Geometry of entanglement in the Bloch sphere

Abstract

Entanglement is an important concept in quantum information, quantum communication, and quantum computing. We provide a geometrical analysis of entanglement and separability for all the rank-2 quantum mixed states: complete analysis for the bipartite states, and partial analysis for the multipartite states. For each rank-2 mixed state, we define its unique Bloch sphere, that is spanned by the eigenstates of its density matrix. We characterize those Bloch spheres into exactly five classes of entanglement and separability, give examples for each class, and prove that those are the only classes.

arXiv comment7 pages; 5 figures

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• Geometry of entanglement and separability in Hilbert subspaces of dimension up to three
• Visualizing Quantum States: A Pilot Study on Problem Solving in Quantum Information Science Education
• Toward a generalization of Kruskal's theorem on tensor decomposition
• Topological order on the Bloch sphere
• Useful variants and perturbations of completely entangled subspaces and spans of unextendible product bases