Information Flow in Secret Sharing Protocols

Abstract: The entangled graph states have emerged as an elegant and powerful quantum resource, indeed almost all multiparty protocols can be written in terms of graph states including measurement based quantum computation (MBQC), error correction and secret sharing amongst others. In addition they are at the forefront in terms of implementations. As such they represent an excellent opportunity to move towards integrated protocols involving many of these elements. In this paper we look at expressing and extending graph state secret sharing and MBQC in a common framework and graphical language related to flow. We do so with two main contributions. First we express in entirely graphical terms which set of players can access which information in graph state secret sharing protocols. These succinct graphical descriptions of access allow us to take known results from graph theory to make statements on the generalisation of the previous schemes to present new secret sharing protocols. Second, we give a set of necessary conditions as to when a graph with flow, i.e. capable of performing a class of unitary operations, can be extended to include vertices which can be ignored, pointless measurements, and hence considered as unauthorised players in terms of secret sharing, or error qubits in terms of fault tolerance. This offers a way to extend existing MBQC patterns to secret sharing protocols. Our characterisation of pointless measurements is believed also to be a useful tool for further integrated measurement based schemes, for example in constructing fault tolerant MBQC schemes.