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On the Density Matrix of Graphs

Kazemi, Mohammad Reza | 2010

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 41719 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Daneshgar, Amir
  7. Abstract:
  8. Density matrix of graphs as defined -for the first time- in [S. Braunstein and et al. The laplacian of a graph as a density matrix, Annals of Combinatorics, (2006)], is obtained through dividing the Laplacian matrix by the degree sum. This matrix is also semi-positive and trace one. Therefore one may talk about the Von Neumann entropy of this matrix. In [F. Passerini, S. Severini. Quantifying complexity in networks: The Von Neumann entropy. IJATS, (2009)], authors have claimed that this quantity can be consisered as a measure of regularity. Here, using a geometric interpretation of Von Neumann entropy, expresed in [G. Mitchison, R. Jozsa, Towards a geometrical interpretation of quantum information compression, Phys. Rev. A, (2004) ] and some methods explained in [B. Mohar, On the laplacian coefficients of acyclic graphs Linear Algebra Appl, (2007)], we have used these operations to increase or decrease the Von Neumann entropy of graphs. Also, we have shown that the entropy does not necessarily increase with the addition of an edge to the graph. Moreover, the density matrix and its corresponding graph could be considered as a quantum system and one may study various problems from Quantum Information in this case, including the Entanglement problem. In this thesis we have also described some previously obtained results in this regard.
  9. Keywords:
  10. Density Matrix ; Graph Laplacian ; Entanglement ; Von Neumann Entropy

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