The strong converse problem
What happens if we try to communicate quantum information through a noisy quantum channel at a rate above its capacity? At first sight, the answer seems obvious: it should not work. After all, this is what we mean by the quantum capacity Q(\mathcal N). Below capacity, there are codes whose error goes to zero as we use the channel more and more times. Above capacity, this is impossible.
But there is an important gap hidden in that statement. The usual, or weak, converse only says that above capacity the error cannot go to zero. It does not rule out the possibility that one could communicate at a rate above capacity if one accepts some loss in fidelity. Maybe the fidelity does not approach one, but it also does not approach zero.
A strong converse says that this cannot happen: as soon as the rate is above capacity, the fidelity must go to zero. An exponential strong converse says even more: it goes to zero exponentially fast in the number of channel uses.
For classical discrete memoryless channels, this sharp threshold has been understood for a very long time. For general quantum channels, however, the corresponding question has stubbornly remained open. There were many results for special classes of channels and various relaxations of the problem, but not for arbitrary finite-dimensional memoryless quantum channels. One fundamental difficulty is that quantum capacity itself is given by a regularized expression,
Q(\mathcal N)=\lim_{n\to\infty}\frac{1}{n}Q^{(1)}(\mathcal N^{\otimes n}),where the optimization for the coherent information, Q^{(1)}(\mathcal N^{\otimes n}), allows states that are entangled across all n channel uses.
This is the problem that two papers we posted this week resolve, by two completely unrelated methods.
The conclusion is simple. For every finite-dimensional memoryless quantum channel and every fixed rate R>Q(\mathcal N), the fidelity of any quantum communication code at rate R goes to zero exponentially fast with the block length.
The Arimoto route
The first project was with Hao-Chung Cheng. The starting point was an old and very natural approach to the strong converse: try to quantumize Arimoto’s classical strong-converse argument.
In fact, most of this argument was already known. For every Rényi parameter \alpha>1, one can define a Rényi version Q_\alpha(\mathcal N) of the regularized coherent information. A standard Rényi-divergence argument gives, for a code of rate R and fidelity F_n,
F_n\leq 2^{-n\frac{\alpha-1}{\alpha}\left(R-Q_\alpha(\mathcal N)\right)}.This is exactly the kind of estimate one wants. If we knew that
\lim_{\alpha\searrow 1}Q_\alpha(\mathcal N)=Q(\mathcal N),then for every R>Q(\mathcal N) we could simply choose \alpha>1 sufficiently close to one so that R>Q_\alpha(\mathcal N). The exponent above would be positive, and the strong converse would follow.
So the entire problem had essentially been reduced to this asymptotic continuity statement. The annoying part is that pointwise continuity at every fixed block length is not enough: one needs a bound that remains uniform as the number of channel uses grows. This is precisely where the regularization in the quantum capacity formula causes trouble.
Hao-Chung asked ChatGPT Astra to investigate this missing step. After a few rounds of prompting it produced a proof. The first version was correct but rather opaque, and it took us some time to understand what was actually going on.
What emerged was a rather surprising connection to an old paper of mine with Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr, and Serge Fehr on quantum Rényi entropies. In that paper, a minimax representation appeared as an intermediate step in establishing duality properties of conditional Rényi entropies. For a pure state \omega_{RBE}, and with s=(\alpha-1)/\alpha, it can be written as the minimax expression
2^{-s\widetilde H_\alpha^\uparrow(R|B)_\omega} = \inf_{\sigma_B}\sup_{\tau_E}\operatorname{Tr} \! \left[\omega_{BE}\left(\sigma_B^{-s}\otimes\tau_E^s\right)\right].The key observation Astra made was to differentiate the optimized expression with respect to the Rényi parameter. Once this is done carefully, these derivatives can be expressed in terms of ordinary von Neumann entropies evaluated on a family of tilted states. We realized that this naturally yields an integral representation of the Rényi coherent information, that is
Q_\alpha^{(1)}(\mathcal M) = \sup_{\rho_A}\frac{1}{s}\int_0^s\left[H(B)_{\zeta(u)}-H(E)_{\zeta(u)}\right] \mathrm{d} u.This changes the problem completely. Instead of directly comparing complicated Rényi quantities at different orders, we can compare ordinary entropy differences.
There is one catch. The tilted states \zeta(u) that appear in the integral need not themselves lie in the image of the Stinespring isometry of the channel. If they did, the expression inside the integral would simply be bounded by the usual coherent information. So the remaining job is to bring these tilted states back to the channel image without changing their entropies too much.
Here another older idea comes in. Debbie Leung and Graeme Smith, in their paper Continuity of quantum channel capacities, developed a telescoping technique that modifies a many-copy state one subsystem at a time. The important point is that the entropy cost at every step depends only on the local dimension, rather than on the dimension of the entire n-fold system.
ChatGPT adapted exactly this philosophy. The tilted states are projected back onto the allowed Stinespring image one channel use at a time, and conditional-entropy continuity bounds control the damage along the way. The total error grows only linearly with n. After dividing by n, we obtain a correction depending on \alpha and the local channel dimensions that vanishes as \alpha\searrow1.
And that is the missing asymptotic continuity.
Put back into the Arimoto bound, it gives the exponential strong converse for quantum capacity. The same strategy also works for the regularized Holevo information, so our first paper actually proves the corresponding exponential strong converse for unrestricted classical communication through arbitrary finite-dimensional quantum channels as well.
What I particularly like about this proof is the chain of ideas. A minimax identity that originally appeared as a technical intermediate step in a paper on Rényi entropy duality, more than a decade ago, turns into an integral representation; that representation turns a Rényi problem into an ordinary entropy problem; and an old continuity technique for channel capacities then finishes the argument.
I would not have guessed this route.
A completely different route
At the same time Salman Beigi and I were working on a somewhat different project.
Our original goal had been much more modest. Kondra, Brinster, Kampermann, Bruß, and Wyderka had recently proved an exponential strong converse for degradable and anti-degradable channels in their paper Sharp Quantum Capacity Thresholds. Salman and I wanted to understand the core argument for anti-degradable channels and rewrite it in a form that was more accessible to us.
This turned into a very useful exercise. We found that many ingredients of the proof could be simplified substantially. At the same time, one ingredient remained essential: a polynomial approximation of the Boolean NOR function. Understanding exactly why this polynomial was needed, and what it was really accomplishing, was crucial.
As usual, we kept our working notes with ChatGPT and repeatedly asked it to reorganize, simplify, and generalize what we had understood. At first we were able to push the argument from anti-degradable channels to Pauli channels. Then, once it had become clear from the parallel project with Hao-Chung that the general strong converse really ought to be in reach, the obvious question became: can this proof also be pushed all the way to a general channel?
ChatGPT started producing some matrix inequalities which, somewhat mysteriously, seemed to do exactly what was needed.
This led to another day and night of prompting, checking, rewriting, throwing away unnecessary notation, and trying to understand what the inequalities were actually telling us. Eventually the conceptual core became clear.
Hidden in the argument was a fully quantum blowing-up lemma.
Blowing up a quantum code
The classical blowing-up lemma is a beautiful idea that has appeared in several strong-converse arguments. Very roughly, suppose an event has a probability that is small, but not exponentially too small, under a product distribution. If we slightly enlarge, or “blow up”, this event, then the enlarged event can have probability close to one. Importantly, the enlargement is small on the scale of the block length.
This allows one to convert a hypothetical code with non-negligible success probability above capacity into another code with success probability close to one and almost the same rate. The weak converse then gives a contradiction.
Our second proof does exactly this, but directly with quantum codes.
Take a code of dimension M and entanglement fidelity F. The fully quantum blowing-up lemma says, roughly, that one can turn it into another code whose fidelity is arbitrarily close to one, while controlling how much code dimension is lost. In its one-shot form, the estimate looks like
\log M' \geq \log M+\log F -2\log\Gamma -O\!\left(\log\frac1\varepsilon\right),where the new code has fidelity at least 1-\varepsilon^2.
The interesting quantity here is \Gamma. It is a projective tensor norm associated with an approximation to the projector onto the image of the channel’s Stinespring isometry, across the receiver-environment cut.
Operationally, this is the quantum analogue of the “cost of blowing up”. If this projective norm is sufficiently small, a code whose fidelity is not exponentially small can be upgraded to a high-fidelity code at almost the same communication rate.
This immediately suggests how to prove the strong converse. Suppose there were a code above capacity whose fidelity was not exponentially small. Blow it up. If the cost of blowing it up is sufficiently small compared with the gap between the communication rate and capacity, we obtain a high-fidelity code that is still above capacity. But the ordinary weak converse says that such codes cannot exist. Therefore the fidelity of the original code must already have been exponentially small.
The remaining question is how to control \Gamma for many uses of an arbitrary channel.
And this is where the polynomial approximation reappears.
If \Pi is the projector onto the Stinespring image for one channel use, then \Pi^{\otimes n} is the corresponding projector for n uses. We approximate this enormous tensor-power projector by applying a low-degree polynomial to the sum of the local “violation” projectors 1-\Pi_i.
The polynomial is chosen so that it is exactly one when there are no violations and exponentially small once there is at least one. At the same time, because its degree is low, every term in its expansion involves only a limited number of channel uses. This is precisely what keeps the projective tensor norm under control.
So the polynomial is balancing two things: it approximates \Pi^{\otimes n} extremely well, but it does so without creating too much entanglement, in an operator-norm sense, across the receiver-environment cut. The first property allows us to amplify fidelity; the second controls how much communication rate we lose in the process.
This proof is completely orthogonal to the Arimoto approach. There are no Rényi capacities and no continuity argument as \alpha approaches one. Instead, it turns a low-fidelity code into a high-fidelity code and then invokes the weak converse. One proof attacks the regularized capacity formula analytically; the other works directly at the level of codes.
That these two routes arrived at the same general theorem essentially simultaneously was quite remarkable.
Doing science with AI
These were two extremely fun, but also extremely intensive, projects. It was a pleasure to work through them with Hao-Chung and Salman.
There is also no question in my mind that the way these projects unfolded would have been impossible, or at least dramatically slower, only a short time ago.
In more normal times, I think each project could easily have taken us many months, even if someone had handed us the crucial new ideas at the beginning. There is simply a huge amount of work between having a promising idea and having a proof that you actually understand: trying variants, checking edge cases, finding the right formulation, rediscovering why some step is necessary, eliminating unnecessary arguments, connecting the proof to the literature, and rewriting everything until the conceptual structure finally becomes visible.
What was striking here is that AI accelerated almost all of these loops.
In the first project, Astra supplied the crucial idea of controlling the Rényi quantity through derivatives and produced the first proof of the missing asymptotic continuity statement. Our task then became to understand it, strip away the opaque parts, and reveal the integral representation and the channel-image projection argument underneath.
In the second project, ChatGPT was initially more of a very active mathematical notebook while we were trying to understand someone else’s proof. But once we started asking it to generalize the argument, it produced the matrix inequalities that ultimately led us to the fully quantum blowing-up lemma. Again, the important work afterwards was to understand what had been found and reorganize it into a proof whose mechanism we could explain.
I think this last part is essential. An AI can produce a derivation that happens to be correct without giving you any reason to feel that you understand the result. For mathematics, that is not enough. At least for us, the goal was not merely to check a sequence of inequalities, but to digest the AI’s input until we could identify the underlying ideas and present the proof in a way that makes those ideas visible.
I hope we succeeded.
But even with that requirement, the speed-up of the scientific process is extraordinary. These two papers are perhaps the clearest experience I have had so far of what research with increasingly capable AI systems can look like: not just faster writing or easier calculations, but a much faster feedback loop between asking questions, exploring mathematical possibilities, finding unexpected connections, rejecting bad formulations, and eventually understanding something new.
And in this particular case, the result is that a problem that had been open for a long time now has not just one solution, but two rather different ones.
The papers
Hao-Chung Cheng and Marco Tomamichel, No information transmission through quantum channels above capacity:
https://arxiv.org/abs/2609.08998
Salman Beigi and Marco Tomamichel, Strong Converse for Quantum Capacity via a Fully Quantum Blowing-Up Lemma:
https://arxiv.org/abs/2609.11771