Quantum information techniques in condensed matter: Quantum equilibration, entanglement typicality, detection of topological order

This thesis presents insights obtained on three questions in condensed matter physics via techniques in quantum information.;The first topic deals with signatures of equilibration in a closed quantum system. Using the Loschmidt echo as a representative observable, that is derived from fidelity---a popular quantity in quantum information, and by studying it's long time statistics the role of quantum criticality in the equilibration dynamics of a closed system is analysed. While off-critical systems under a quantum quench are shown to equilibrate well, critical systems are shown to do so poorly signified by relatively large fluctuations of the echo around it's long time average.;The second topic deals with typicality of entanglement in the physical Hilbert space. Here, a general framework for studying statistical moments of physically relevant quantities in ensembles of quantum states generated by Local Random Quantum Circuits (LRQC) is outlined. These ensembles are constructed by finite-length random quantum circuits acting on the (hyper)edges of an underlying (hyper)graph structure. The latter are designed to encode for the locality structure associated with finite-time quantum evolutions generated by physical i.e. local Hamiltonians. Physical properties of typical states in these ensembles, in particular purity as a proxy of quantum entanglement is studied. The problem is formulated in terms of matrix elements of superoperators which depend on the graph structure, choice of probability measure over the local unitaries and circuit length. We consider different families of LRQCs and study their typical entanglement properties for finite-time as well as their asymptotic behavior. In particular, for a model of LRQC that resembles closely the Trotter scheme of discretizing quantum evolutions with local Hamiltonians, we find that the area law holds in average and that the volume law is a typical property (that is, it holds in average and the fluctuations around the average are vanishing for the large system) of physical states. The area law arises when the evolution time is O(1) with respect to the size L of the system, while the volume law arises as typical when the evolution time scales like O(L).;The final topic deals with the perturbative response of the set of Renyi entropies of a subsystem when the entire system is in a state displaying some quantum order. The characteristic behavior of the entropies is shown to be able to identify topologically non-trivial and trivial phases in the case of quantum double models. The implications of the response towards the possibility of simulating the adiabatic evolution within a phase using the protocol of Local Operations and Classical Communications are discussed.