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In the PIR problem, a user wants to retrieve one of the $K$ messages, stored in $N$ servers, privately by sending queries to servers. In the quantum PIR (QPIR) problem, we allow $N$ servers to share some state prior to the initialization of the scheme, which they are able to locally operate on, and the download links from the servers to the user has been replaced with an ideal quantum channel, i.e., a channel that can send a quantum state without disturbing it or its extension. For our first problem, we consider a QPIR setting with an eavesdropper that has access to some number of queries sent by the user, and also to the answers by the servers. PIR version of this has been completely solved, as the optimal download cost is completely characterized; therefore, the capacity region is known. We propose an achievable scheme for the QPIR version which improves the classical capacity region.

In the second problem, we consider the QPIR problem when some of the servers are Byzantine. Classically, Byzantine servers are servers that send some noise to the user with unknown probabilistic characteristics, instead of the actual answer characterized by the scheme. In the QPIR problem, as we allowed our servers to share some prior quantum state, we instead define Byzantine servers by the encoding operations they apply to this state. We first consider a restricted case in which we limit the encoding operations of the Byzantine servers to some set, and show an achievable region. Then, we show how the general case of any encoding operation allowed follows by showing an equivalence of the general case to the restricted case under our models of encodings and decodings.

In the secure aggregation problem, we consider secure membership aggregation, from which the general case follows due to the linearity of the scheme. In secure membership aggregation, there are $N$ users, with each of them having a subset of a fixed universal set known to all of them, and they want to know the frequency of the elements in the universal set, without publicly revealing their subset. We propose a scheme that allows $N$ users to share some prior quantum state, and by proper encodings and measurements, it is shown that a scheme that is quantum information theoretically secure for this problem exists.

Lastly, we consider single-user quantum communication for classical messages. In this setting, there is a single transmitter which wants to send a classical message through a quantum channel to a single receiver, with many uses of the channel being allowed. The capacity region with asymptotically vanishing error probability has been well studied and is characterized by an optimization problem. We consider the empirical output quantum state induced by capacity-achieving codes (good codes) to this problem with vanishing or non-vanishing error probability, and compare it with the solution to the optimization problem. The classical case, in which the quantum channel is replaced with classical channels, has been known, and we show how similar results hold in the quantum world as well, when the quantum systems we have are of finite dimensions.

We present four different proposed works. For our first proposed work, we consider how the output of good encoded states when entanglement between receiver and transmitter is allowed in classical communication will compare, in normalized divergence, to the capacity achieving output of the entanglement-assisted classical capacity, similarly, in the context of both vanishing and non-vanishing error probability for the codes considered. Our second proposed work is regarding quantum communication for quantum messages, i.e., a subspace of an Hilbert space that is to be transmitted over the channel. We first define what is meant by the output state in such codes, as there is no classical analogue for this case and the operational problem at hand is more open to interpretation, and then we consider how this output state will compare to the capacity achieving output of the corresponding quantum capacity of the channel. In the third proposed work, we opt to focus on quantum network information theory, and we propose to investigate the empirical distribution, that is the $k$-letter type distribution, of output of good codes for some well studied classical quantum network channels such as multiple access or degraded broadcast channel. The last work proposed considers good quantum codes for classical messages when the channels used are for continuous variable systems, such as quantum Gaussian channels, as opposed to finite dimensional systems we investigated before. We propose to check the asymptotic convergence properties between good codes and the capacity achieving output state in such channels, first in the vanishing error probability case, and then try to obtain the more powerful non-vanishing error case.