Aspects of quantum resources: coherence, measurements, and network correlations

This thesis is devoted to different aspects in quantum information theory. We will put forward new results on the topics of coherence theory, correlations in quantum networks, measurement incompatibility, quantification of quantum resources, as well as a connection between channel compatibility and the quantum marginal problem.

First, we will introduce the notion of genuine correlated coherence, which is defined as the amount of coherence that remains after applying global incoherent unitaries, which are deemed to be free in a resource theoretic approach to coherence. This contributes to an ongoing discussion on the possible free operations in a resource theory of multipartite coherence and reveals a connection to genuine multilevel entanglement.

Then, we will derive monogamy relations that capture the trade-off in the coherence that can exist between multiple orthogonal subspaces. Such a trade-off puts limits on the distinguishability of quantum states under unitary time evolution when measurements are restricted to subspaces. Moreover, this will allow us to derive criteria detecting genuine multisubspace coherence of the density matrix, which has applications in, e.g., the characterization of quantum networks.

Next, we will turn our focus to correlations in quantum networks. We will show how the structure of the network limits the distribution of entanglement, focusing on the so-called triangle network. We derive several necessary criteria for states to be preparable in the triangle network, based on the independence of the sources, entanglement monogamy and constraints on the local ranks. Then, we will consider a different approach based on the coherence properties of covariance matrices that arise from performing measurements on a network state. We will use the theory of coherence to analyze the relevant properties of the covariance matrices. This allows us to witness probability distributions that are incompatible with the structure of the network.

Another large part of this thesis is concerned with the quantification of quantum resources. We will first show that incompatible measurements provide an advantage over all compatible measurements in certain instances of quantum state discrimination. This provides an operational characterization of measurement incompatibility and opens a possibility of its semi-device-independent verification. The result is based on properties of the so-called incompatibility robustness.

Subsequently, we will show that such a result is a rather generic feature of the generalized robustness. More precisely, we will show that in any convex resource theory of states, measurements, channels, and collections of those, the robustness with respect to the set of free elements quantifies the advantage of a resourceful element over all free ones, in a task that can be derived from the duality theory of conic optimization.

Finally, we will put forward a connection between the compatibility of channels and certain instances of the quantum marginal problem, which allows us to translate many structural results between the two fields.