Laplacian matrices of weighted digraphs represented as quantum states

  • Bibhas Adhikari
  • Subhashish Banerjee
  • Satyabrata Adhikari
  • Atul Kumar


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.


Combinatorial Laplacian Signless Laplacian Eigenvalues Pure and mixed states Density matrix Quantum entanglement 



This work is partially supported by CSIR (Council of Scientific and Industrial Research) Grant No. 25(0210)/13/EMR-II, New Delhi, India. SB wishes to thank Supriyo Dutta for his help in carefully reading the manuscript and suggesting a number of changes.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsIIT KharagpurKharagpurIndia
  2. 2.Department of PhysicsIIT JodhpurJodhpurIndia
  3. 3.Department of MathematicsBIT MesraRanchiIndia
  4. 4.Department of ChemistryIIT JodhpurJodhpurIndia

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