Multiparticle entanglement and its applications to cryptography

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 nonlocal 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 subsystem. These “hidden nonlocalities” will allow us to exhibit a class of multipartite states that cannot be distinguished from each other by any party. Generalizing a result of Bennett, Popescu, Rohrlich, Smolin, and Thapliyal, 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 $\mathrm{}\left(\right(n,k\left)\right)$-threshold schemes].