Applications of Kronecker states in quantum information theory

Abstract

In quantum information theory, maximally entangled states (MES) are key to implement many well known protocols such as quantum teleportation, quantum error correction, quantum key distribution, among others. The representation theory of the symmetric group provides a mechanism that allows one to generate a wide class of maximally entangled multipartite states. Such states, which we call Kronecker states, belong to the invariant subspace of products of irreducible representations of Sn. The reduced density matrices of such states in each individual subspace are completely mixed, proving that Kronecker states are MES. The purpose of this work is to better understand Kronecker states and their applications in quantum information theory. In particular, we will implement them in four cases: "Superadditivity of classical channel capacity in quantum channels: Hastings, used the invariance of MES in bipartite systems under the action of some elements of composed channels to prove that the minimum output entropy of such channels is subadditive, therefore there is superadditivity of classical channel capacity. We want to build channels where it is possible to exploit the natural invariant property of Kronecker states to achieve superadditivity expanding this behaviour when composing a system with more than two channels. Quantum error correction: In the 5 qubit protocol, codewords can be understood as particular examples of Kronecker states. We want to study the possibility of generalizing the protocol to generic Kronecker states with possible applications to different noise models. Quantum secret sharing: One of the most important characteristics of the codification in this schemes is that different individuals must get no information without help of others; or in density matrix language, reduced density matrices have to be completely mixed, the salient property of Kronecker states. We want to study a possible generalization of the threshold scheme (t,n) that allows one to recover the initial information to selected groups when the codification is done using Kronecker states. Entanglement concentration: Mejía and Botero demonstrated that the entanglement of tripartite states in the W class can be concentrated in Kronecker states of high dimension. We want to better understand the structure of these states and to study how they appear in the entanglement concentration of multipartite states. We expect that the study of these applications will help us to understand in a more general way Kronecker states, pointing to a systematic formalism that establish the generalities behind their applications and draw a path to extend the study to invariant elements in other groups