Bloch vectors for qudits

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 twoqudit state and decompose it according to each basis. In 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.

Composite parameterization and Haar measure for all unitary and special unitary groups

“…This and the observation (31) imply that when we choose the following orthogonal operator basis and order…”

SU(1, 1|N) superconformal mechanics with fermionic gauge symmetry

“…Using the fact that (3.3) along with the unity matrix define a basis in the space of diagonal N × N matrices, one can establish the identity [47] |j j| = 1…”

On rationalR-matrices with adjoint SU(n) symmetry

“…For the orthonormal basis {I a } one can, up to a normalisation, use the generalised Gell-Mann matrices [17]. We recall that in such a basis the commutator and anticommutator of two elements can be written as…”