these parameters that describe the noise process, probability of getting output Y, given that you chose input X. And you'd like to transmit with low probability of error as many messages as you possibly can, given access to many independent copies of this channel. Right? And the beautiful thing is that this has a simple answer optimized over all these arbitrarily complicated communication schemes. And it turns out that there's just a simple formula for what's the best rate you can communicate at, given the channel. 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 the mutual information. You don't have to worry too much about that. And this is sort of a reasonably complete and really beautiful theory. And it's the sort of thing that we'd like to mimic or develop in the realm of transmitting quantum information and dealing with quantum noise. Okay. And let's see. So specifically, one nice thing about this theory is that it tells us which channels are useless. So this is a channel that I created in my lab at IBM. And, you know, it's got a sender here, receiver somewhere out that way. And there's no correlation between what the sender tries to transmit and what the receiver gets out here. So this channel is totally useless. And actually, in some sense, this is the only useless channel. Okay. So any channel that has some correlation between sender and receiver can actually be used via error correction to transmit noiselessly classical information. Okay. And what I want to 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. There's sort of, well, basically because quantum information is a lot more delicate, there are a lot more things that can go wrong. Or there are a lot more delicate, a lot more subtle things that can happen to your quantum state that will make it impossible to transmit quantum information than simply having no correlation between input and output. Okay. So this is my little picture of the zero quantum capacity channels that we really want to understand. So, you know, if this was zero classical capacity channels, then we would just have a single point. It's the uncorrelated channels. But here there's something more complicated going on. So there are two different sets of zero quantum capacity channels. They're both convex. This one, it's called symmetric channels or anti-degradable channels. And these are just the channels that, there's sort of a reason that they have no quantum capacity. The reason is that, you know, channels have environments and the environment of these channels, these symmetric ones, actually can simulate the output of the channel. So if there was any capacity for these symmetric channels, you'd be able to generate, you'd be able to clone the input state, right? So these guys have no quantum capacity because of the no cloning theorem. And sort of the main point of this talk, or at least the starting point when we started thinking about these ideas, was this other class of channels. They're called PPT for positive partial transpose. And I wanted to find some way to explain in some operational language or just something that didn't involve writing down a bunch of matrices and appealing to complicated mathematics what these PPT channels are and why they have zero quantum capacity. Okay, so the point of this slide is basically zero quantum capacity channels, very complicated set. We have these two known classes of zero quantum capacity channels. There could be other ones outside these sets, but we don't know. And we know that actually these guys can interact in very unusual ways. So you can take one of these PPT channels which has zero quantum capacity. You can mix it together with one of these symmetric channels that has zero quantum capacity. And the joint channel actually can develop quite a large amount of quantum capacity. Okay, this is the object that we're going to try to understand. Okay, here's how we're going to go about doing it. Yep, okay. So first I'll just sort of review these two different classes of zero capacity channels, the anti-degradable ones, and these positive partial transpose channels. And then I want to show you a new proof that these PPT channels have no capacity that sort of appeals to a more physical reasoning than the usual more mathematical thing. Like I'm not going to mention anything called LOCC, so, you know, I don't know. Some people don't like LOCC. LOCC is way worse than group theory, right? Okay, anyway. But then I'm going to take this kind of simple proof that these PPT channels have no quantum capacity and try to generalize it to search for a general way to detect whether a channel has any capacity. And the nice thing is that eventually we'll make it so general that in fact it includes these anti-degradable channels and these PPT channels. So we have this nice kind of framework for thinking about tests for quantum capacity. Okay, and the motivation is to try to unify our thinking about incapacity and try to appeal to some physical principles. Another motivation when we were starting this work was there are sort of non-quantum theories that you'd like to ask questions about, information theoretic questions, like these generalized probabilistic theories you might have heard of, or something by Schumacher and Westmoreland called modal quantum theory, kind of a nice generalization of, or a nice toy model of quantum mechanics where you have a vector space over a finite field rather than complex numbers. And it wasn't so clear how we could ask questions about this PPT criterion in these settings. But in this more sort of operational way of thinking about positive partial transport, it becomes a little bit clearer how to think about these things. Another motivation is that I like to be able to give sort of very simple explanations for why PPT, or for why things happen. So we're going to get a nice simple explanation for why these PPT channels have no quantum capacity. Any questions at this point? Okay. All right, so this is just a reminder of what the quantum capacity is. The idea is I have access, I have a sender over here, and I have a receiver over here, and she's trying to transmit some qubits by encoding them into the inputs of some noisy channel. This is a bunch of independent uses 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 qubits from the left to the right? And because I said it's a much harder than it is classically, I'm willing to accept much less. And actually I want to understand, I just want to be able to answer, well, can I send any qubits or not? I want to understand whether I have positive quantum capacity or not. And this is still actually quite a difficult question. Okay, so here's what we know about the quantum capacity at this point. There's something called the 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 channel's environment. I should remind you, I guess, that I can think of any noise process that maps psi to some input psi to an output b as a unitary interaction with some inaccessible environment that starts off in some pure state zero. And this turns out to be a useful way to think about these quantum channels when we're thinking about transmitting qubits. Okay, so this coherent information, which is kind of the analog of the mutual information from the Shannon formula for the capacity of a classical channel, is just this maximum difference in entropy of b and e, maximized overall input states to the channel. Okay, the reason I say it's the analog is that if you talk about random coding for a 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 quantum capacity, and in fact, if you do this thing called regularization, if you take this coherent information and evaluate it for many uses of the channel and normalize by how many channel uses you've got, then in fact that gives you a formal expression for the quantum capacity. I mean, unfortunately, this is a disaster if you try to calculate it. We know nothing about how this limit converges. We know that it does converge. And it turns out that, excuse me, that actually this q1, this coherent information can be strictly smaller than the quantum capacity, and in fact it can be zero when the quantum capacity is quite large. So it can be a very bad approximation. Alright, the point of 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 know how to answer. Okay, so here's a nice example of zero quantum capacity channel called symmetric channel. So what's a symmetric channel? I have an isometric extension of my channel, so I have some input state psi. It interacts with this environment that starts out in state zero. It's a unitary, and there's some output B and some environment E. And I call the channel symmetric. For any psi I put in, this state on BE is symmetric, so I might as well just swap the two. So if swapping the environment and the output of the channel leaves everything alone, then I call the channel symmetric. And the example to have in your head is a 50% attenuation channel. Okay, so I have some fiber, it's got an input mode. The environment it interacts with is some vacuum. Okay, and I ring these things off a beam splitter, and it's 50-50, so half the signal goes down to the environment, half the signal goes to the output. Alright, and this is more or less symmetric. Okay, so now I want to show you a nice old argument for why symmetric channels have zero quantum capacity. Okay, so let's suppose that I was wrong and that they did have some quantum capacity. Well, the first thing to note is that quantum capacity is not a question about a single use of the channel, it's a question about many uses of the channel. You get asymptotically many, and you have to do some encoding across many uses, and you get to use the best encoding and decoding schemes. Okay, so let's say I had an encoder that I could send through many uses of the channel, and I could take the output of the channel and do a decoding and get psi out. Okay, so now transmitted some, say, qubit state from here over to there. Fine, I want to get a contradiction, though. So, remember this guy's symmetric, so who says I have to decode here? This guy's, I could just as easily decode over here. I could just as easily decode on both sides. And now what have I done? I've taken my input state psi through some encoding, sent it through the channel that's symmetric, done decodings on both sides, and I get two copies of psi out. Right, this is a violation of no cloning. Okay, this is actually not a linear operation. It's completely unphysical. 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 arc, because you can give the illusion of understanding to somebody who asks you why is the symmetric channel have no capacity. The reason is otherwise you'd be violating no cloning, and everybody knows about no cloning, right? And it's obviously impossible. Okay, so I want to try to do something similar to this, but for this positive partial transpose criterion. Okay, so what is P P T? So, basically, remember what transpose is. 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 it, you know, flip it symmetrically. Okay, so partial transpose is just, well, it's an operation defined on a bipartite system now, on AB. And basically what I do is I take the transpose of the B system, but not of the A system. Okay, so partial transpose, it's linear. It takes IJKL to IJ on A, K on B. Okay, so this is, well, first of all, this is not a physical map. Okay, this is actually, it's positive, but not completely positive. Okay, so it gives a nice test for whether a state has any sort of useful entanglement in it. So specifically, if I do partial transpose to a bipartite state, rho AB, and I find that it's not positive, right? So the original rho AB is, of course, positive. It's a physical state. But when I do this non-physical operation, partial transpose to it, it may not be positive anymore. In that case, the state is definitely entangled. Okay, but if it is positive, okay, it may be entangled and it may not be. If it's separable, it's definitely positive, but there do exist these entangled states that have positive partial transpose. Okay, and whether it's entangled, well, even if it's actually very, very noisy, and it's so noisy that the entanglement in it is bound, which means that there's no, you can't do any distillation protocol to make sort of perfect pure entanglement again out of the entanglement in this rho AB. Okay, and a PPT channel is just a channel that enforces PPT between its output and the purification of its input. Okay, said another way, it has a Choi matrix that is PPT. So if I take some state and put it into the channel, the resulting state is always PPT. Okay, and in fact, because of this bound entangled property of PPT states, any channel that is PPT actually has zero quantum capacity. Okay, so now what I want to do is give you a short proof that PPT channels have no quantum capacity. Okay, so first of all, I'm just going to give you some notation. So let's let T of rho just be the transpose of rho. And you can check from what I told you before that any channel is PPT exactly when T composed with the channel is completely positive. So it has to be physical. All right? So you should note, first of all, not every channel is going to be PPT. For example, if I let the channel just be the identity, T composed with the identity is just T, which is not physical. It's not completely positive. Okay, so this identifies a certain subclass of channels that are very noisy, and I want to show have no capacity for transmitting quantum information. Okay, so let's suppose that we could transmit quantum information with the channel. That is PPT. Okay, so what does that mean? That just means that there's some encoder. I can feed my state into the, I can take the output of the encoder, feed it into the channel. I can take the output of the channel, feed it into a decoder, and get the state back again. Okay, so that's just what it means to transmit quantum information. So now what I want to do is act on both sides of this equation up here with this transpose operation. Excuse me. On the left, I just get transpose of the state, and then on the right, I get transpose composed with decoder, composed with channel, composed with encoder acting on the state. And transpose has this nice property that actually I can commute it through this decoding operation. So I'll say a little more on the next slide, but the idea is there's another physical map associated with this D. Actually, it's just got the same crowd operators, but you take the complex conjugate of them. And if I apply transpose, or say the channel followed by the transpose, I get the same thing as if I had applied transpose first and apply this D star, or the complex conjugate of the channel second. So that's the sort of nice property of transpose that I want to use. But now we're in business because actually on the left-hand side, what have I got? I've got some encoder. I've got this joint channel, so it's the transpose composed with the channel. That's a physical operation by hypothesis, and it's composed with this complex conjugate channel, which is also physical. So on the right-hand side, what I've got is some physical operation. And on the left-hand side, what I've got is the transpose of the state. So this is the contradiction that we're looking for. So assuming that there was an encoding and decoding pair actually allows us to derive this contradiction as long as the channel is PPT, and we're done. And if you want to talk about capacities, actually you need to talk about many uses of the channel, and you have to worry about this being only approximately true. But because transpose is continuous and actually many copies of a channel is PPT exactly when the channel is PPT, we're in business. This is a good proof that the quantum capacity of a PPT channel is zero. Todd, how long do I have? Okay, so now what I want to do is try to generalize this idea, this sort of simple proof of the zero capacity of PPT channels. So, uh-oh, good, good. I spoiled the surprise, but anyway, basically I want to talk about, I want to formalize this argument about PPT channels so that I can include different kinds of tests for quantum capacity. And actually I want to be able to take any unphysical, or almost any unphysical map on quantum states and turn that into some test for whether, or some test for channels having any quantum capacity. Okay, so I do it with a picture, and can anybody tell me the name of this picture? John Seem, we give this talk before, that's how he knew. Oh, I guess it's the first time. Yeah, good. This is the house diagram, and I really like it. And here's the idea. This is the kind of property that we want to have, want our unphysical channels, our unphysical map, okay? Just focus on the bottom part of the house, the foundation, okay? What I want is, I want a way to sort of commute my unphysical channel, my unphysical map through any physical channel. So, here I've got R composed with D, and I want to turn that into some other D star composed with R, right? So, that's what this diagram shows me, shows happening. Here I have, I apply R, let me see, let me see, let me see. Okay, here I can either apply D and then apply R, or I can apply R first, and then apply some second channel that's related to D called D star, and get to the same place over here, okay? So, the point is, think about starting here, you can either apply D and then R, or you can apply R and then D star, okay? And any unphysical map that has this property actually, well, P commuting, okay? And the idea is that this unphysical map can be commuted through any physical operation with some additional modification on the operation on the left-hand side, okay? And the little lemma that you can prove is that if R is unphysical on some set of states, S, and R composed with your channel is physical, then the channel can't be used to reliably transmit states from this set, S, okay? And the proof of the lemma involves both the foundation of the house and also the roof. And what we're going to do is go through the roof. Okay, so here's how it goes. So, this top part says, well, let's suppose I have my state at node one, and there's some way for me to encode it so that when I send it through the channel and then decode to the original state that I encoded. Okay, so what I want to do is use the fact that this entire diagram commutes to actually show you, based on that assumption, based on the assumption that this guy exists, and also that N composed with this R is physical, that in fact I can implement physically this unphysical operation R, okay? So here's the idea. We have the encoder, channel, decoder, that gets me here to psi, okay? But actually what I'm interested in doing is getting down to number five, jumping from one to five, that's unphysical. However, if this roof exists, then I can apply my encoder, I can jump from two down to four because this R composed with N is physical, okay? And then I can upd star of the encoder to get all the way to five, right? So the point is, if this leg is physical, then these guys can't exist, otherwise I could implement this unphysical operation, okay? And actually for any, for any, for any peak commuting unphysical operation like R, I can get a criterion now for having zero quantum capacity. And phrased in this way, I can now apply it to theories of modal quantum mechanics or whatever. Maybe it's not the right crowd for that, but in terms of foundations, it may be interesting to ask about what are the capacities of noisy channels in other non-quantum theories. Okay. So I told you this whole story about peak commutation and giving a general criterion for weather, for incapacity. And you might ask, well, is this more general? Because the only example you've talked about so far is this transpose operation. And it turns out that in fact, well, a priori it's more general, but if you do a little work, you can actually show that if you're talking about linear maps, okay? Now, linear unphysical maps that in fact transposes it, okay? And here's a little lemma that basically says that, with a little bit more math. And the basic idea is from group theory, and it concerns faithful representations of the projective unitary group. Basically, there are only two of them of the right dimension, and one of them corresponds to the identity or sort of the perfect channel, which is physical, and the other one corresponds to the transpose. And as a result, if you're interested in using sort of peak commuting unphysical maps that are linear, you're kind of sunk. That's kind of sad to me because there are lots of unphysical maps that are linear that I was hoping to use, these indecomposable positive but not completely positive maps. And it turns out that this argument is not going to work to show that one of those, well, will show that those guys give criterion for quantum communication capacity. Okay, but so the answer is no, it's not quite more general, but I can make it more general. I'm just going to refine my definition. And the idea is because I have to get away from these linear unphysical maps, it gets very difficult to make sure that things are peak commutative. Which means, well, you remember I wanted to sort of be able to push my unphysical map through my physical map in such a way that instead of applying the unphysical map to the output of the decoder, I could apply the unphysical map to the input of the decoder and apply some modified decoding operation instead. Okay, so we can actually generalize this idea of peak commutation. The idea is right here, basically I only have to, well, this is my unphysical map, r, it's maybe nonlinear. And I now have, well, I have the same equation here, I can commute this r sub d through d, but now I've given myself a little bit more freedom basically because I allow this r sub d to depend on which decoder it is that I'm trying to commute through. And now all of these guys have to be unphysical maps. And that gives me now a new criterion for whether a channel can quantum information. It's, again, of the same form. If this unphysical map r, composed with my physical map n, is physical itself, then n can't send quantum information. And I should take some time to introduce some, I think, useful terminology again. When this happens, we say that n is so noisy that it physicalizes r. Okay? Or you could say that r is so unphysical that it incapacitates n. Okay, these are, I think, useful, useful pieces of terminology. Okay? So now what I want to do with this more general idea of peak commutation, I want to 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 are these anti-degradable or symmetric channels. And the idea is, well, this again just says in kind of in math talk, 10 minutes, okay, that, okay, first let me just remind you what an anti-degradable channel is. It's one of these symmetric things, right? It's like a channel n is anti-degradable if there's a second physical channel, n1,2, that has two outputs. And when I trace out the second output, I get the original channel. Or if I trace out the first output, I also get the original channel. Okay, so this is just a slightly more mathy way of saying this symmetric extension thing. Okay, and now, given one of these channels, I can find a map that's unphysical that actually incapacitates the channel. Okay, how do I, roughly 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 operation. Okay, and it's a little calculation you can do, but this map R is not physical. And the way you see it is that it can clone. So it can take its input to this map. If you trace out the two system, the right hand system of this output, actually is just the identity, but it's also the identity if you trace out the left hand system. Okay, so this is a map that clones. You might be a little worried about this, but just, I'll say one word, which is pseudo-inverse, and this gets a little more complicated, but basically you can make this work. Okay, and then we can actually define this star operation that we needed, because what we want to do is show that this R map, which is unphysical, also has this P-commutative property, which is that if I apply R to a channel, I can equally well apply R first and some second channel second. Okay, that's the idea that we want R to be able to commute through any decoder in this house diagram. And for unitaries, this is easy. You just let it be u-tensor u. And in fact, you end up with, well, this is the sort of u-dependent R that we had for the more general motion of P-commutation. And basically, the short story is that this is how it works. This works. Okay, so now we can draw this house picture and we can talk about the reason for symmetric or anti-degradable channels having zero quantum capacity. Well, the reason is otherwise it would have no cloning. We can talk about the reason for P-P-T channels having no capacity. Otherwise, it would give a physical implementation of this transpose operation, which is unphysical. And there's this nice sort of unified picture for describing both of them. Now, this is sort of a challenge. So the point of this story was to try to better understand positive partial transpose and why these P-P-T channels have no quantum capacity. But here is, I think, an 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 tell you about it and we'll see whether you like the idea. Okay, so first this is a picture of teleportation. I have some state psi and I want to teleport it up to here. So what do I do? I take some maximally entangled state. Psi comes along, interacts with one half of the maximally entangled state. We do a measurement. I send some classical information along to the guy who has the other half of the maximally entangled state. Conditioned on that classical information, he does some rotation and out pops psi. So that's how regular teleportation works. Okay, and a way that one could think about this is, well, I take this psi, I do this measurement. I split the information about psi up into two pieces. One is classical information that gets transmitted up here. And the other is quantum information and it sort of creeps along back in time down around this bend and up over to here. And you put them back together again and out pops psi again. Okay, so now we can actually talk about doing teleportation with noisy states here. And the rate at which you can transmit information from the left to the right is related to the capacity of the channel that's associated with this noisy state. Okay, and you can tell the same story. The psi comes in, you do some measurement. Classical information goes that way. Quantum information sort of goes back in time, goes around the bend, up pops out over here and out comes psi again. Okay, but now suppose that this state rho A B is actually partial transposon variant. Okay, so if I do a partial transposon B, let's suppose, or A over here, let's say that the state pops out. Well, the state is the resulting state is the same as before. First of all, this state has to have no distillable entanglement. So I actually can't do any teleportation through the state if it's partial transposon variant because it means it's also PPT. Okay, so why is, here's the story that I want to tell you. Well, so we have this information, it's coming along, and we do this measurement, and the classical information goes along forward just fine. But this quantum information, it's supposed to go back in time, but because this state is actually a partial transposon variant and transposed corresponds to time reversal, in fact the information gets stuck in here, it gets confused, it doesn't know which way is back in time and which way is forward. So it can't wrap around the bend and pop up out to here again. So that's, I think, an even more intuitive way of explaining why PPT states have no distillable entanglement and PPT channels have no quantum capacity, but I didn't really see a way to make that more rigorous. So if you have any ideas, that would be great. All right, quick summary. So there are these channels with zero quantum capacity. They're not trivial in the way the classical channels with zero quantum capacity are, and it's actually a difficult problem to characterize, which noisy channels can transmit information and which can't. There are these two known tests for incapacity, symmetric extension and PPT, and you can understand both of them as special cases of this house diagram. Okay, and the thing I like about this is that it gives an operational proof of PPT channels having zero quantum capacity, because otherwise if they had some quantum capacity, we could implement this on physical time reversal operation. And to go beyond PPT, we actually need nonlinear R, and by doing that we can actually recover this symmetric extension. Okay, I'll just leave the questions, and maybe you have some too. I always admire someone who finds the most intuitive explanation, one that involves sending information backwards in time, so kudos. Any questions? We have time for maybe one or two. So you're using this teleportation thing as a stand-in for your channel, for your PPT channel, is that the right idea? So what's the obstacle you think would make it more generally rigorous? Well, because it was the part where I said, well, if the state is in very another partial transpose, the information gets confused about going back in time. So that's the hand-wavy bit? Yeah. Okay, but I mean the picture here is very similar to these so-called post-selected CTC setups. Yeah. So anyway, I don't know if that makes any intuitive difference, but okay, any other questions? Okay, could we have the next speaker come up? Let's thank Graham again.
 So this is Graham Smith who's curiously appropriately named talk is detecting incapacity.  I think we found it.  Beautiful.  There we go.  Okay.  Sorry about that.  Okay.  So I want to thank the organizers for letting me come and give this talk.  And like Todd said, what I want to talk about is detecting incapacity.  This is joint work with John Smollin.  We're still got, I got a bet going on about whether I'm actually going to be allowed to put that in the title in some real journal incapacity.  If you're the referee, if you allow it, it's going to get me a bottle of wine.  Okay.  All right. So the problem that I want to talk about today is more about communications than doing logic.  But I think it's still a really nice problem.  So here's the idea.  So classically, if you have some, you know, you have this whole theory of information found by Shannon in 1948.  So you have some access to some noisy communication channel.  It's got some input X.  It gets mapped noise-elitism and output Y.  And you have these parameters that describe the noise process, probability of getting output Y, given that you chose input X.  And you'd like to transmit with low probability of error as many messages as you possibly can,  given access to many independent copies of this channel.  Right?  And the beautiful thing is that this has a simple answer optimized over all these arbitrarily complicated communication schemes.  And it turns out that there's just a simple formula for what's the best rate you can communicate at, given the channel.  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 the mutual information.  You don't have to worry too much about that.  And this is sort of a reasonably complete and really beautiful theory.  And it's the sort of thing that we'd like to mimic or develop in the realm of transmitting quantum information and dealing with quantum noise.  Okay.  And let's see.  So specifically, one nice thing about this theory is that it tells us which channels are useless.  So this is a channel that I created in my lab at IBM.  And, you know, it's got a sender here, receiver somewhere out that way.  And there's no correlation between what the sender tries to transmit and what the receiver gets out here.  So this channel is totally useless.  And actually, in some sense, this is the only useless channel.  Okay. So any channel that has some correlation between sender and receiver can actually be used via error correction to transmit noiselessly classical information.  Okay.  And what I want to 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.  There's sort of, well, basically because quantum information is a lot more delicate, there are a lot more things that can go wrong.  Or there are a lot more delicate, a lot more subtle things that can happen to your quantum state that will make it impossible to transmit quantum information than simply having no correlation between input and output.  Okay.  So this is my little picture of the zero quantum capacity channels that we really want to understand.  So, you know, if this was zero classical capacity channels, then we would just have a single point.  It's the uncorrelated channels.  But here there's something more complicated going on.  So there are two different sets of zero quantum capacity channels.  They're both convex.  This one, it's called symmetric channels or anti-degradable channels.  And these are just the channels that, there's sort of a reason that they have no quantum capacity.  The reason is that, you know, channels have environments and the environment of these channels, these symmetric ones, actually can simulate the output of the channel.  So if there was any capacity for these symmetric channels, you'd be able to generate, you'd be able to clone the input state, right?  So these guys have no quantum capacity because of the no cloning theorem.  And sort of the main point of this talk, or at least the starting point when we started thinking about these ideas, was this other class of channels.  They're called PPT for positive partial transpose.  And I wanted to find some way to explain in some operational language or just something that didn't involve writing down a bunch of matrices and appealing to complicated mathematics what these PPT channels are and why they have zero quantum capacity.  Okay, so the point of this slide is basically zero quantum capacity channels, very complicated set.  We have these two known classes of zero quantum capacity channels.  There could be other ones outside these sets, but we don't know.  And we know that actually these guys can interact in very unusual ways.  So you can take one of these PPT channels which has zero quantum capacity.  You can mix it together with one of these symmetric channels that has zero quantum capacity.  And the joint channel actually can develop quite a large amount of quantum capacity.  Okay, this is the object that we're going to try to understand.  Okay, here's how we're going to go about doing it.  Yep, okay.  So first I'll just sort of review these two different classes of zero capacity channels, the anti-degradable ones, and these positive partial transpose channels.  And then I want to show you a new proof that these PPT channels have no capacity that sort of appeals to a more physical reasoning than the usual more mathematical thing.  Like I'm not going to mention anything called LOCC, so, you know, I don't know.  Some people don't like LOCC.  LOCC is way worse than group theory, right?  Okay, anyway.  But then I'm going to take this kind of simple proof that these PPT channels have no quantum capacity and try to generalize it to search for a general way to detect whether a channel has any capacity.  And the nice thing is that eventually we'll make it so general that in fact it includes these anti-degradable channels and these PPT channels.  So we have this nice kind of framework for thinking about tests for quantum capacity.  Okay, and the motivation is to try to unify our thinking about incapacity and try to appeal to some physical principles.  Another motivation when we were starting this work was there are sort of non-quantum theories that you'd like to ask questions about, information theoretic questions,  like these generalized probabilistic theories you might have heard of, or something by Schumacher and Westmoreland called modal quantum theory,  kind of a nice generalization of, or a nice toy model of quantum mechanics where you have a vector space over a finite field rather than complex numbers.  And it wasn't so clear how we could ask questions about this PPT criterion in these settings.  But in this more sort of operational way of thinking about positive partial transport, it becomes a little bit clearer how to think about these things.  Another motivation is that I like to be able to give sort of very simple explanations for why PPT, or for why things happen.  So we're going to get a nice simple explanation for why these PPT channels have no quantum capacity.  Any questions at this point?  Okay. All right, so this is just a reminder of what the quantum capacity is.  The idea is I have access, I have a sender over here, and I have a receiver over here, and she's trying to transmit some qubits by encoding them into the inputs of some noisy channel.  This is a bunch of independent uses 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 qubits from the left to the right?  And because I said it's a much harder than it is classically, I'm willing to accept much less.  And actually I want to understand, I just want to be able to answer, well, can I send any qubits or not?  I want to understand whether I have positive quantum capacity or not.  And this is still actually quite a difficult question.  Okay, so here's what we know about the quantum capacity at this point.  There's something called the 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 channel's environment.  I should remind you, I guess, that I can think of any noise process that maps psi to some input psi to an output b as a unitary interaction with some inaccessible environment that starts off in some pure state zero.  And this turns out to be a useful way to think about these quantum channels when we're thinking about transmitting qubits.  Okay, so this coherent information, which is kind of the analog of the mutual information from the Shannon formula for the capacity of a classical channel,  is just this maximum difference in entropy of b and e, maximized overall input states to the channel.  Okay, the reason I say it's the analog is that if you talk about random coding for a 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 quantum capacity, and in fact, if you do this thing called regularization,  if you take this coherent information and evaluate it for many uses of the channel and normalize by how many channel uses you've got,  then in fact that gives you a formal expression for the quantum capacity.  I mean, unfortunately, this is a disaster if you try to calculate it.  We know nothing about how this limit converges. We know that it does converge.  And it turns out that, excuse me, that actually this q1, this coherent information can be strictly smaller than the quantum capacity,  and in fact it can be zero when the quantum capacity is quite large. So it can be a very bad approximation.  Alright, the point of 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 know how to answer.  Okay, so here's a nice example of zero quantum capacity channel called symmetric channel.  So what's a symmetric channel? I have an isometric extension of my channel, so I have some input state psi.  It interacts with this environment that starts out in state zero. It's a unitary, and there's some output B and some environment E.  And I call the channel symmetric. For any psi I put in, this state on BE is symmetric, so I might as well just swap the two.  So if swapping the environment and the output of the channel leaves everything alone, then I call the channel symmetric.  And the example to have in your head is a 50% attenuation channel.  Okay, so I have some fiber, it's got an input mode. The environment it interacts with is some vacuum.  Okay, and I ring these things off a beam splitter, and it's 50-50, so half the signal goes down to the environment, half the signal goes to the output.  Alright, and this is more or less symmetric.  Okay, so now I want to show you a nice old argument for why symmetric channels have zero quantum capacity.  Okay, so let's suppose that I was wrong and that they did have some quantum capacity.  Well, the first thing to note is that quantum capacity is not a question about a single use of the channel, it's a question about many uses of the channel.  You get asymptotically many, and you have to do some encoding across many uses, and you get to use the best encoding and decoding schemes.  Okay, so let's say I had an encoder that I could send through many uses of the channel, and I could take the output of the channel and do a decoding and get psi out.  Okay, so now transmitted some, say, qubit state from here over to there.  Fine, I want to get a contradiction, though.  So, remember this guy's symmetric, so who says I have to decode here?  This guy's, I could just as easily decode over here.  I could just as easily decode on both sides.  And now what have I done?  I've taken my input state psi through some encoding, sent it through the channel that's symmetric, done decodings on both sides, and I get two copies of psi out.  Right, this is a violation of no cloning.  Okay, this is actually not a linear operation.  It's completely unphysical.  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 arc, because you can give the illusion of understanding to somebody who asks you why is the symmetric channel have no capacity.  The reason is otherwise you'd be violating no cloning, and everybody knows about no cloning, right?  And it's obviously impossible.  Okay, so I want to try to do something similar to this, but for this positive partial transpose criterion.  Okay, so what is P P T?  So, basically, remember what transpose is.  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 it, you know, flip it symmetrically.  Okay, so partial transpose is just, well, it's an operation defined on a bipartite system now, on AB.  And basically what I do is I take the transpose of the B system, but not of the A system.  Okay, so partial transpose, it's linear.  It takes IJKL to IJ on A, K on B.  Okay, so this is, well, first of all, this is not a physical map.  Okay, this is actually, it's positive, but not completely positive.  Okay, so it gives a nice test for whether a state has any sort of useful entanglement in it.  So specifically, if I do partial transpose to a bipartite state, rho AB, and I find that it's not positive, right?  So the original rho AB is, of course, positive. It's a physical state.  But when I do this non-physical operation, partial transpose to it, it may not be positive anymore.  In that case, the state is definitely entangled.  Okay, but if it is positive, okay, it may be entangled and it may not be.  If it's separable, it's definitely positive, but there do exist these entangled states that have positive partial transpose.  Okay, and whether it's entangled, well, even if it's actually very, very noisy,  and it's so noisy that the entanglement in it is bound, which means that there's no,  you can't do any distillation protocol to make sort of perfect pure entanglement again out of the entanglement in this rho AB.  Okay, and a PPT channel is just a channel that enforces PPT between its output and the purification of its input.  Okay, said another way, it has a Choi matrix that is PPT.  So if I take some state and put it into the channel, the resulting state is always PPT.  Okay, and in fact, because of this bound entangled property of PPT states,  any channel that is PPT actually has zero quantum capacity.  Okay, so now what I want to do is give you a short proof that PPT channels have no quantum capacity.  Okay, so first of all, I'm just going to give you some notation.  So let's let T of rho just be the transpose of rho.  And you can check from what I told you before that any channel is PPT exactly when T composed with the channel is completely positive.  So it has to be physical.  All right?  So you should note, first of all, not every channel is going to be PPT.  For example, if I let the channel just be the identity, T composed with the identity is just T, which is not physical.  It's not completely positive.  Okay, so this identifies a certain subclass of channels that are very noisy,  and I want to show have no capacity for transmitting quantum information.  Okay, so let's suppose that we could transmit quantum information with the channel.  That is PPT.  Okay, so what does that mean?  That just means that there's some encoder.  I can feed my state into the, I can take the output of the encoder, feed it into the channel.  I can take the output of the channel, feed it into a decoder, and get the state back again.  Okay, so that's just what it means to transmit quantum information.  So now what I want to do is act on both sides of this equation up here with this transpose operation.  Excuse me.  On the left, I just get transpose of the state, and then on the right, I get transpose composed with decoder,  composed with channel, composed with encoder acting on the state.  And transpose has this nice property that actually I can commute it through this decoding operation.  So I'll say a little more on the next slide, but the idea is there's another physical map associated with this D.  Actually, it's just got the same crowd operators, but you take the complex conjugate of them.  And if I apply transpose, or say the channel followed by the transpose,  I get the same thing as if I had applied transpose first and apply this D star, or the complex conjugate of the channel second.  So that's the sort of nice property of transpose that I want to use.  But now we're in business because actually on the left-hand side, what have I got?  I've got some encoder. I've got this joint channel, so it's the transpose composed with the channel.  That's a physical operation by hypothesis, and it's composed with this complex conjugate channel, which is also physical.  So on the right-hand side, what I've got is some physical operation.  And on the left-hand side, what I've got is the transpose of the state.  So this is the contradiction that we're looking for.  So assuming that there was an encoding and decoding pair actually allows us to derive this contradiction as long as the channel is PPT, and we're done.  And if you want to talk about capacities, actually you need to talk about many uses of the channel,  and you have to worry about this being only approximately true.  But because transpose is continuous and actually many copies of a channel is PPT exactly when the channel is PPT,  we're in business. This is a good proof that the quantum capacity of a PPT channel is zero.  Todd, how long do I have?  Okay, so now what I want to do is try to generalize this idea, this sort of simple proof of the zero capacity of PPT channels.  So, uh-oh, good, good. I spoiled the surprise, but anyway, basically I want to talk about, I want to formalize this argument about PPT channels so that I can include different kinds of tests for quantum capacity.  And actually I want to be able to take any unphysical, or almost any unphysical map on quantum states and turn that into some test for whether, or some test for channels having any quantum capacity.  Okay, so I do it with a picture, and can anybody tell me the name of this picture? John Seem, we give this talk before, that's how he knew.  Oh, I guess it's the first time.  Yeah, good. This is the house diagram, and I really like it. And here's the idea. This is the kind of property that we want to have, want our unphysical channels, our unphysical map, okay?  Just focus on the bottom part of the house, the foundation, okay? What I want is, I want a way to sort of commute my unphysical channel, my unphysical map through any physical channel.  So, here I've got R composed with D, and I want to turn that into some other D star composed with R, right?  So, that's what this diagram shows me, shows happening. Here I have, I apply R, let me see, let me see, let me see.  Okay, here I can either apply D and then apply R, or I can apply R first, and then apply some second channel that's related to D called D star, and get to the same place over here, okay?  So, the point is, think about starting here, you can either apply D and then R, or you can apply R and then D star, okay?  And any unphysical map that has this property actually, well, P commuting, okay?  And the idea is that this unphysical map can be commuted through any physical operation with some additional modification on the operation on the left-hand side, okay?  And the little lemma that you can prove is that if R is unphysical on some set of states, S, and R composed with your channel is physical,  then the channel can't be used to reliably transmit states from this set, S, okay?  And the proof of the lemma involves both the foundation of the house and also the roof.  And what we're going to do is go through the roof.  Okay, so here's how it goes.  So, this top part says, well, let's suppose I have my state at node one, and there's some way for me to encode it so that when I send it through the channel and then decode to the original state that I encoded.  Okay, so what I want to do is use the fact that this entire diagram commutes to actually show you, based on that assumption, based on the assumption that this guy exists,  and also that N composed with this R is physical, that in fact I can implement physically this unphysical operation R, okay?  So here's the idea. We have the encoder, channel, decoder, that gets me here to psi, okay?  But actually what I'm interested in doing is getting down to number five, jumping from one to five, that's unphysical.  However, if this roof exists, then I can apply my encoder, I can jump from two down to four because this R composed with N is physical, okay?  And then I can upd star of the encoder to get all the way to five, right?  So the point is, if this leg is physical, then these guys can't exist, otherwise I could implement this unphysical operation, okay?  And actually for any, for any, for any peak commuting unphysical operation like R, I can get a criterion now for having zero quantum capacity.  And phrased in this way, I can now apply it to theories of modal quantum mechanics or whatever.  Maybe it's not the right crowd for that, but in terms of foundations, it may be interesting to ask about what are the capacities of noisy channels in other non-quantum theories.  Okay. So I told you this whole story about peak commutation and giving a general criterion for weather, for incapacity.  And you might ask, well, is this more general?  Because the only example you've talked about so far is this transpose operation.  And it turns out that in fact, well, a priori it's more general, but if you do a little work, you can actually show that if you're talking about linear maps, okay?  Now, linear unphysical maps that in fact transposes it, okay?  And here's a little lemma that basically says that, with a little bit more math.  And the basic idea is from group theory, and it concerns faithful representations of the projective unitary group.  Basically, there are only two of them of the right dimension, and one of them corresponds to the identity or sort of the perfect channel, which is physical, and the other one corresponds to the transpose.  And as a result, if you're interested in using sort of peak commuting unphysical maps that are linear, you're kind of sunk.  That's kind of sad to me because there are lots of unphysical maps that are linear that I was hoping to use, these indecomposable positive but not completely positive maps.  And it turns out that this argument is not going to work to show that one of those, well, will show that those guys give criterion for quantum communication capacity.  Okay, but so the answer is no, it's not quite more general, but I can make it more general.  I'm just going to refine my definition.  And the idea is because I have to get away from these linear unphysical maps, it gets very difficult to make sure that things are peak commutative.  Which means, well, you remember I wanted to sort of be able to push my unphysical map through my physical map in such a way that instead of applying the unphysical map to the output of the decoder,  I could apply the unphysical map to the input of the decoder and apply some modified decoding operation instead.  Okay, so we can actually generalize this idea of peak commutation.  The idea is right here, basically I only have to, well, this is my unphysical map, r, it's maybe nonlinear.  And I now have, well, I have the same equation here, I can commute this r sub d through d, but now I've given myself a little bit more freedom basically because I allow this r sub d to depend on which decoder it is that I'm trying to commute through.  And now all of these guys have to be unphysical maps.  And that gives me now a new criterion for whether a channel can quantum information.  It's, again, of the same form.  If this unphysical map r, composed with my physical map n, is physical itself, then n can't send quantum information.  And I should take some time to introduce some, I think, useful terminology again.  When this happens, we say that n is so noisy that it physicalizes r.  Okay?  Or you could say that r is so unphysical that it incapacitates n.  Okay, these are, I think, useful, useful pieces of terminology.  Okay?  So now what I want to do with this more general idea of peak commutation, I want to 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 are these anti-degradable or symmetric channels.  And the idea is, well, this again just says in kind of in math talk,  10 minutes, okay, that, okay, first let me just remind you what an anti-degradable channel is.  It's one of these symmetric things, right?  It's like a channel n is anti-degradable if there's a second physical channel, n1,2, that has two outputs.  And when I trace out the second output, I get the original channel.  Or if I trace out the first output, I also get the original channel.  Okay, so this is just a slightly more mathy way of saying this symmetric extension thing.  Okay, and now, given one of these channels, I can find a map that's unphysical that actually incapacitates the channel.  Okay, how do I, roughly 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 operation.  Okay, and it's a little calculation you can do, but this map R is not physical.  And the way you see it is that it can clone.  So it can take its input to this map.  If you trace out the two system, the right hand system of this output,  actually is just the identity, but it's also the identity if you trace out the left hand system.  Okay, so this is a map that clones.  You might be a little worried about this, but just, I'll say one word, which is pseudo-inverse,  and this gets a little more complicated, but basically you can make this work.  Okay, and then we can actually define this star operation that we needed,  because what we want to do is show that this R map, which is unphysical, also has this P-commutative property,  which is that if I apply R to a channel, I can equally well apply R first and some second channel second.  Okay, that's the idea that we want R to be able to commute through any decoder in this house diagram.  And for unitaries, this is easy. You just let it be u-tensor u.  And in fact, you end up with, well, this is the sort of u-dependent R that we had for the more general motion of P-commutation.  And basically, the short story is that this is how it works. This works.  Okay, so now we can draw this house picture and we can talk about the reason for symmetric or anti-degradable channels having zero quantum capacity.  Well, the reason is otherwise it would have no cloning.  We can talk about the reason for P-P-T channels having no capacity.  Otherwise, it would give a physical implementation of this transpose operation, which is unphysical.  And there's this nice sort of unified picture for describing both of them.  Now, this is sort of a challenge.  So the point of this story was to try to better understand positive partial transpose and why these P-P-T channels have no quantum capacity.  But here is, I think, an 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 tell you about it and we'll see whether you like the idea.  Okay, so first this is a picture of teleportation.  I have some state psi and I want to teleport it up to here.  So what do I do? I take some maximally entangled state.  Psi comes along, interacts with one half of the maximally entangled state.  We do a measurement.  I send some classical information along to the guy who has the other half of the maximally entangled state.  Conditioned on that classical information, he does some rotation and out pops psi.  So that's how regular teleportation works.  Okay, and a way that one could think about this is, well, I take this psi, I do this measurement.  I split the information about psi up into two pieces.  One is classical information that gets transmitted up here.  And the other is quantum information and it sort of creeps along back in time down around this bend and up over to here.  And you put them back together again and out pops psi again.  Okay, so now we can actually talk about doing teleportation with noisy states here.  And the rate at which you can transmit information from the left to the right is related to the capacity of the channel that's associated with this noisy state.  Okay, and you can tell the same story. The psi comes in, you do some measurement.  Classical information goes that way.  Quantum information sort of goes back in time, goes around the bend, up pops out over here and out comes psi again.  Okay, but now suppose that this state rho A B is actually partial transposon variant.  Okay, so if I do a partial transposon B, let's suppose, or A over here, let's say that the state pops out.  Well, the state is the resulting state is the same as before.  First of all, this state has to have no distillable entanglement.  So I actually can't do any teleportation through the state if it's partial transposon variant because it means it's also PPT.  Okay, so why is, here's the story that I want to tell you.  Well, so we have this information, it's coming along, and we do this measurement, and the classical information goes along forward just fine.  But this quantum information, it's supposed to go back in time, but because this state is actually a partial transposon variant and transposed corresponds to time reversal,  in fact the information gets stuck in here, it gets confused, it doesn't know which way is back in time and which way is forward.  So it can't wrap around the bend and pop up out to here again.  So that's, I think, an even more intuitive way of explaining why PPT states have no distillable entanglement and PPT channels have no quantum capacity,  but I didn't really see a way to make that more rigorous.  So if you have any ideas, that would be great.  All right, quick summary.  So there are these channels with zero quantum capacity.  They're not trivial in the way the classical channels with zero quantum capacity are,  and it's actually a difficult problem to characterize, which noisy channels can transmit information and which can't.  There are these two known tests for incapacity, symmetric extension and PPT,  and you can understand both of them as special cases of this house diagram.  Okay, and the thing I like about this is that it gives an operational proof of PPT channels having zero quantum capacity,  because otherwise if they had some quantum capacity, we could implement this on physical time reversal operation.  And to go beyond PPT, we actually need nonlinear R,  and by doing that we can actually recover this symmetric extension.  Okay, I'll just leave the questions, and maybe you have some too.  I always admire someone who finds the most intuitive explanation,  one that involves sending information backwards in time, so kudos.  Any questions? We have time for maybe one or two.  So you're using this teleportation thing as a stand-in for your channel, for your PPT channel, is that the right idea?  So what's the obstacle you think would make it more generally rigorous?  Well, because it was the part where I said, well, if the state is in very another partial transpose,  the information gets confused about going back in time.  So that's the hand-wavy bit?  Yeah.  Okay, but I mean the picture here is very similar to these so-called post-selected CTC setups.  Yeah.  So anyway, I don't know if that makes any intuitive difference, but okay, any other questions?  Okay, could we have the next speaker come up? Let's thank Graham again.
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