Quantum entanglement in finite-dimensional Hilbert spaces

Abstract

In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.

Ph.D. dissertation, 162 pages. It covers, it slightly improves, and it presents the material of the papers http://arxiv.org/abs/0807.1804, http://arxiv.org/abs/1101.3256, http://arxiv.org/abs/1105.3086 and http://arxiv.org/abs/1206.6253 in a unified treatment. Apart from those, a review of 40 pages on qualification and quantification of entanglement is given as introduction

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• Tensor product methods and entanglement optimization for ab initio quantum chemistry
• Multipartite entanglement measures
• Entanglement patterns and generalized correlation functions in quantum many body systems
• The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group SL(3,C)
• Stretching the limits of multiparticle entanglement
• Spatial and polarization entanglement of three photons in a discrete-time quantum walk