Multi-particle entanglement and its applications to cryptography

Abstract

Entanglement between three or more parties exhibits a realm of properties unknown to two-party states. Bipartite states are easily classified using the Schmidt decomposition. The Schmidt coefficients of a bipartite pure state encompass all the non-local properties of the state and can be "seen" by looking at one party's density matrix only. Pure states of three and more parties however lack such a simple form. They have more invariants under local unitary transformations than any one party can "see" on their sub-system. These "hidden non-localities" will allow us to exhibit a class of multipartite states that cannot be distinguished from each other by any party. Generalizing a result of BPRST and using a recent result by Nielsen we will show that these states cannot be transformed into each other by local actions and classical communication. Furthermore we will use an orthogonal subset of such states to hint at applications to cryptography and illustrate an extension to quantum secret sharing (using recently suggested ((n,k))-threshold schemes).