Executive Summary : | This project is based on a realistic formulation of certain questions from quantum physics using the language of graph theory. Its aim is to help quantum physicists to produce a high-dimensional multi-photonic Greenberger-Horne-Zeilinger (GHZ) state. GHZ state is studied in the area of quanutm information theory, and is a certain type of entangled quantum state, having extremely non-classical properties. They are useful in protocols in quantum communication and cryptography. The PI's interest in this problem arises from the work of Dr. Mario Krenn, a research group leader, from Max-Planck Institute for the Science of light, Germany. He has very neatly formulated the question of producing high-dimensional GHZ states in graph theoretic terms. The formulation captures all the details required. After studying the problem out of curiosity for several months, the PI has realised that the questions they ask, involve some deep structural phenomena of graphs. Considering the fact that the problems were formulated by a physicist rather than a mathematician, it should be the depth of the physical phenomena that gets converted to the mathematical depth. In Krenn's formulation, a graph corresponds to a quantum optical experiment using probabilistic photon-pair sources and linear optics; vertices to single-photon detectors in the output of some photon path; Edges to photon pairs that emerge from two-photon paths; edge weights to the amplitude of the corresponding photon pair; edge colours to the mode numbers of the two photons in the path defined by the vertices at the end point of the edge. Now using this correspondence and using concepts like perfect matching, inherited vertex coloring etc he defines a certain type of edge-weighted edge-colored graphs which may be called perfectly monochromatic graphs to which a parameter called matching index can be associated. (The technical details are attached in a separate pdf file; unfortunately it is not an easy to state problem since it captures all the required details, unlike typical graph theory problems, where usually the problems are stated in couple of sentences!) A perfectly monochromatic graph corresponds to a GHZ state and its matching index would correspond to its dimension. It is a problem of significant importance in quantum physics to produce high dimension GHZ states. In view of the graph theoretic approach discussed here, our objective in this project is to construct a perfectly monochromatic graph with as high a matching index as possible. As of now except one graph, all other known perfectly monochromatic graphs have matching index at most 2. A proof that matching index cannot be greater than 2 (except for the one exception graph) also would have serious implications to quantum physics. We will also work on the set of other related graph theoretic problems (the approximate version, the generalised version etc.) formulated by Krenn, as part of this project. |