In Shannon's classical information theory, the mutual information between two systems gives the answer to the question of how much you can learn about one system if you hold the other system. What happens when we try to extend this to quantum systems? To make things simpler, we start by considering an ensemble of (possibly mixed) quantum states, each with a classical label. Our first question will be: suppose that we have access to the quantum state, how much can we learn about the label? This quantity is called the accessible information. For the essentially classical scenario where all the quantum states commute, the accessible information is equal to the Shannon mutual information between the label and the state. Maximizing this over all probability distributions for the quantum ensemble yields the capacity of a channel where, for each transmission, Alice (the sender) sends Bob (the receiver) one of the quantum states in the ensemble. For quantum states, in contrast, the capacity achievable in the channel where Alice communicates by sending Bob one of a set of quantum states can be strictly larger than the accessible information for this ensemble. What gives rise to this difference?

One difference between quantum states and classical states in this setting is that for classical (i.e., commuting) states, if Alice sends Bob some additional classical information C about the label of the state she sent (and depending only on the label), it can only decrease the amount of information that Bob can extract from the state; i.e., the mutual information of Alice and Bob's state, conditioned on the extra information C, decreases. For quantum states, this is not true. By sending more information about the label, Alice can increase the accessible information of Bob's remaining state. This gives rise to a new measure of the amount of information in a quantum ensemble, which we call the adaptively accessible information: the amount of the information extractable from a quantum ensemble by Bob with the help of additional communications from Alice that give information about the label. This new information quantity is not additive, so the tensor product of two copies of a quantum ensemble may contain more than twice the adaptively accessible information of a single copy. In the limit of the tensor product of many copies, we show that this quantity approaches the Holevo bound for the quantum ensemble, which is known to be the information capacity for this ensemble.

If, instead of considering the capacity of ensembles of quantum states (where Alice chooses which of the set of states to send to Bob), we consider the capacity of quantum channels, not only do we discover analogs of the several capacities discussed above, but more capacities also appear. In particular, we will also discuss the entanglement-assisted capacity, which is the capacity of a channel when Alice and Bob are allowed to use prior entanglement as an extra resource in their protocols. We give the formula for entanglement-assisted capacity, and show how it relates to the other capacities discussed in this talk.