2 00 6 The Structure of Bipartite Quantum States

Currently, a rethinking of the fundamental properties of quantum mechanical systems in the light of quantum computation and quantum cryptography is taking place. In this PhD thesis, I wish to contribute to this effort with a study of the bipartite quantum state. Bipartite quantum-mechanical systems are made up of just two subsystems, A and B, yet, the quantum states that describe these systems have a rich structure. The focus is two-fold: Part I studies the relations between the spectra of the joint and the reduced states, and in part II, I will analyse the amount of entanglement, or quantum correlations, present in a given state. In part I, the mathematical tools from group theory play an important role, mainly drawing on the representation theory of finite and Lie groups and the Schur-Weyl duality. This duality will be used to derive a one-to-one relation between the spectra of a joint quantum system AB and its parts A and B, and the Kronecker coefficients of the symmetric group. In this way the two problems are connected for the first time, which makes it possible to transfer solutions and gain insights that illuminate both problems. Part II of this thesis is guided by the question: How can we measure the strength of entanglement in bipartite quantum states? The search for an answer starts with an extensive review of the literature on entanglement measures. I will then approach the subject from a cryptographic point of view. The parts A and B of a bipartite quantum state are given to the cooperative players Alice and Bob, whereas a purifying system is given to the eavesdropper Eve, who aims at reducing the correlation between Alice and Bob. The result is a new measure for entanglement: squashed entanglement. Squashed entanglement is the only known strongly superadditive, additive and asymptotically continuous entanglement measure. These properties, as well as the simplicity of their proofs, position squashed entanglement as a unique tool to study entanglement.