Bloch vectors for qudits and geometry of entanglement

Abstract

We present three different matrix bases that can be used to decompose density matrices of d--dimensional quantum systems, so-called qudits: the generalized Gell-Mann matrix basis, the polarization operator basis, and the Weyl operator basis. Such a decomposition can be identified with a vector --the Bloch vector, i.e. a generalization of the well known qubit case-- and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We consider the important case of an isotropic two--qudit state and decompose it according to each basis. Investigating the geometry of entanglement of special parameterized two--qubit and two--qutrit states, in particular we calculate the Hilbert--Schmidt measure of entanglement, we find that the Weyl operator basis is the optimal choice since it is closely connected to the entanglement of the considered states.

30 pages, 4 figures

Cited by in corpus (5)

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• A simplex of bound entangled multipartite qubit states
• Inequalities detecting entanglement for arbitrary bipartite systems
• Structure constants of $su(2S+1)$ algebra and the decomplexification of the Liouville-von Neumann equation
• Classifying coherence with a finite set of witnesses