Quantum Inf Process (2016) 15:2193–2212
A graph theoretical approach to states and unitary
· Bibhas Adhikari
Received: 15 September 2015 / Accepted: 13 January 2016 / Published online: 4 February 2016
© Springer Science+Business Media New York 2016
Abstract Building upon our previous work, on graphical representation of a quantum
state by signless Laplacian matrix, we pose the following question. If a local unitary
operation is applied to a quantum state, represented by a signless Laplacian matrix,
what would be the corresponding graph and how does one implement local unitary
transformations graphically? We answer this question by developing the notion of
local unitary equivalent graphs. We illustrate our method by a few, well known, local
unitary transformations implemented by single-qubit Pauli and Hadamard gates. We
also show how graph switching can be used to implement the action of the C
resulting in a graphical description of Bell state generation.
Keywords Signless Laplacian of a combinatorial graph · Graph switching · Pauli
matrices · Local unitary operators
This work is supported by CSIR (Council of Scientiﬁc and Industrial Research) Grant No.
25(0210)/13/EMR-II, New Delhi, India.
Department of Mathematics, IIT Jodhpur, Jodhpur, India
Department of Mathematics, IIT Kharagpur, Kharagpur, India
Department of Physics, IIT Jodhpur, Jodhpur, India