I will lecture on the asymptotic theory of quantum communication. The goal is to determine how much information can, in principle, be encoded into a collection of noisy quantum systems so that it can be retrieved with negligible error as the number of systems grows. The noise is modeled by a completely positive, trace preserving linear map on density matrices, otherwise known as a quantum channel, and we will be interested in the case where the same channel acts independently on each of the systems. I will primarily focus on the quantum capacity of a given channel, which measures the number of qubits per transmission that can be reliably protected. Main topics to be covered are:

1. Computing the quantum capacity. For certain classes of channels, we know how to write down an easily computable, closed-form formula for the quantum capacity. In other cases, the best we have is an open-form "regularized" expression, which is too unwieldy to be more than formally useful. I will outline what is known here and show that quantum capacity has some surprising behavior as well, such being a non-additive of the channel.

2. I will outline a proof of the LSD (Lloyd, Shor, Devetak) coding theorem, which shows that asymptotically good codes can be constructed by selecting a random subspace of the inputs of the channels, provided the communication rate is less than the quantum capacity. Two subtopics that will be necessary to cover are:

2.1 Approximate error correction. The usual theory of quantum error correction focuses on perfectly correcting some set (actually, a

subspace) of quantum errors. When the noise is modeled by a quantum channel, one can speak of approximately correcting encoded information in a meaningful and quantitative way.

2.2 The method of types. A basic tool coming from classical information theory, statistics and large deviation theory aiding the analysis of sequences of i.i.d. (independent and identically

distributed) random variables. It is an indispensable tool for studying channel capacities.

3. Time permitting, I will also discuss capacities for transmitting classical and private information, and how they are related to the quantum capacity.