The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography

TL;DR

This thesis explores the structure of bipartite quantum states using group theory and cryptography, revealing spectral relations and introducing a new entanglement measure called squashed entanglement.

Contribution

It establishes a connection between quantum state spectra and symmetric group coefficients and proposes squashed entanglement as a novel measure of quantum entanglement.

Findings

Spectral relations between bipartite states and their reductions

Relation between spectra and Kronecker coefficients

Introduction of squashed entanglement as an entanglement measure

Abstract

This thesis presents a study of the structure of bipartite quantum states. In the first part, the representation theory of the unitary and symmetric groups is used to analyse the spectra of quantum states. In particular, it is shown how to derive a one-to-one relation between the spectra of a bipartite quantum state and its reduced states, and the Kronecker coefficients of the symmetric group. In the second part, the focus lies on the entanglement of bipartite quantum states. Drawing on an analogy between entanglement distillation and secret-key agreement in classical cryptography, a new entanglement measure, `squashed entanglement', is introduced.