Bloch vectors for qudits

TLDR: A new method to decompose density matrices via so-called standard matrices is presented, and a representation of an entanglement witness in terms of expectation values of spin-1 measurements is shown, appropriate for an experimental realization.

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 present a new method to decompose density matrices via so-called standard matrices, consider the important case of an isotropic two-qudit state and decompose it according to each basis. In the case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin-1 measurements, which is appropriate for an experimental realization.