Pretty good indeed

Ciara Morgan gave a very clear QIP-like kind of talk. In a work in collaboration with Andreas Winter they show a pretty good converse for the quantum capacity of degradable channels. Did it sound complicated to you? Well, it’s in fact a hard topic, but also a very important one. Let me try to explain it a bit.

The point is that in the classical communication world Shannon showed that when Alice wants to transmit information to Bob by using a noisy classical channel, in the asymptotic limit of many uses of the channel if the rate of information is below a certain threshold, then there exists an encoding and decoding of the information that allows for communication with exponentially small error. Shannon went even further, and showed the converse of that, i.e., if the rate is above this threshold then the error goes to one, and no communication is possible. Given the importance of this sharp threshold it was even given a name: channel capacity.

What Ciara and Andreas are trying to show is that the same sharp threshold exists when Alice and Bob want to communicate quantum information through an entangling quantum channel. As in the Shannon case, the converse part of the theorem is the hard one. What they can show up to now is that for degradable channels if the rate is above the threshold then the error goes to 1/\sqrt{2}. It may sound like this is not quite the deal, but before that we only knew that the error was bounded away from zero. I guess that allows Ciara and Andreas calling their result a pretty good converse!

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