Random Unitary Matrices Associated to a Graph

P. Kondratiuka , and K. Życzkowskib, c aFaculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland bInstitute of Physics, Jagiellonian University, W.S. Reymonta 4, 30-059 Kraków, Poland cCenter for Theoretical Physics, PAS, al. Lotników 32/44, 02-668 Warszawa, Poland

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We analyze composed quantum systems consisting of k subsystems, each described by states in the n-dimensional Hilbert space. Interaction between subsystems can be represented by a graph, with vertices corresponding to individual subsystems and edges denoting a generic interaction, modeled by random unitary matrices of order n2. The global evolution operator is represented by a unitary matrix of size N = nk. We investigate statistical properties of such matrices and show that they display spectral properties characteristic to the Haar random unitary matrices provided the corresponding graph is connected. Thus basing on random unitary matrices of a small size n2 one can construct a fair approximation of large random unitary matrices of size nk. Graph-structured random unitary matrices investigated here allow one to define the corresponding structured ensembles of random pure states.

DOI: 10.12693/APhysPolA.124.1098 PACS numbers: 02.10.Yn, 03.65.Aa, 03.67.Mn, 05.45.Mt, 05.45.Pq