Laplacian matrices of weighted digraphs represented as quantum states
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Publication:2412612
DOI10.1007/S11128-017-1530-1zbMATH Open1373.81044arXiv1205.2747OpenAlexW2587179651MaRDI QIDQ2412612
Author name not available (Why is that?)
Publication date: 23 October 2017
Published in: (Search for Journal in Brave)
Abstract: Representing graphs as quantum states is becoming an increasingly important approach to study entanglement of mixed states, alternate to the standard linear algebraic density matrix-based approach of study. In this paper, we propose a general weighted directed graph framework for investigating properties of a large class of quantum states which are defined by three types of Laplacian matrices associated with such graphs. We generalize the standard framework of defining density matrices from simple connected graphs to density matrices using both combinatorial and signless Laplacian matrices associated with weighted directed graphs with complex edge weights and with/without self-loops. We also introduce a new notion of Laplacian matrix, which we call signed Laplacian matrix associated with such graphs. We produce necessary and/or sufficient conditions for such graphs to correspond to pure and mixed quantum states. Using these criteria, we finally determine the graphs whose corresponding density matrices represent entangled pure states which are well known and important for quantum computation applications. We observe that all these entangled pure states share a common combinatorial structure.
Full work available at URL: https://arxiv.org/abs/1205.2747
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