Symmetries and self-similarity of many-body wavefunctions
  This PhD thesis is dedicated to the study of the interplay between symmetries of quantum states and their self-similar properties. It consists of three connected threads of research: polynomial invariants for multiphoton states, visualization schemes for quantum many-body systems and a complex networks approach to quantum walks on a graph.   First, we study the problem of which many-photon states are equivalent up to the action of passive linear optics. We prove that it can be converted into the problem of equivalence of two permutation-symmetric states, not necessarily restricted to the same operation on all parties. We show that the problem can be formulated in terms of symmetries of complex polynomials of many variables, and provide two families of invariants, which are straightforward to compute and provide analytical results.   Second, we study a family of recursive visualization schemes for many-particle systems, for which we have coined the name 'qubism'. While all many-qudit states can be plotted with qubism, it is especially useful for spin chains and one-dimensional translationally invariant states. This symmetry results in self-similarity of the plot, making it more comprehensible and allowing to discover certain structures.   Third, we study quantum walks of a single particle on graphs, which are classical analogues of random walks. Our focus is on the long-time limit of the probability distribution and we study how (especially in the long-time limit) off-diagonal elements of the density matrix behave. We use them to perform quantum community detection - splitting of a graph into subgraphs in such a way that the coherence between them is small. Our method captures properties that classical methods cannot - the impact of constructive and destructive interference, as well as the dependence of the results on the tunneling phase. 
Piotr Migda{\l}
quant-ph