A Baker-Campbell-Hausdorff disentanglement relation for Lie groups

Parameter differentiation and quantum state decomposition for time varying Schrödinger equations

“…compute the Jordan form by similarity transformations, but this requires the knowledge of eigenvalues and eigenvectors. The method we use here is taken from [14]. It relies on the Cayley-Hamilton theorem and consists in expressing the series expansion of e ad A i in terms of the first n − 1 powers of ad Ai with suitable coefficients depending on the coefficients of the characteristic polynomial of ad Ai and on γ i .…”

“…To obtain an explicit closed form expression for (9), we use the method of [14] which is based on the Cayley-Hamilton theorem i.e. on expressing the exponential of a n × n matrix in terms of its first n − 1 powers.…”

“…, a n−1 , γ). In [14], the values β k are obtained via a Laplace transform method that is briefly recapitulated here. From ( 9) and ( 10), one can write β n−1 as the infinite series…”

“…See [12,13,16] for a few examples of explicit choices on SU (N ), [3,5,11,17] for SU (3) and [4,9] for the related issue of how to uniformly parameterize the rotation groups. Such decompositions are very useful for example in the generation of elementary gates in quantum computing or more generally in the control of driven dynamics [15,19], but also in quantum state disentanglement [14], in the study of coherent states [11,13], in the solution of the Liouville-von Neumann equation and in the design of schemes for the numerical integration of differential equations on Lie groups [8].…”

“…exponentials of the matrices of the adjoint representation of any linear Lie algebra. Such technique is adapted from [14]. As an example, we compute two Wei-Norman formulae for su(2) for the same basis obtained from the Pauli matrices but for different ordering of the basis elements (i.e.…”